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

Optimal. Leaf size=71 \[ \frac{a^3}{6 b^4 \left (a+b x^2\right )^3}-\frac{3 a^2}{4 b^4 \left (a+b x^2\right )^2}+\frac{3 a}{2 b^4 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0636041, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{a^3}{6 b^4 \left (a+b x^2\right )^3}-\frac{3 a^2}{4 b^4 \left (a+b x^2\right )^2}+\frac{3 a}{2 b^4 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [A]  time = 0.0184551, size = 50, normalized size = 0.7 \[ \frac{\frac{a \left (11 a^2+27 a b x^2+18 b^2 x^4\right )}{\left (a+b x^2\right )^3}+6 \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(11*a^2 + 27*a*b*x^2 + 18*b^2*x^4))/(a + b*x^2)^3 + 6*Log[a + b*x^2])/(12*b^4)

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Maple [A]  time = 0.049, size = 64, normalized size = 0.9 \begin{align*}{\frac{{a}^{3}}{6\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{3\,{a}^{2}}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,a}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{\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^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.00506, size = 104, normalized size = 1.46 \begin{align*} \frac{18 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 11 \, a^{3}}{12 \,{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} + \frac{\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^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

1/12*(18*a*b^2*x^4 + 27*a^2*b*x^2 + 11*a^3)/(b^7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + a^3*b^4) + 1/2*log(b*x^2
+ a)/b^4

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Fricas [A]  time = 1.72319, size = 213, normalized size = 3. \begin{align*} \frac{18 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 11 \, a^{3} + 6 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} 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)^2,x, algorithm="fricas")

[Out]

1/12*(18*a*b^2*x^4 + 27*a^2*b*x^2 + 11*a^3 + 6*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*log(b*x^2 + a))/(b^
7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + a^3*b^4)

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Sympy [A]  time = 0.654259, size = 76, normalized size = 1.07 \begin{align*} \frac{11 a^{3} + 27 a^{2} b x^{2} + 18 a b^{2} x^{4}}{12 a^{3} b^{4} + 36 a^{2} b^{5} x^{2} + 36 a b^{6} x^{4} + 12 b^{7} x^{6}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{4}} \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)**2,x)

[Out]

(11*a**3 + 27*a**2*b*x**2 + 18*a*b**2*x**4)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6)
+ log(a + b*x**2)/(2*b**4)

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Giac [A]  time = 1.16371, size = 72, normalized size = 1.01 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} - \frac{11 \, b^{2} x^{6} + 15 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \,{\left (b x^{2} + a\right )}^{3} 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)^2,x, algorithm="giac")

[Out]

1/2*log(abs(b*x^2 + a))/b^4 - 1/12*(11*b^2*x^6 + 15*a*b*x^4 + 6*a^2*x^2)/((b*x^2 + a)^3*b^3)