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

Optimal. Leaf size=53 \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.0350458, antiderivative size = 53, 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} \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0053087, size = 53, normalized size = 1. \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 46, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{4}}{4\,{b}^{2}}}+{\frac{{x}^{6}}{6\,b}}-{\frac{{a}^{3}\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),x)

[Out]

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

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Maxima [A]  time = 2.59227, size = 62, normalized size = 1.17 \begin{align*} -\frac{a^{3} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.2818, size = 99, normalized size = 1.87 \begin{align*} \frac{2 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} - 6 \, a^{3} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.305953, size = 44, normalized size = 0.83 \begin{align*} - \frac{a^{3} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{a x^{4}}{4 b^{2}} + \frac{x^{6}}{6 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.38341, size = 63, normalized size = 1.19 \begin{align*} -\frac{a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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