3.319 \(\int \frac{x^5}{a+b x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0169171, antiderivative size = 27, 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} \[ \frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^3),x]

[Out]

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

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^5}{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0040328, size = 27, normalized size = 1. \[ \frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.001, size = 24, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,b}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 0.984372, size = 31, normalized size = 1.15 \begin{align*} \frac{x^{3}}{3 \, b} - \frac{a \log \left (b x^{3} + a\right )}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a),x, algorithm="maxima")

[Out]

1/3*x^3/b - 1/3*a*log(b*x^3 + a)/b^2

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Fricas [A]  time = 1.40539, size = 49, normalized size = 1.81 \begin{align*} \frac{b x^{3} - a \log \left (b x^{3} + a\right )}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a),x, algorithm="fricas")

[Out]

1/3*(b*x^3 - a*log(b*x^3 + a))/b^2

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Sympy [A]  time = 0.357183, size = 20, normalized size = 0.74 \begin{align*} - \frac{a \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{x^{3}}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**3+a),x)

[Out]

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

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Giac [A]  time = 1.12267, size = 32, normalized size = 1.19 \begin{align*} \frac{x^{3}}{3 \, b} - \frac{a \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a),x, algorithm="giac")

[Out]

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