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

Optimal. Leaf size=252 \[ \frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (10*a^3*b^2*x^3*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a^2*b^3*x^6*Sqrt[a^2 + 2*a*b*
x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a*b^4*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/
(9*(a + b*x^3)) + (b^5*x^12*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*(a + b*x^3)) +
(5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi [A]  time = 0.185517, antiderivative size = 252, 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 \[ \frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (10*a^3*b^2*x^3*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a^2*b^3*x^6*Sqrt[a^2 + 2*a*b*
x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a*b^4*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/
(9*(a + b*x^3)) + (b^5*x^12*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*(a + b*x^3)) +
(5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi in Sympy [A]  time = 26.1648, size = 199, normalized size = 0.79 \[ \frac{5 a^{4} b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} + \frac{5 a^{3} b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3} + \frac{5 a^{2} b \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{6} + \frac{5 a b \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{9} - \frac{5 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{12 x^{3}} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{12 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**4*b*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(x)/(a + b*x**3) + 5*a**3*b*sqrt
(a**2 + 2*a*b*x**3 + b**2*x**6)/3 + 5*a**2*b*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3
+ b**2*x**6)/6 + 5*a*b*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/9 - 5*a*(a + b*x*
*3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(12*x**3) + (a**2 + 2*a*b*x**3 + b**2
*x**6)**(5/2)/(12*x**3)

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Mathematica [A]  time = 0.0517115, size = 85, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-12 a^5+180 a^4 b x^3 \log (x)+120 a^3 b^2 x^6+60 a^2 b^3 x^9+20 a b^4 x^{12}+3 b^5 x^{15}\right )}{36 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-12*a^5 + 120*a^3*b^2*x^6 + 60*a^2*b^3*x^9 + 20*a*b^4*x^12
+ 3*b^5*x^15 + 180*a^4*b*x^3*Log[x]))/(36*x^3*(a + b*x^3))

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Maple [A]  time = 0.019, size = 82, normalized size = 0.3 \[{\frac{3\,{b}^{5}{x}^{15}+20\,a{b}^{4}{x}^{12}+60\,{a}^{2}{b}^{3}{x}^{9}+120\,{a}^{3}{b}^{2}{x}^{6}+180\,{a}^{4}b\ln \left ( x \right ){x}^{3}-12\,{a}^{5}}{36\, \left ( b{x}^{3}+a \right ) ^{5}{x}^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*((b*x^3+a)^2)^(5/2)*(3*b^5*x^15+20*a*b^4*x^12+60*a^2*b^3*x^9+120*a^3*b^2*x^
6+180*a^4*b*ln(x)*x^3-12*a^5)/(b*x^3+a)^5/x^3

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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((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 = 0.261885, size = 82, normalized size = 0.33 \[ \frac{3 \, b^{5} x^{15} + 20 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 120 \, a^{3} b^{2} x^{6} + 180 \, a^{4} b x^{3} \log \left (x\right ) - 12 \, a^{5}}{36 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*(3*b^5*x^15 + 20*a*b^4*x^12 + 60*a^2*b^3*x^9 + 120*a^3*b^2*x^6 + 180*a^4*b*
x^3*log(x) - 12*a^5)/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.282666, size = 167, normalized size = 0.66 \[ \frac{1}{12} \, b^{5} x^{12}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{9} \, a b^{4} x^{9}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a^{4} b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) - \frac{5 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm sign}\left (b x^{3} + a\right )}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*b^5*x^12*sign(b*x^3 + a) + 5/9*a*b^4*x^9*sign(b*x^3 + a) + 5/3*a^2*b^3*x^6*
sign(b*x^3 + a) + 10/3*a^3*b^2*x^3*sign(b*x^3 + a) + 5*a^4*b*ln(abs(x))*sign(b*x
^3 + a) - 1/3*(5*a^4*b*x^3*sign(b*x^3 + a) + a^5*sign(b*x^3 + a))/x^3