3.117 \(\int \frac{1}{x^4 (a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\)

Optimal. Leaf size=269 \[ -\frac{b}{2 a^4 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b}{12 a^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{4 b}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 b \log (x) \left (a+b x^3\right )}{a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.142528, antiderivative size = 269, 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} \[ -\frac{b}{2 a^4 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b}{12 a^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{4 b}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 b \log (x) \left (a+b x^3\right )}{a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.0522613, size = 119, normalized size = 0.44 \[ \frac{-a \left (260 a^2 b^2 x^6+125 a^3 b x^3+12 a^4+210 a b^3 x^9+60 b^4 x^{12}\right )-180 b x^3 \log (x) \left (a+b x^3\right )^4+60 b x^3 \left (a+b x^3\right )^4 \log \left (a+b x^3\right )}{36 a^6 x^3 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

(-(a*(12*a^4 + 125*a^3*b*x^3 + 260*a^2*b^2*x^6 + 210*a*b^3*x^9 + 60*b^4*x^12)) - 180*b*x^3*(a + b*x^3)^4*Log[x
] + 60*b*x^3*(a + b*x^3)^4*Log[a + b*x^3])/(36*a^6*x^3*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.02, size = 219, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 180\,{b}^{5}\ln \left ( x \right ){x}^{15}-60\,\ln \left ( b{x}^{3}+a \right ){x}^{15}{b}^{5}+720\,a{b}^{4}\ln \left ( x \right ){x}^{12}-240\,\ln \left ( b{x}^{3}+a \right ){x}^{12}a{b}^{4}+60\,a{b}^{4}{x}^{12}+1080\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{9}-360\,\ln \left ( b{x}^{3}+a \right ){x}^{9}{a}^{2}{b}^{3}+210\,{a}^{2}{b}^{3}{x}^{9}+720\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{6}-240\,\ln \left ( b{x}^{3}+a \right ){x}^{6}{a}^{3}{b}^{2}+260\,{a}^{3}{b}^{2}{x}^{6}+180\,{a}^{4}b\ln \left ( x \right ){x}^{3}-60\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{4}b+125\,{a}^{4}b{x}^{3}+12\,{a}^{5} \right ) \left ( b{x}^{3}+a \right ) }{36\,{x}^{3}{a}^{6}} \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^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

-1/36*(180*b^5*ln(x)*x^15-60*ln(b*x^3+a)*x^15*b^5+720*a*b^4*ln(x)*x^12-240*ln(b*x^3+a)*x^12*a*b^4+60*a*b^4*x^1
2+1080*a^2*b^3*ln(x)*x^9-360*ln(b*x^3+a)*x^9*a^2*b^3+210*a^2*b^3*x^9+720*a^3*b^2*ln(x)*x^6-240*ln(b*x^3+a)*x^6
*a^3*b^2+260*a^3*b^2*x^6+180*a^4*b*ln(x)*x^3-60*ln(b*x^3+a)*x^3*a^4*b+125*a^4*b*x^3+12*a^5)*(b*x^3+a)/x^3/a^6/
((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^4/(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 = 1.61363, size = 446, normalized size = 1.66 \begin{align*} -\frac{60 \, a b^{4} x^{12} + 210 \, a^{2} b^{3} x^{9} + 260 \, a^{3} b^{2} x^{6} + 125 \, a^{4} b x^{3} + 12 \, a^{5} - 60 \,{\left (b^{5} x^{15} + 4 \, a b^{4} x^{12} + 6 \, a^{2} b^{3} x^{9} + 4 \, a^{3} b^{2} x^{6} + a^{4} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 180 \,{\left (b^{5} x^{15} + 4 \, a b^{4} x^{12} + 6 \, a^{2} b^{3} x^{9} + 4 \, a^{3} b^{2} x^{6} + a^{4} b x^{3}\right )} \log \left (x\right )}{36 \,{\left (a^{6} b^{4} x^{15} + 4 \, a^{7} b^{3} x^{12} + 6 \, a^{8} b^{2} x^{9} + 4 \, a^{9} b x^{6} + a^{10} 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)^(5/2),x, algorithm="fricas")

[Out]

-1/36*(60*a*b^4*x^12 + 210*a^2*b^3*x^9 + 260*a^3*b^2*x^6 + 125*a^4*b*x^3 + 12*a^5 - 60*(b^5*x^15 + 4*a*b^4*x^1
2 + 6*a^2*b^3*x^9 + 4*a^3*b^2*x^6 + a^4*b*x^3)*log(b*x^3 + a) + 180*(b^5*x^15 + 4*a*b^4*x^12 + 6*a^2*b^3*x^9 +
4*a^3*b^2*x^6 + a^4*b*x^3)*log(x))/(a^6*b^4*x^15 + 4*a^7*b^3*x^12 + 6*a^8*b^2*x^9 + 4*a^9*b*x^6 + a^10*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{5}{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)**(5/2),x)

[Out]

Integral(1/(x**4*((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^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x