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

Optimal. Leaf size=440 \[ \frac{x^{5/2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{80 c^2 d^3}-\frac{\sqrt{x} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{16 c d^4}-\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{17/4}}-\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} d^{17/4}}-\frac{x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{9/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

-((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Sqrt[x])/(16*c*d^4) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*x^(5/2)
)/(80*c^2*d^3) + ((b*c - a*d)^2*x^(9/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(17*b*c - a*d)*x^(9/2))/(16*c^
2*d^2*(c + d*x^2)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(3/4)*d^(17/4)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d
^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[
c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(17/4))

________________________________________________________________________________________

Rubi [A]  time = 0.380913, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {463, 457, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{x^{5/2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{80 c^2 d^3}-\frac{\sqrt{x} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{16 c d^4}-\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{17/4}}-\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} d^{17/4}}-\frac{x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{9/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Sqrt[x])/(16*c*d^4) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*x^(5/2)
)/(80*c^2*d^3) + ((b*c - a*d)^2*x^(9/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(17*b*c - a*d)*x^(9/2))/(16*c^
2*d^2*(c + d*x^2)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(3/4)*d^(17/4)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d
^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[
c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(17/4))

Rule 463

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 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 321

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 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 211

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 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]

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 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 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])

Rubi steps

\begin{align*} \int \frac{x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{\int \frac{x^{7/2} \left (\frac{1}{2} \left (-8 a^2 d^2+9 (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^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac{x^{7/2}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac{x^{3/2}}{c+d x^2} \, dx}{32 c d^3}\\ &=-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt{x}}{16 c d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{32 d^4}\\ &=-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt{x}}{16 c d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 d^4}\\ &=-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt{x}}{16 c d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (117 b^2 c^2-90 a b c d+5 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 \sqrt{c} d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 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 \sqrt{c} d^4}\\ &=-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt{x}}{16 c d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (117 b^2 c^2-90 a b c d+5 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 \sqrt{c} d^{9/2}}+\frac{\left (117 b^2 c^2-90 a b c d+5 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 \sqrt{c} d^{9/2}}-\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}-\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}\\ &=-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt{x}}{16 c d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}+\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}+\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}-\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}\\ &=-\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt{x}}{16 c d^4}+\frac{\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac{(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}+\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}-\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}+\frac{\left (117 b^2 c^2-90 a b c d+5 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^{3/4} d^{17/4}}\\ \end{align*}

Mathematica [A]  time = 0.36706, size = 383, normalized size = 0.87 \[ \frac{-\frac{40 \sqrt [4]{d} \sqrt{x} \left (9 a^2 d^2-34 a b c d+25 b^2 c^2\right )}{c+d x^2}-\frac{5 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}+\frac{5 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}-\frac{10 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{3/4}}+\frac{10 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{3/4}}+\frac{160 c \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}-1280 b \sqrt [4]{d} \sqrt{x} (3 b c-2 a d)+256 b^2 d^{5/4} x^{5/2}}{640 d^{17/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1280*b*d^(1/4)*(3*b*c - 2*a*d)*Sqrt[x] + 256*b^2*d^(5/4)*x^(5/2) + (160*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c
+ d*x^2)^2 - (40*d^(1/4)*(25*b^2*c^2 - 34*a*b*c*d + 9*a^2*d^2)*Sqrt[x])/(c + d*x^2) - (10*Sqrt[2]*(117*b^2*c^2
- 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) + (10*Sqrt[2]*(117*b^2*c^2 -
90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) - (5*Sqrt[2]*(117*b^2*c^2 - 90
*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4) + (5*Sqrt[2]*(117*b^
2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4))/(640*d^(1
7/4))

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Maple [A]  time = 0.019, size = 590, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.965665, size = 3676, normalized size = 8.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*(20*(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c
^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^
2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(1/4)*arctan((sqrt(c^2*d^8*sqrt(-(187388721*b^8*c^8 - 576
580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739
000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17)) + (13689*b^4*c^4
- 21060*a*b^3*c^3*d + 9270*a^2*b^2*c^2*d^2 - 900*a^3*b*c*d^3 + 25*a^4*d^4)*x)*c^2*d^13*(-(187388721*b^8*c^8 -
576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17
739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(3/4) - (117*b
^2*c^4*d^13 - 90*a*b*c^3*d^14 + 5*a^2*c^2*d^15)*sqrt(x)*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 6973176
60*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 127350
0*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(3/4))/(187388721*b^8*c^8 - 576580680*a*b^7*c
^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^
3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)) + 5*(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)*(-(1
87388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a
^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^
17))^(1/4)*log(c*d^4*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*
b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d
^7 + 625*a^8*d^8)/(c^3*d^17))^(1/4) + (117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*sqrt(x)) - 5*(d^6*x^4 + 2*c*d^5*x
^2 + c^2*d^4)*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5
*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 62
5*a^8*d^8)/(c^3*d^17))^(1/4)*log(-c*d^4*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d
^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^
6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(1/4) + (117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*sqrt(x)) + 4*(
32*b^2*d^3*x^6 - 585*b^2*c^3 + 450*a*b*c^2*d - 25*a^2*c*d^2 - 32*(13*b^2*c*d^2 - 10*a*b*d^3)*x^4 - 9*(117*b^2*
c^2*d - 90*a*b*c*d^2 + 5*a^2*d^3)*x^2)*sqrt(x))/(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.19657, size = 609, normalized size = 1.38 \begin{align*} \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \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 d^{5}} + \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \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 d^{5}} + \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \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 d^{5}} - \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \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 d^{5}} - \frac{25 \, b^{2} c^{2} d x^{\frac{5}{2}} - 34 \, a b c d^{2} x^{\frac{5}{2}} + 9 \, a^{2} d^{3} x^{\frac{5}{2}} + 21 \, b^{2} c^{3} \sqrt{x} - 26 \, a b c^{2} d \sqrt{x} + 5 \, a^{2} c d^{2} \sqrt{x}}{16 \,{\left (d x^{2} + c\right )}^{2} d^{4}} + \frac{2 \,{\left (b^{2} d^{12} x^{\frac{5}{2}} - 15 \, b^{2} c d^{11} \sqrt{x} + 10 \, a b d^{12} \sqrt{x}\right )}}{5 \, d^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^3)^(1/4)*a*b*c*d + 5*(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*d^5) + 1/64*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^
3)^(1/4)*a*b*c*d + 5*(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*d^5) + 1/128*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(
sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^5) - 1/128*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^3)^(
1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^5) - 1/16*(25*b
^2*c^2*d*x^(5/2) - 34*a*b*c*d^2*x^(5/2) + 9*a^2*d^3*x^(5/2) + 21*b^2*c^3*sqrt(x) - 26*a*b*c^2*d*sqrt(x) + 5*a^
2*c*d^2*sqrt(x))/((d*x^2 + c)^2*d^4) + 2/5*(b^2*d^12*x^(5/2) - 15*b^2*c*d^11*sqrt(x) + 10*a*b*d^12*sqrt(x))/d^
15