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

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

[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 = 0.311404, 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 \[ -\frac{b}{12 a^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 b \log (x) \left (a+b x^3\right )}{a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{4 b}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b}{2 a^4 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

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])

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Rubi in Sympy [A]  time = 46.0664, size = 264, normalized size = 0.98 \[ \frac{2 a + 2 b x^{3}}{24 a x^{3} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}} + \frac{5}{36 a^{2} x^{3} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} + \frac{5 \left (2 a + 2 b x^{3}\right )}{36 a^{3} x^{3} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} + \frac{5}{6 a^{4} x^{3} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}} - \frac{5 b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x^{3} \right )}}{3 a^{6} \left (a + b x^{3}\right )} + \frac{5 b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (a + b x^{3} \right )}}{3 a^{6} \left (a + b x^{3}\right )} - \frac{5 \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3 a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x**3)/(24*a*x**3*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)) + 5/(36*a**2
*x**3*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)) + 5*(2*a + 2*b*x**3)/(36*a**3*x**3
*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)) + 5/(6*a**4*x**3*sqrt(a**2 + 2*a*b*x**3
+ b**2*x**6)) - 5*b*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(x**3)/(3*a**6*(a +
b*x**3)) + 5*b*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(a + b*x**3)/(3*a**6*(a +
b*x**3)) - 5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(3*a**6*x**3)

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Mathematica [A]  time = 0.0958391, size = 119, normalized size = 0.44 \[ \frac{-a \left (12 a^4+125 a^3 b x^3+260 a^2 b^2 x^6+210 a b^3 x^9+60 b^4 x^{12}\right )-180 b x^3 \log (x) \left (a+b x^3\right )^4+60 b x^3 \left (a+b x^3\right )^4 \log \left (a+b x^3\right )}{36 a^6 x^3 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]

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 = 0.026, size = 219, normalized size = 0.8 \[{\frac{ \left ( 60\,\ln \left ( b{x}^{3}+a \right ){x}^{15}{b}^{5}-180\,{b}^{5}\ln \left ( x \right ){x}^{15}+240\,\ln \left ( b{x}^{3}+a \right ){x}^{12}a{b}^{4}-720\,a{b}^{4}\ln \left ( x \right ){x}^{12}-60\,a{b}^{4}{x}^{12}+360\,\ln \left ( b{x}^{3}+a \right ){x}^{9}{a}^{2}{b}^{3}-1080\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{9}-210\,{a}^{2}{b}^{3}{x}^{9}+240\,\ln \left ( b{x}^{3}+a \right ){x}^{6}{a}^{3}{b}^{2}-720\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{6}-260\,{a}^{3}{b}^{2}{x}^{6}+60\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{4}b-180\,{a}^{4}b\ln \left ( x \right ){x}^{3}-125\,{a}^{4}b{x}^{3}-12\,{a}^{5} \right ) \left ( b{x}^{3}+a \right ) }{36\,{x}^{3}{a}^{6}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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*(60*ln(b*x^3+a)*x^15*b^5-180*b^5*ln(x)*x^15+240*ln(b*x^3+a)*x^12*a*b^4-720*
a*b^4*ln(x)*x^12-60*a*b^4*x^12+360*ln(b*x^3+a)*x^9*a^2*b^3-1080*a^2*b^3*ln(x)*x^
9-210*a^2*b^3*x^9+240*ln(b*x^3+a)*x^6*a^3*b^2-720*a^3*b^2*ln(x)*x^6-260*a^3*b^2*
x^6+60*ln(b*x^3+a)*x^3*a^4*b-180*a^4*b*ln(x)*x^3-125*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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26863, size = 279, normalized size = 1.04 \[ -\frac{60 \, a b^{4} x^{12} + 210 \, a^{2} b^{3} x^{9} + 260 \, a^{3} b^{2} x^{6} + 125 \, a^{4} b x^{3} + 12 \, a^{5} - 60 \,{\left (b^{5} x^{15} + 4 \, a b^{4} x^{12} + 6 \, a^{2} b^{3} x^{9} + 4 \, a^{3} b^{2} x^{6} + a^{4} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 180 \,{\left (b^{5} x^{15} + 4 \, a b^{4} x^{12} + 6 \, a^{2} b^{3} x^{9} + 4 \, a^{3} b^{2} x^{6} + a^{4} b x^{3}\right )} \log \left (x\right )}{36 \,{\left (a^{6} b^{4} x^{15} + 4 \, a^{7} b^{3} x^{12} + 6 \, a^{8} b^{2} x^{9} + 4 \, a^{9} b x^{6} + a^{10} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^4),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^12 + 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. \[ \int \frac{1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

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/XCAS [A]  time = 0.659481, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x