3.106 \(\int \frac{1}{x^4 (a^2+2 a b x^3+b^2 x^6)^{3/2}} \, dx\)

Optimal. Leaf size=188 \[ -\frac{b}{6 a^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.0972588, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 44} \[ -\frac{b}{6 a^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]

[Out]

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

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0339763, size = 97, normalized size = 0.52 \[ \frac{-a \left (2 a^2+9 a b x^3+6 b^2 x^6\right )-18 b x^3 \log (x) \left (a+b x^3\right )^2+6 b x^3 \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 a^4 x^3 \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]

[Out]

(-(a*(2*a^2 + 9*a*b*x^3 + 6*b^2*x^6)) - 18*b*x^3*(a + b*x^3)^2*Log[x] + 6*b*x^3*(a + b*x^3)^2*Log[a + b*x^3])/
(6*a^4*x^3*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.02, size = 133, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 18\,{b}^{3}\ln \left ( x \right ){x}^{9}-6\,\ln \left ( b{x}^{3}+a \right ){x}^{9}{b}^{3}+36\,a{b}^{2}\ln \left ( x \right ){x}^{6}-12\,\ln \left ( b{x}^{3}+a \right ){x}^{6}a{b}^{2}+6\,a{b}^{2}{x}^{6}+18\,{a}^{2}b\ln \left ( x \right ){x}^{3}-6\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{2}b+9\,{a}^{2}b{x}^{3}+2\,{a}^{3} \right ) \left ( b{x}^{3}+a \right ) }{6\,{x}^{3}{a}^{4}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

-1/6*(18*b^3*ln(x)*x^9-6*ln(b*x^3+a)*x^9*b^3+36*a*b^2*ln(x)*x^6-12*ln(b*x^3+a)*x^6*a*b^2+6*a*b^2*x^6+18*a^2*b*
ln(x)*x^3-6*ln(b*x^3+a)*x^3*a^2*b+9*a^2*b*x^3+2*a^3)*(b*x^3+a)/x^3/a^4/((b*x^3+a)^2)^(3/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79788, size = 247, normalized size = 1.31 \begin{align*} -\frac{6 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3} - 6 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(6*a*b^2*x^6 + 9*a^2*b*x^3 + 2*a^3 - 6*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b*x^3)*log(b*x^3 + a) + 18*(b^3*x^9 +
2*a*b^2*x^6 + a^2*b*x^3)*log(x))/(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral(1/(x**4*((a + b*x**3)**2)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x