3.91 \(\int \frac{x^2}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=44 \[ \frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.034893, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1352, 608, 31} \[ \frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

Rule 1352

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

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0083088, size = 35, normalized size = 0.8 \[ \frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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Maple [A]  time = 0.007, size = 32, normalized size = 0.7 \begin{align*}{\frac{ \left ( b{x}^{3}+a \right ) \ln \left ( b{x}^{3}+a \right ) }{3\,b}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((b*x^3+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.07148, size = 23, normalized size = 0.52 \begin{align*} \frac{1}{3} \, \sqrt{\frac{1}{b^{2}}} \log \left (x^{3} + \frac{a}{b}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/((b*x^3+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(b^(-2))*log(x^3 + a/b)

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Fricas [A]  time = 1.59779, size = 30, normalized size = 0.68 \begin{align*} \frac{\log \left (b x^{3} + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*log(b*x^3 + a)/b

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Sympy [A]  time = 0.168491, size = 10, normalized size = 0.23 \begin{align*} \frac{\log{\left (a + b x^{3} \right )}}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/((b*x**3+a)**2)**(1/2),x)

[Out]

log(a + b*x**3)/(3*b)

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Giac [A]  time = 1.12937, size = 30, normalized size = 0.68 \begin{align*} \frac{\log \left ({\left | b x^{3} + a \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

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