3.5.39 \(\int \frac {(a+b x^2)^2}{x^{5/2} (c+d x^2)^3} \, dx\) [439]

Optimal. Leaf size=402 \[ -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \sqrt {x}}{48 c^3 d \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}} \]

[Out]

-2/3*a^2/c/x^(3/2)/(d*x^2+c)^2-1/64*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4)
)/c^(15/4)/d^(5/4)*2^(1/2)+1/64*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^
(15/4)/d^(5/4)*2^(1/2)-1/128*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^
(1/2))/c^(15/4)/d^(5/4)*2^(1/2)+1/128*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2
^(1/2)*x^(1/2))/c^(15/4)/d^(5/4)*2^(1/2)-1/12*(11*a^2*d^2-6*a*b*c*d+3*b^2*c^2)*x^(1/2)/c^2/d/(d*x^2+c)^2+1/48*
(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*x^(1/2)/c^3/d/(d*x^2+c)

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Rubi [A]
time = 0.26, antiderivative size = 398, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {473, 468, 296, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{12 c \left (c+d x^2\right )^2}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\sqrt {x} \left (\frac {7 a (6 b c-11 a d)}{c^2}+\frac {3 b^2}{d}\right )}{48 c \left (c+d x^2\right )}-\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^3),x]

[Out]

(-2*a^2)/(3*c*x^(3/2)*(c + d*x^2)^2) + ((6*a*b - (3*b^2*c)/d - (11*a^2*d)/c)*Sqrt[x])/(12*c*(c + d*x^2)^2) + (
((3*b^2)/d + (7*a*(6*b*c - 11*a*d))/c^2)*Sqrt[x])/(48*c*(c + d*x^2)) - ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*A
rcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*d^(5/4)) + ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*
a*d))*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*d^(5/4)) - ((3*b^2*c^2 + 7*a*d*(6*b*
c - 11*a*d))*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*d^(5/4)) + ((3*b
^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15
/4)*d^(5/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 473

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[c^2*(e*x)^(m
+ 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {2 \int \frac {\frac {1}{2} a (6 b c-11 a d)+\frac {3}{2} b^2 c x^2}{\sqrt {x} \left (c+d x^2\right )^3} \, dx}{3 c}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {1}{24} \left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )^2} \, dx\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{32 c}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2}}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2}}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{7/2} d^{3/2}}+\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{7/2} d^{3/2}}-\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 241, normalized size = 0.60 \begin {gather*} \frac {-\frac {4 c^{3/4} \sqrt [4]{d} \left (3 b^2 c^2 x^2 \left (3 c-d x^2\right )-6 a b c d x^2 \left (11 c+7 d x^2\right )+a^2 d \left (32 c^2+121 c d x^2+77 d^2 x^4\right )\right )}{x^{3/2} \left (c+d x^2\right )^2}-3 \sqrt {2} \left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+3 \sqrt {2} \left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{192 c^{15/4} d^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^3),x]

[Out]

((-4*c^(3/4)*d^(1/4)*(3*b^2*c^2*x^2*(3*c - d*x^2) - 6*a*b*c*d*x^2*(11*c + 7*d*x^2) + a^2*d*(32*c^2 + 121*c*d*x
^2 + 77*d^2*x^4)))/(x^(3/2)*(c + d*x^2)^2) - 3*Sqrt[2]*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*ArcTan[(Sqrt[c] -
Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + 3*Sqrt[2]*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*ArcTanh[(Sqrt
[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(192*c^(15/4)*d^(5/4))

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Maple [A]
time = 0.15, size = 220, normalized size = 0.55

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\left (\frac {15}{32} a^{2} d^{2}-\frac {7}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-22 a b c d +3 b^{2} c^{2}\right ) \sqrt {x}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-42 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d c}\right )}{c^{3}}-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}}\)	\(220\)
default	\(-\frac {2 \left (\frac {\left (\frac {15}{32} a^{2} d^{2}-\frac {7}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-22 a b c d +3 b^{2} c^{2}\right ) \sqrt {x}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-42 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d c}\right )}{c^{3}}-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}}\)	\(220\)
risch	\(-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}}-\frac {15 x^{\frac {5}{2}} a^{2} d^{2}}{16 c^{3} \left (d \,x^{2}+c \right )^{2}}+\frac {7 x^{\frac {5}{2}} a b d}{8 c^{2} \left (d \,x^{2}+c \right )^{2}}+\frac {x^{\frac {5}{2}} b^{2}}{16 c \left (d \,x^{2}+c \right )^{2}}-\frac {19 d \sqrt {x}\, a^{2}}{16 c^{2} \left (d \,x^{2}+c \right )^{2}}+\frac {11 \sqrt {x}\, a b}{8 c \left (d \,x^{2}+c \right )^{2}}-\frac {3 \sqrt {x}\, b^{2}}{16 \left (d \,x^{2}+c \right )^{2} d}-\frac {77 d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{64 c^{4}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{32 c^{3}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{64 c^{2} d}-\frac {77 d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{64 c^{4}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{32 c^{3}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2}}{64 c^{2} d}-\frac {77 d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a^{2}}{128 c^{4}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a b}{64 c^{3}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) b^{2}}{128 c^{2} d}\)	\(562\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(5/2)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-2/c^3*(((15/32*a^2*d^2-7/16*a*b*c*d-1/32*b^2*c^2)*x^(5/2)+1/32*c*(19*a^2*d^2-22*a*b*c*d+3*b^2*c^2)/d*x^(1/2))
/(d*x^2+c)^2+1/256*(77*a^2*d^2-42*a*b*c*d-3*b^2*c^2)/d*(c/d)^(1/4)/c*2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2
)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2
^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))-2/3*a^2/c^3/x^(3/2)

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Maxima [A]
time = 0.52, size = 377, normalized size = 0.94 \begin {gather*} -\frac {32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 42 \, a b c d^{2} - 77 \, a^{2} d^{3}\right )} x^{4} + {\left (9 \, b^{2} c^{3} - 66 \, a b c^{2} d + 121 \, a^{2} c d^{2}\right )} x^{2}}{48 \, {\left (c^{3} d^{3} x^{\frac {11}{2}} + 2 \, c^{4} d^{2} x^{\frac {7}{2}} + c^{5} d x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, c^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(5/2)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/48*(32*a^2*c^2*d - (3*b^2*c^2*d + 42*a*b*c*d^2 - 77*a^2*d^3)*x^4 + (9*b^2*c^3 - 66*a*b*c^2*d + 121*a^2*c*d^
2)*x^2)/(c^3*d^3*x^(11/2) + 2*c^4*d^2*x^(7/2) + c^5*d*x^(3/2)) + 1/128*(2*sqrt(2)*(3*b^2*c^2 + 42*a*b*c*d - 77
*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqr
t(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(
1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(3*b^2*c^2 + 42*a*b
*c*d - 77*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(3*b
^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)
))/(c^3*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1433 vs. \(2 (318) = 636\).
time = 1.18, size = 1433, normalized size = 3.56 \begin {gather*} -\frac {12 \, {\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )} \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{8} d^{2} \sqrt {-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}} + {\left (9 \, b^{4} c^{4} + 252 \, a b^{3} c^{3} d + 1302 \, a^{2} b^{2} c^{2} d^{2} - 6468 \, a^{3} b c d^{3} + 5929 \, a^{4} d^{4}\right )} x} c^{11} d^{4} \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {3}{4}} + {\left (3 \, b^{2} c^{13} d^{4} + 42 \, a b c^{12} d^{5} - 77 \, a^{2} c^{11} d^{6}\right )} \sqrt {x} \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {3}{4}}}{81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}\right ) + 3 \, {\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )} \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {1}{4}} \log \left (c^{4} d \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {1}{4}} - {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \sqrt {x}\right ) - 3 \, {\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )} \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {1}{4}} \log \left (-c^{4} d \left (-\frac {81 \, b^{8} c^{8} + 4536 \, a b^{7} c^{7} d + 86940 \, a^{2} b^{6} c^{6} d^{2} + 539784 \, a^{3} b^{5} c^{5} d^{3} - 1457946 \, a^{4} b^{4} c^{4} d^{4} - 13854456 \, a^{5} b^{3} c^{3} d^{5} + 57274140 \, a^{6} b^{2} c^{2} d^{6} - 76697544 \, a^{7} b c d^{7} + 35153041 \, a^{8} d^{8}}{c^{15} d^{5}}\right )^{\frac {1}{4}} - {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \sqrt {x}\right ) + 4 \, {\left (32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 42 \, a b c d^{2} - 77 \, a^{2} d^{3}\right )} x^{4} + {\left (9 \, b^{2} c^{3} - 66 \, a b c^{2} d + 121 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {x}}{192 \, {\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(5/2)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/192*(12*(c^3*d^3*x^6 + 2*c^4*d^2*x^4 + c^5*d*x^2)*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2
+ 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6 - 766
97544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4)*arctan((sqrt(c^8*d^2*sqrt(-(81*b^8*c^8 + 4536*a*b^7*c^
7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57
274140*a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5)) + (9*b^4*c^4 + 252*a*b^3*c^3*d +
1302*a^2*b^2*c^2*d^2 - 6468*a^3*b*c*d^3 + 5929*a^4*d^4)*x)*c^11*d^4*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*
a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b
^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(3/4) + (3*b^2*c^13*d^4 + 42*a*b*c^12*d^5 -
77*a^2*c^11*d^6)*sqrt(x)*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1
457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041
*a^8*d^8)/(c^15*d^5))^(3/4))/(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 -
1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 351530
41*a^8*d^8)) + 3*(c^3*d^3*x^6 + 2*c^4*d^2*x^4 + c^5*d*x^2)*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^
6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6
- 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4)*log(c^4*d*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 869
40*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^
6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4) - (3*b^2*c^2 + 42*a*b*c*d - 77*a^2*
d^2)*sqrt(x)) - 3*(c^3*d^3*x^6 + 2*c^4*d^2*x^4 + c^5*d*x^2)*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c
^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^
6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4)*log(-c^4*d*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 8
6940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*
a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4) - (3*b^2*c^2 + 42*a*b*c*d - 77*a^
2*d^2)*sqrt(x)) + 4*(32*a^2*c^2*d - (3*b^2*c^2*d + 42*a*b*c*d^2 - 77*a^2*d^3)*x^4 + (9*b^2*c^3 - 66*a*b*c^2*d
+ 121*a^2*c*d^2)*x^2)*sqrt(x))/(c^3*d^3*x^6 + 2*c^4*d^2*x^4 + c^5*d*x^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(5/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.41, size = 426, normalized size = 1.06 \begin {gather*} -\frac {2 \, a^{2}}{3 \, c^{3} x^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{2}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{2}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{2}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{2}} + \frac {b^{2} c^{2} d x^{\frac {5}{2}} + 14 \, a b c d^{2} x^{\frac {5}{2}} - 15 \, a^{2} d^{3} x^{\frac {5}{2}} - 3 \, b^{2} c^{3} \sqrt {x} + 22 \, a b c^{2} d \sqrt {x} - 19 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(5/2)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-2/3*a^2/(c^3*x^(3/2)) + 1/64*sqrt(2)*(3*(c*d^3)^(1/4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 77*(c*d^3)^(1/4)*a
^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^2) + 1/64*sqrt(2)*(3*(c*d^3)^
(1/4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 77*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4)
- 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^2) + 1/128*sqrt(2)*(3*(c*d^3)^(1/4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 77*(
c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^4*d^2) - 1/128*sqrt(2)*(3*(c*d^3)^(1
/4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 77*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt
(c/d))/(c^4*d^2) + 1/16*(b^2*c^2*d*x^(5/2) + 14*a*b*c*d^2*x^(5/2) - 15*a^2*d^3*x^(5/2) - 3*b^2*c^3*sqrt(x) + 2
2*a*b*c^2*d*sqrt(x) - 19*a^2*c*d^2*sqrt(x))/((d*x^2 + c)^2*c^3*d)

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Mupad [B]
time = 0.35, size = 1508, normalized size = 3.75 \begin {gather*} -\frac {\frac {2\,a^2}{3\,c}-\frac {x^4\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{48\,c^3}+\frac {x^2\,\left (121\,a^2\,d^2-66\,a\,b\,c\,d+9\,b^2\,c^2\right )}{48\,c^2\,d}}{c^2\,x^{3/2}+d^2\,x^{11/2}+2\,c\,d\,x^{7/2}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )-\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}+\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )+\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}}{\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )-\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}-\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )+\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,1{}\mathrm {i}}{32\,{\left (-c\right )}^{15/4}\,d^{5/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )-\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}+\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )+\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}}{\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )-\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}-\frac {\left (\sqrt {x}\,\left (97140736\,a^4\,c^9\,d^{10}-105971712\,a^3\,b\,c^{10}\,d^9+21331968\,a^2\,b^2\,c^{11}\,d^8+4128768\,a\,b^3\,c^{12}\,d^7+147456\,b^4\,c^{13}\,d^6\right )+\frac {\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (-80740352\,a^2\,c^{13}\,d^9+44040192\,a\,b\,c^{14}\,d^8+3145728\,b^2\,c^{15}\,d^7\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{5/4}}}\right )\,\left (-77\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{15/4}\,d^{5/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^3),x)

[Out]

(atan((((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 105971712*a^3*b*c^10*
d^9 + 21331968*a^2*b^2*c^11*d^8) - ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2
*c^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*1i)/(64*(-
c)^(15/4)*d^(5/4)) + ((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 1059717
12*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) + ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7
- 80740352*a^2*c^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c
*d)*1i)/(64*(-c)^(15/4)*d^(5/4)))/(((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12
*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) - ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728
*b^2*c^15*d^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*
d^2 + 42*a*b*c*d))/(64*(-c)^(15/4)*d^(5/4)) - ((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768
*a*b^3*c^12*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) + ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*
d)*(3145728*b^2*c^15*d^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^
2 - 77*a^2*d^2 + 42*a*b*c*d))/(64*(-c)^(15/4)*d^(5/4))))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*1i)/(32*(-c)^(1
5/4)*d^(5/4)) - ((2*a^2)/(3*c) - (x^4*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d))/(48*c^3) + (x^2*(121*a^2*d^2 + 9*
b^2*c^2 - 66*a*b*c*d))/(48*c^2*d))/(c^2*x^(3/2) + d^2*x^(11/2) + 2*c*d*x^(7/2)) + (atan((((x^(1/2)*(97140736*a
^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*
d^8) - ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^1
4*d^8)*1i)/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d))/(64*(-c)^(15/4)*d^(5/4)) + ((x^(1/
2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968
*a^2*b^2*c^11*d^8) + ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2*c^13*d^9 + 44
040192*a*b*c^14*d^8)*1i)/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d))/(64*(-c)^(15/4)*d^(5
/4)))/(((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 105971712*a^3*b*c^10*
d^9 + 21331968*a^2*b^2*c^11*d^8) - ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2
*c^13*d^9 + 44040192*a*b*c^14*d^8)*1i)/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*1i)/(64
*(-c)^(15/4)*d^(5/4)) - ((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 1059
71712*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) + ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d
^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^14*d^8)*1i)/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42
*a*b*c*d)*1i)/(64*(-c)^(15/4)*d^(5/4))))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d))/(32*(-c)^(15/4)*d^(5/4))

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