3.440 \(\int \frac {(a+b x^2)^2}{x^{7/2} (c+d x^2)^3} \, dx\)

Optimal. Leaf size=439 \[ -\frac {13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt {x}}-\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{80 c^3 d \sqrt {x} \left (c+d x^2\right )} \]

[Out]

-2/5*a^2/c/x^(5/2)/(d*x^2+c)^2-1/64*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4
))/c^(17/4)/d^(3/4)*2^(1/2)+1/64*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/
c^(17/4)/d^(3/4)*2^(1/2)+1/128*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)
*x^(1/2))/c^(17/4)/d^(3/4)*2^(1/2)-1/128*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/
4)*2^(1/2)*x^(1/2))/c^(17/4)/d^(3/4)*2^(1/2)+1/16*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))/c^4/d/x^(1/2)+1/20*(-13*a
^2*d^2+10*a*b*c*d-5*b^2*c^2)/c^2/d/(d*x^2+c)^2/x^(1/2)+1/80*(-5*b^2*c^2+9*a*d*(-13*a*d+10*b*c))/c^3/d/(d*x^2+c
)/x^(1/2)

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Rubi [A]  time = 0.45, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {462, 457, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]

[Out]

(5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))/(16*c^4*d*Sqrt[x]) - (2*a^2)/(5*c*x^(5/2)*(c + d*x^2)^2) - (5*b^2*c^2 -
10*a*b*c*d + 13*a^2*d^2)/(20*c^2*d*Sqrt[x]*(c + d*x^2)^2) - ((5*b^2)/d - (9*a*(10*b*c - 13*a*d))/c^2)/(80*c*Sq
rt[x]*(c + d*x^2)) - ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32
*Sqrt[2]*c^(17/4)*d^(3/4)) + ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/
4)])/(32*Sqrt[2]*c^(17/4)*d^(3/4)) + ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1
/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*d^(3/4)) - ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*Log[Sqrt[c] +
Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*d^(3/4))

Rule 204

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

Rule 290

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] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

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

Rule 325

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

Rule 329

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 457

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 462

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 617

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 628

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 1162

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 1165

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^{7/2} \left (c+d x^2\right )^3} \, dx &=-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {2 \int \frac {\frac {1}{2} a (10 b c-13 a d)+\frac {5}{2} b^2 c x^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx}{5 c}\\ &=-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}+\frac {1}{40} \left (-\frac {5 b^2}{d}+\frac {9 a (10 b c-13 a d)}{c^2}\right ) \int \frac {1}{x^{3/2} \left (c+d x^2\right )^2} \, dx\\ &=-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}\right ) \int \frac {1}{x^{3/2} \left (c+d x^2\right )} \, dx}{32 c}\\ &=\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c^4}\\ &=\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^4}\\ &=\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^4 \sqrt {d}}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^4 \sqrt {d}}\\ &=\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {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^4 d}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {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^4 d}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {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^{17/4} d^{3/4}}\\ &=\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {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^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \operatorname {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^{17/4} d^{3/4}}\\ &=\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 382, normalized size = 0.87 \[ \frac {\frac {40 \sqrt [4]{c} x^{3/2} \left (21 a^2 d^2-26 a b c d+5 b^2 c^2\right )}{c+d x^2}+\frac {5 \sqrt {2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{d^{3/4}}-\frac {5 \sqrt {2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{d^{3/4}}-\frac {10 \sqrt {2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{d^{3/4}}+\frac {10 \sqrt {2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{d^{3/4}}-\frac {256 a^2 c^{5/4}}{x^{5/2}}+\frac {160 c^{5/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {1280 a \sqrt [4]{c} (3 a d-2 b c)}{\sqrt {x}}}{640 c^{17/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]

[Out]

((-256*a^2*c^(5/4))/x^(5/2) + (1280*a*c^(1/4)*(-2*b*c + 3*a*d))/Sqrt[x] + (160*c^(5/4)*(b*c - a*d)^2*x^(3/2))/
(c + d*x^2)^2 + (40*c^(1/4)*(5*b^2*c^2 - 26*a*b*c*d + 21*a^2*d^2)*x^(3/2))/(c + d*x^2) - (10*Sqrt[2]*(5*b^2*c^
2 - 90*a*b*c*d + 117*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (10*Sqrt[2]*(5*b^2*c^2
- 90*a*b*c*d + 117*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (5*Sqrt[2]*(5*b^2*c^2 - 9
0*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4) - (5*Sqrt[2]*(5*b
^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4))/(640*c
^(17/4))

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fricas [B]  time = 0.60, size = 1832, normalized size = 4.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(7/2)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/320*(20*(c^4*d^2*x^7 + 2*c^5*d*x^5 + c^6*x^3)*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2
- 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6
- 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(1/4)*arctan((sqrt((15625*b^12*c^12 - 1687500*a*b^11
*c^11*d + 78131250*a^2*b^10*c^10*d^2 - 2019937500*a^3*b^9*c^9*d^3 + 31839834375*a^4*b^8*c^8*d^4 - 314326575000
*a^5*b^7*c^7*d^5 + 1936382557500*a^6*b^6*c^6*d^6 - 7355241855000*a^7*b^5*c^5*d^7 + 17434219710375*a^8*b^4*c^4*
d^8 - 25881265273500*a^9*b^3*c^3*d^9 + 23425464012210*a^10*b^2*c^2*d^10 - 11839219392780*a^11*b*c*d^11 + 25651
64201769*a^12*d^12)*x - (625*b^8*c^17*d - 45000*a*b^7*c^16*d^2 + 1273500*a^2*b^6*c^15*d^3 - 17739000*a^3*b^5*c
^14*d^4 + 124525350*a^4*b^4*c^13*d^5 - 415092600*a^5*b^3*c^12*d^6 + 697317660*a^6*b^2*c^11*d^7 - 576580680*a^7
*b*c^10*d^8 + 187388721*a^8*c^9*d^9)*sqrt(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 177390
00*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 57658
0680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3)))*c^4*d*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*
c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2
*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(1/4) - (125*b^6*c^10*d - 6750*a*b^5*c^9*d^2
+ 130275*a^2*b^4*c^8*d^3 - 1044900*a^3*b^3*c^7*d^4 + 3048435*a^4*b^2*c^6*d^5 - 3696030*a^5*b*c^5*d^6 + 160161
3*a^6*c^4*d^7)*sqrt(x)*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3
+ 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 +
187388721*a^8*d^8)/(c^17*d^3))^(1/4))/(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a
^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680
*a^7*b*c*d^7 + 187388721*a^8*d^8)) - 5*(c^4*d^2*x^7 + 2*c^5*d*x^5 + c^6*x^3)*(-(625*b^8*c^8 - 45000*a*b^7*c^7*
d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5
+ 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(1/4)*log(c^13*d^2*(-(62
5*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4
- 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^
3))^(3/4) + (125*b^6*c^6 - 6750*a*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 1044900*a^3*b^3*c^3*d^3 + 3048435*a^4*b
^2*c^2*d^4 - 3696030*a^5*b*c*d^5 + 1601613*a^6*d^6)*sqrt(x)) + 5*(c^4*d^2*x^7 + 2*c^5*d*x^5 + c^6*x^3)*(-(625*
b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 -
415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3)
)^(1/4)*log(-c^13*d^2*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3
+ 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 +
187388721*a^8*d^8)/(c^17*d^3))^(3/4) + (125*b^6*c^6 - 6750*a*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 1044900*a^3*
b^3*c^3*d^3 + 3048435*a^4*b^2*c^2*d^4 - 3696030*a^5*b*c*d^5 + 1601613*a^6*d^6)*sqrt(x)) - 4*(5*(5*b^2*c^2*d -
90*a*b*c*d^2 + 117*a^2*d^3)*x^6 - 32*a^2*c^3 + 9*(5*b^2*c^3 - 90*a*b*c^2*d + 117*a^2*c*d^2)*x^4 - 32*(10*a*b*c
^3 - 13*a^2*c^2*d)*x^2)*sqrt(x))/(c^4*d^2*x^7 + 2*c^5*d*x^5 + c^6*x^3)

________________________________________________________________________________________

giac [A]  time = 0.53, size = 444, normalized size = 1.01 \[ \frac {5 \, b^{2} c^{2} d x^{\frac {7}{2}} - 26 \, a b c d^{2} x^{\frac {7}{2}} + 21 \, a^{2} d^{3} x^{\frac {7}{2}} + 9 \, b^{2} c^{3} x^{\frac {3}{2}} - 34 \, a b c^{2} d x^{\frac {3}{2}} + 25 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{4}} - \frac {2 \, {\left (10 \, a b c x^{2} - 15 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{4} x^{\frac {5}{2}}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{5} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(7/2)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/16*(5*b^2*c^2*d*x^(7/2) - 26*a*b*c*d^2*x^(7/2) + 21*a^2*d^3*x^(7/2) + 9*b^2*c^3*x^(3/2) - 34*a*b*c^2*d*x^(3/
2) + 25*a^2*c*d^2*x^(3/2))/((d*x^2 + c)^2*c^4) - 2/5*(10*a*b*c*x^2 - 15*a^2*d*x^2 + a^2*c)/(c^4*x^(5/2)) + 1/6
4*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*
(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^5*d^3) + 1/64*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^
(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/
(c^5*d^3) - 1/128*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*log
(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^5*d^3) + 1/128*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^
(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^5*d^3)

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maple [A]  time = 0.03, size = 590, normalized size = 1.34 \[ \frac {21 a^{2} d^{3} x^{\frac {7}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c^{4}}-\frac {13 a b \,d^{2} x^{\frac {7}{2}}}{8 \left (d \,x^{2}+c \right )^{2} c^{3}}+\frac {5 b^{2} d \,x^{\frac {7}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {25 a^{2} d^{2} x^{\frac {3}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c^{3}}-\frac {17 a b d \,x^{\frac {3}{2}}}{8 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {9 b^{2} x^{\frac {3}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c}+\frac {117 \sqrt {2}\, a^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{4}}+\frac {117 \sqrt {2}\, a^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{4}}+\frac {117 \sqrt {2}\, a^{2} d \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{128 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{4}}-\frac {45 \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{32 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{3}}-\frac {45 \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{32 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{3}}-\frac {45 \sqrt {2}\, a b \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{3}}+\frac {5 \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2} d}+\frac {5 \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2} d}+\frac {5 \sqrt {2}\, b^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{128 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2} d}+\frac {6 a^{2} d}{c^{4} \sqrt {x}}-\frac {4 a b}{c^{3} \sqrt {x}}-\frac {2 a^{2}}{5 c^{3} x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(7/2)/(d*x^2+c)^3,x)

[Out]

21/16/c^4/(d*x^2+c)^2*x^(7/2)*a^2*d^3-13/8/c^3/(d*x^2+c)^2*x^(7/2)*a*b*d^2+5/16/c^2/(d*x^2+c)^2*x^(7/2)*b^2*d+
25/16/c^3/(d*x^2+c)^2*x^(3/2)*a^2*d^2-17/8/c^2/(d*x^2+c)^2*x^(3/2)*a*b*d+9/16/c/(d*x^2+c)^2*x^(3/2)*b^2+117/12
8/c^4*d/(c/d)^(1/4)*2^(1/2)*a^2*ln((x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+
(c/d)^(1/2)))+117/64/c^4*d/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+117/64/c^4*d/(c/d)^(1
/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-45/64/c^3/(c/d)^(1/4)*2^(1/2)*a*b*ln((x-(c/d)^(1/4)*2^(1
/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))-45/32/c^3/(c/d)^(1/4)*2^(1/2)*a*b*arctan
(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-45/32/c^3/(c/d)^(1/4)*2^(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+5/128/
c^2/d/(c/d)^(1/4)*2^(1/2)*b^2*ln((x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c
/d)^(1/2)))+5/64/c^2/d/(c/d)^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+5/64/c^2/d/(c/d)^(1/4)*2^
(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/5*a^2/c^3/x^(5/2)+6*a^2/c^4/x^(1/2)*d-4*a/c^3/x^(1/2)*b

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maxima [A]  time = 2.41, size = 324, normalized size = 0.74 \[ \frac {5 \, {\left (5 \, b^{2} c^{2} d - 90 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{6} - 32 \, a^{2} c^{3} + 9 \, {\left (5 \, b^{2} c^{3} - 90 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} x^{4} - 32 \, {\left (10 \, a b c^{3} - 13 \, a^{2} c^{2} d\right )} x^{2}}{80 \, {\left (c^{4} d^{2} x^{\frac {13}{2}} + 2 \, c^{5} d x^{\frac {9}{2}} + c^{6} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, b^{2} c^{2} - 90 \, a b c d + 117 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(7/2)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/80*(5*(5*b^2*c^2*d - 90*a*b*c*d^2 + 117*a^2*d^3)*x^6 - 32*a^2*c^3 + 9*(5*b^2*c^3 - 90*a*b*c^2*d + 117*a^2*c*
d^2)*x^4 - 32*(10*a*b*c^3 - 13*a^2*c^2*d)*x^2)/(c^4*d^2*x^(13/2) + 2*c^5*d*x^(9/2) + c^6*x^(5/2)) + 1/128*(5*b
^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*(2*sqrt(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(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(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(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d
^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(
d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/c^4

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mupad [B]  time = 0.24, size = 208, normalized size = 0.47 \[ \frac {\frac {9\,x^4\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{80\,c^3}-\frac {2\,a^2}{5\,c}+\frac {2\,a\,x^2\,\left (13\,a\,d-10\,b\,c\right )}{5\,c^2}+\frac {d\,x^6\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{16\,c^4}}{c^2\,x^{5/2}+d^2\,x^{13/2}+2\,c\,d\,x^{9/2}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{17/4}\,d^{3/4}}-\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{17/4}\,d^{3/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x)

[Out]

((9*x^4*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(80*c^3) - (2*a^2)/(5*c) + (2*a*x^2*(13*a*d - 10*b*c))/(5*c^2)
+ (d*x^6*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(16*c^4))/(c^2*x^(5/2) + d^2*x^(13/2) + 2*c*d*x^(9/2)) + (at
an((d^(1/4)*x^(1/2))/(-c)^(1/4))*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(32*(-c)^(17/4)*d^(3/4)) - (atanh((d^
(1/4)*x^(1/2))/(-c)^(1/4))*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(32*(-c)^(17/4)*d^(3/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(7/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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