3.634 \(\int \frac{x^7}{(a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{a^3}{4 b^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a^2}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^2 \left (a+b x^2\right )}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-3*a^2)/(2*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + a^3/(4*b^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (
x^2*(a + b*x^2))/(2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*a*(a + b*x^2)*Log[a + b*x^2])/(2*b^4*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.131324, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \[ \frac{a^3}{4 b^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a^2}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^2 \left (a+b x^2\right )}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(-3*a^2)/(2*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + a^3/(4*b^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (
x^2*(a + b*x^2))/(2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*a*(a + b*x^2)*Log[a + b*x^2])/(2*b^4*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])

Rule 1111

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && Integ
erQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^7}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^3}{\left (a b+b^2 x\right )^3} \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \left (\frac{1}{b^6}-\frac{a^3}{b^6 (a+b x)^3}+\frac{3 a^2}{b^6 (a+b x)^2}-\frac{3 a}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{3 a^2}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^3}{4 b^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^2 \left (a+b x^2\right )}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0295263, size = 81, normalized size = 0.51 \[ \frac{-4 a^2 b x^2-5 a^3+4 a b^2 x^4-6 a \left (a+b x^2\right )^2 \log \left (a+b x^2\right )+2 b^3 x^6}{4 b^4 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(-5*a^3 - 4*a^2*b*x^2 + 4*a*b^2*x^4 + 2*b^3*x^6 - 6*a*(a + b*x^2)^2*Log[a + b*x^2])/(4*b^4*(a + b*x^2)*Sqrt[(a
+ b*x^2)^2])

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Maple [A]  time = 0.227, size = 103, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2\,{b}^{3}{x}^{6}+6\,\ln \left ( b{x}^{2}+a \right ){x}^{4}a{b}^{2}-4\,a{x}^{4}{b}^{2}+12\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{2}b+4\,{a}^{2}b{x}^{2}+6\,\ln \left ( b{x}^{2}+a \right ){a}^{3}+5\,{a}^{3} \right ) \left ( b{x}^{2}+a \right ) }{4\,{b}^{4}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

-1/4*(-2*b^3*x^6+6*ln(b*x^2+a)*x^4*a*b^2-4*a*x^4*b^2+12*ln(b*x^2+a)*x^2*a^2*b+4*a^2*b*x^2+6*ln(b*x^2+a)*a^3+5*
a^3)*(b*x^2+a)/b^4/((b*x^2+a)^2)^(3/2)

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Maxima [A]  time = 1.32425, size = 198, normalized size = 1.25 \begin{align*} \frac{x^{4}}{2 \, \sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}} b^{2}} - \frac{3 \, a^{2} x^{2}}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2}} - \frac{3 \, a \log \left (x^{2} + \frac{a}{b}\right )}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} - \frac{9 \, a^{3} b}{4 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2}} + \frac{a^{2}}{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}} b^{4}} - \frac{a^{3}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/2*x^4/(sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)*b^2) - 3*a^2*x^2/((b^2)^(5/2)*(x^2 + a/b)^2) - 3/2*a*log(x^2 + a/b)/(
(b^2)^(3/2)*b) - 9/4*a^3*b/((b^2)^(7/2)*(x^2 + a/b)^2) + a^2/(sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)*b^4) - 1/2*a^3/(
(b^2)^(3/2)*(x^2 + a/b)^2*b^3)

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Fricas [A]  time = 1.34379, size = 186, normalized size = 1.18 \begin{align*} \frac{2 \, b^{3} x^{6} + 4 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*(2*b^3*x^6 + 4*a*b^2*x^4 - 4*a^2*b*x^2 - 5*a^3 - 6*(a*b^2*x^4 + 2*a^2*b*x^2 + a^3)*log(b*x^2 + a))/(b^6*x^
4 + 2*a*b^5*x^2 + a^2*b^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(x**7/((a + b*x**2)**2)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x