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

Optimal. Leaf size=65 \[ \frac{a^3}{4 b^4 \left (a+b x^2\right )^2}-\frac{3 a^2}{2 b^4 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0458384, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^3}{4 b^4 \left (a+b x^2\right )^2}-\frac{3 a^2}{2 b^4 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^2)^3,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A]  time = 0.0544749, size = 48, normalized size = 0.74 \[ -\frac{\frac{a^2 \left (5 a+6 b x^2\right )}{\left (a+b x^2\right )^2}+6 a \log \left (a+b x^2\right )-2 b x^2}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 58, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{3}}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,{a}^{2}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.33507, size = 89, normalized size = 1.37 \begin{align*} -\frac{6 \, a^{2} b x^{2} + 5 \, a^{3}}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac{x^{2}}{2 \, b^{3}} - \frac{3 \, a \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.19138, size = 186, normalized size = 2.86 \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*x^2+a)^3,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 [A]  time = 0.538994, size = 66, normalized size = 1.02 \begin{align*} - \frac{3 a \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{5 a^{3} + 6 a^{2} b x^{2}}{4 a^{2} b^{4} + 8 a b^{5} x^{2} + 4 b^{6} x^{4}} + \frac{x^{2}}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a*log(a + b*x**2)/(2*b**4) - (5*a**3 + 6*a**2*b*x**2)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) + x**2/(2
*b**3)

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Giac [A]  time = 1.71611, size = 84, normalized size = 1.29 \begin{align*} \frac{x^{2}}{2 \, b^{3}} - \frac{3 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{9 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} + 4 \, a^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/b^3 - 3/2*a*log(abs(b*x^2 + a))/b^4 + 1/4*(9*a*b^2*x^4 + 12*a^2*b*x^2 + 4*a^3)/((b*x^2 + a)^2*b^4)