3.5.34 \(\int \frac {x^{5/2} (a+b x^2)^2}{(c+d x^2)^3} \, dx\) [434]

Optimal. Leaf size=401 \[ -\frac {\left (42 a b-\frac {77 b^2 c}{d}+\frac {3 a^2 d}{c}\right ) x^{3/2}}{48 c d^2}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}} \]

[Out]

-1/48*(42*a*b-77*b^2*c/d+3*a^2*d/c)*x^(3/2)/c/d^2+1/4*(-a*d+b*c)^2*x^(7/2)/c/d^2/(d*x^2+c)^2-1/16*(-a*d+b*c)*(
a*d+15*b*c)*x^(7/2)/c^2/d^2/(d*x^2+c)+1/64*(-3*a^2*d^2-42*a*b*c*d+77*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)
/c^(1/4))/c^(5/4)/d^(15/4)*2^(1/2)-1/64*(-3*a^2*d^2-42*a*b*c*d+77*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^
(1/4))/c^(5/4)/d^(15/4)*2^(1/2)-1/128*(-3*a^2*d^2-42*a*b*c*d+77*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*
2^(1/2)*x^(1/2))/c^(5/4)/d^(15/4)*2^(1/2)+1/128*(-3*a^2*d^2-42*a*b*c*d+77*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4
)*d^(1/4)*2^(1/2)*x^(1/2))/c^(5/4)/d^(15/4)*2^(1/2)

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Rubi [A]
time = 0.23, antiderivative size = 401, normalized size of antiderivative = 1.00, 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 = {474, 468, 327, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (-3 a^2 d^2-42 a b c d+77 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^{5/4} d^{15/4}}-\frac {\left (-3 a^2 d^2-42 a b c d+77 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^{5/4} d^{15/4}}-\frac {\left (-3 a^2 d^2-42 a b c d+77 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^{5/4} d^{15/4}}+\frac {\left (-3 a^2 d^2-42 a b c d+77 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^{5/4} d^{15/4}}-\frac {x^{3/2} \left (\frac {3 a^2 d}{c}+42 a b-\frac {77 b^2 c}{d}\right )}{48 c d^2}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/48*((42*a*b - (77*b^2*c)/d + (3*a^2*d)/c)*x^(3/2))/(c*d^2) + ((b*c - a*d)^2*x^(7/2))/(4*c*d^2*(c + d*x^2)^2
) - ((b*c - a*d)*(15*b*c + a*d)*x^(7/2))/(16*c^2*d^2*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Arc
Tan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*
d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d
- 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(15/4)) + ((77*
b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5
/4)*d^(15/4))

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 303

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 474

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

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 {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^{5/2} \left (\frac {1}{2} \left (-8 a^2 d^2+7 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \int \frac {x^{5/2}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c d^3}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c d^3}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c d^{7/2}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c d^{7/2}}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 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 d^4}-\frac {\left (77 b^2 c^2-42 a b c d-3 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 d^4}-\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 235, normalized size = 0.59 \begin {gather*} \frac {\frac {4 \sqrt [4]{c} d^{3/4} x^{3/2} \left (3 a^2 d^2 \left (-c+3 d x^2\right )-6 a b c d \left (7 c+11 d x^2\right )+b^2 c \left (77 c^2+121 c d x^2+32 d^2 x^4\right )\right )}{\left (c+d x^2\right )^2}+3 \sqrt {2} \left (77 b^2 c^2-42 a b c d-3 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 (77 b^2 c^2-42 a b c d-3 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^{5/4} d^{15/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 306, normalized size = 0.76 \begin {gather*} \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{3}} + \frac {{\left (19 \, b^{2} c^{2} d - 22 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{\frac {7}{2}} + {\left (15 \, b^{2} c^{3} - 14 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}} - \frac {{\left (77 \, b^{2} c^{2} - 42 \, a b c d - 3 \, 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 d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*x^(3/2)/d^3 + 1/16*((19*b^2*c^2*d - 22*a*b*c*d^2 + 3*a^2*d^3)*x^(7/2) + (15*b^2*c^3 - 14*a*b*c^2*d - a
^2*c*d^2)*x^(3/2))/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3) - 1/128*(77*b^2*c^2 - 42*a*b*c*d - 3*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*d^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1813 vs. \(2 (317) = 634\).
time = 1.20, size = 1813, normalized size = 4.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(12*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6
*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6
+ 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*arctan((sqrt((208422380089*b^12*c^12 - 682109607564*a*b^11
*c^11*d + 881427350034*a^2*b^10*c^10*d^2 - 543593843100*a^3*b^9*c^9*d^3 + 136525986135*a^4*b^8*c^8*d^4 + 83346
77736*a^5*b^7*c^7*d^5 - 7849956996*a^6*b^6*c^6*d^6 - 324727704*a^7*b^5*c^5*d^7 + 207241335*a^8*b^4*c^4*d^8 + 3
2148900*a^9*b^3*c^3*d^9 + 2030994*a^10*b^2*c^2*d^10 + 61236*a^11*b*c*d^11 + 729*a^12*d^12)*x - (35153041*b^8*c
^11*d^7 - 76697544*a*b^7*c^10*d^8 + 57274140*a^2*b^6*c^9*d^9 - 13854456*a^3*b^5*c^8*d^10 - 1457946*a^4*b^4*c^7
*d^11 + 539784*a^5*b^3*c^6*d^12 + 86940*a^6*b^2*c^5*d^13 + 4536*a^7*b*c^4*d^14 + 81*a^8*c^3*d^15)*sqrt(-(35153
041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4
*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15)))*c*d^4*(-(3
5153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4
*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4) +
(456533*b^6*c^7*d^4 - 747054*a*b^5*c^6*d^5 + 354123*a^2*b^4*c^5*d^6 - 15876*a^3*b^3*c^4*d^7 - 13797*a^4*b^2*c
^3*d^8 - 1134*a^5*b*c^2*d^9 - 27*a^6*c*d^10)*sqrt(x)*(-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2
*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2
*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4))/(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a
^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c
^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)) - 3*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697
544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b
^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(c^4*d^11*(-(35153041
*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^
4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (45653
3*b^6*c^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 13797*a^4*b^2*c^2*d^4 - 1134
*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) + 3*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a
*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^
3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(-c^4*d^11*(-(35153041*b^8
*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 +
539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (456533*b^
6*c^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 13797*a^4*b^2*c^2*d^4 - 1134*a^5
*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) - 4*(32*b^2*c*d^2*x^5 + (121*b^2*c^2*d - 66*a*b*c*d^2 + 9*a^2*d^3)*x^3 + (77*b
^2*c^3 - 42*a*b*c^2*d - 3*a^2*c*d^2)*x)*sqrt(x))/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)

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

[Out]

Timed out

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Giac [A]
time = 0.62, size = 427, normalized size = 1.06 \begin {gather*} \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{3}} + \frac {19 \, b^{2} c^{2} d x^{\frac {7}{2}} - 22 \, a b c d^{2} x^{\frac {7}{2}} + 3 \, a^{2} d^{3} x^{\frac {7}{2}} + 15 \, b^{2} c^{3} x^{\frac {3}{2}} - 14 \, a b c^{2} d x^{\frac {3}{2}} - a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c d^{3}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{2} d^{6}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{2} d^{6}} + \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{2} d^{6}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{2} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*x^(3/2)/d^3 + 1/16*(19*b^2*c^2*d*x^(7/2) - 22*a*b*c*d^2*x^(7/2) + 3*a^2*d^3*x^(7/2) + 15*b^2*c^3*x^(3/
2) - 14*a*b*c^2*d*x^(3/2) - a^2*c*d^2*x^(3/2))/((d*x^2 + c)^2*c*d^3) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2
- 42*(c*d^3)^(3/4)*a*b*c*d - 3*(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^2*d^6) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(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^2*d^6) + 1/128*sqrt(2)*(77*(c*d^3)^
(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt
(c/d))/(c^2*d^6) - 1/128*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^
2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^6)

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Mupad [B]
time = 0.22, size = 197, normalized size = 0.49 \begin {gather*} \frac {2\,b^2\,x^{3/2}}{3\,d^3}-\frac {x^{3/2}\,\left (\frac {a^2\,d^2}{16}+\frac {7\,a\,b\,c\,d}{8}-\frac {15\,b^2\,c^2}{16}\right )-\frac {x^{7/2}\,\left (3\,a^2\,d^3-22\,a\,b\,c\,d^2+19\,b^2\,c^2\,d\right )}{16\,c}}{c^2\,d^3+2\,c\,d^4\,x^2+d^5\,x^4}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (3\,a^2\,d^2+42\,a\,b\,c\,d-77\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{5/4}\,d^{15/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (3\,a^2\,d^2+42\,a\,b\,c\,d-77\,b^2\,c^2\right )\,1{}\mathrm {i}}{32\,{\left (-c\right )}^{5/4}\,d^{15/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^2*x^(3/2))/(3*d^3) - (x^(3/2)*((a^2*d^2)/16 - (15*b^2*c^2)/16 + (7*a*b*c*d)/8) - (x^(7/2)*(3*a^2*d^3 + 19
*b^2*c^2*d - 22*a*b*c*d^2))/(16*c))/(c^2*d^3 + d^5*x^4 + 2*c*d^4*x^2) - (atan((d^(1/4)*x^(1/2))/(-c)^(1/4))*(3
*a^2*d^2 - 77*b^2*c^2 + 42*a*b*c*d))/(32*(-c)^(5/4)*d^(15/4)) - (atan((d^(1/4)*x^(1/2)*1i)/(-c)^(1/4))*(3*a^2*
d^2 - 77*b^2*c^2 + 42*a*b*c*d)*1i)/(32*(-c)^(5/4)*d^(15/4))

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