3.114 \(\int \frac{1}{x (a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\)

Optimal. Leaf size=223 \[ \frac{1}{6 a^3 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{9 a^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{3 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\log (x) \left (a+b x^3\right )}{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

1/(3*a^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(9*a^2*
(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a^3*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + ((a +
b*x^3)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - ((a + b*x^3)*Log[a + b*x^3])/(3*a^5*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6])

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Rubi [A]  time = 0.123905, antiderivative size = 223, 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} \[ \frac{1}{6 a^3 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{9 a^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{3 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\log (x) \left (a+b x^3\right )}{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

1/(3*a^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(9*a^2*
(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a^3*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + ((a +
b*x^3)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - ((a + b*x^3)*Log[a + b*x^3])/(3*a^5*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 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x \left (a b+b^2 x^3\right )^5} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (a b+b^2 x\right )^5} \, dx,x,x^3\right )}{3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^5 b^5 x}-\frac{1}{a b^4 (a+b x)^5}-\frac{1}{a^2 b^4 (a+b x)^4}-\frac{1}{a^3 b^4 (a+b x)^3}-\frac{1}{a^4 b^4 (a+b x)^2}-\frac{1}{a^5 b^4 (a+b x)}\right ) \, dx,x,x^3\right )}{3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{3 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{9 a^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a^3 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log (x)}{a^5 \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 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0417819, size = 96, normalized size = 0.43 \[ \frac{a \left (52 a^2 b x^3+25 a^3+42 a b^2 x^6+12 b^3 x^9\right )+36 \log (x) \left (a+b x^3\right )^4-12 \left (a+b x^3\right )^4 \log \left (a+b x^3\right )}{36 a^5 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

(a*(25*a^3 + 52*a^2*b*x^3 + 42*a*b^2*x^6 + 12*b^3*x^9) + 36*(a + b*x^3)^4*Log[x] - 12*(a + b*x^3)^4*Log[a + b*
x^3])/(36*a^5*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.019, size = 193, normalized size = 0.9 \begin{align*}{\frac{ \left ( 36\,\ln \left ( x \right ){x}^{12}{b}^{4}-12\,\ln \left ( b{x}^{3}+a \right ){x}^{12}{b}^{4}+144\,\ln \left ( x \right ){x}^{9}a{b}^{3}-48\,\ln \left ( b{x}^{3}+a \right ){x}^{9}a{b}^{3}+12\,{x}^{9}a{b}^{3}+216\,\ln \left ( x \right ){x}^{6}{a}^{2}{b}^{2}-72\,\ln \left ( b{x}^{3}+a \right ){x}^{6}{a}^{2}{b}^{2}+42\,{x}^{6}{a}^{2}{b}^{2}+144\,\ln \left ( x \right ){x}^{3}{a}^{3}b-48\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{3}b+52\,{x}^{3}{a}^{3}b+36\,\ln \left ( x \right ){a}^{4}-12\,\ln \left ( b{x}^{3}+a \right ){a}^{4}+25\,{a}^{4} \right ) \left ( b{x}^{3}+a \right ) }{36\,{a}^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/36*(36*ln(x)*x^12*b^4-12*ln(b*x^3+a)*x^12*b^4+144*ln(x)*x^9*a*b^3-48*ln(b*x^3+a)*x^9*a*b^3+12*x^9*a*b^3+216*
ln(x)*x^6*a^2*b^2-72*ln(b*x^3+a)*x^6*a^2*b^2+42*x^6*a^2*b^2+144*ln(x)*x^3*a^3*b-48*ln(b*x^3+a)*x^3*a^3*b+52*x^
3*a^3*b+36*ln(x)*a^4-12*ln(b*x^3+a)*a^4+25*a^4)*(b*x^3+a)/a^5/((b*x^3+a)^2)^(5/2)

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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/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55085, size = 382, normalized size = 1.71 \begin{align*} \frac{12 \, a b^{3} x^{9} + 42 \, a^{2} b^{2} x^{6} + 52 \, a^{3} b x^{3} + 25 \, a^{4} - 12 \,{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (b x^{3} + a\right ) + 36 \,{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (x\right )}{36 \,{\left (a^{5} b^{4} x^{12} + 4 \, a^{6} b^{3} x^{9} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{3} + a^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/36*(12*a*b^3*x^9 + 42*a^2*b^2*x^6 + 52*a^3*b*x^3 + 25*a^4 - 12*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a
^3*b*x^3 + a^4)*log(b*x^3 + a) + 36*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*log(x))/(a^5*
b^4*x^12 + 4*a^6*b^3*x^9 + 6*a^7*b^2*x^6 + 4*a^8*b*x^3 + a^9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(1/(x*((a + b*x**3)**2)**(5/2)), 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/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x