3.70 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^7} \, dx\)

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*x^6*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (10*a^2*b^3*x^3*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a*b^4*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6
*(a + b*x^3)) + (b^5*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (10*
a^3*b^2*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 = 0.180803, 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 \[ \frac{b^5 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a b^4 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^7,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*x^6*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (10*a^2*b^3*x^3*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a*b^4*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6
*(a + b*x^3)) + (b^5*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (10*
a^3*b^2*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 = 27.1344, size = 201, normalized size = 0.8 \[ \frac{10 a^{3} b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} + \frac{10 a^{2} b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3} + \frac{5 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3} + \frac{5 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{6 x^{6}} + \frac{10 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{9} - \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{x^{6}} \]

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**7,x)

[Out]

10*a**3*b**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(x)/(a + b*x**3) + 10*a**2*b
**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/3 + 5*a*b**2*(a + b*x**3)*sqrt(a**2 + 2*
a*b*x**3 + b**2*x**6)/3 + 5*a*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2
)/(6*x**6) + 10*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/9 - (a**2 + 2*a*b*x*
*3 + b**2*x**6)**(5/2)/x**6

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Mathematica [A]  time = 0.0492585, size = 85, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-3 a^5-30 a^4 b x^3+180 a^3 b^2 x^6 \log (x)+60 a^2 b^3 x^9+15 a b^4 x^{12}+2 b^5 x^{15}\right )}{18 x^6 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^7,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(-3*a^5 - 30*a^4*b*x^3 + 60*a^2*b^3*x^9 + 15*a*b^4*x^12 + 2
*b^5*x^15 + 180*a^3*b^2*x^6*Log[x]))/(18*x^6*(a + b*x^3))

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Maple [A]  time = 0.019, size = 82, normalized size = 0.3 \[{\frac{2\,{b}^{5}{x}^{15}+15\,a{b}^{4}{x}^{12}+60\,{a}^{2}{b}^{3}{x}^{9}+180\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{6}-30\,{a}^{4}b{x}^{3}-3\,{a}^{5}}{18\, \left ( b{x}^{3}+a \right ) ^{5}{x}^{6}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^7,x)

[Out]

1/18*((b*x^3+a)^2)^(5/2)*(2*b^5*x^15+15*a*b^4*x^12+60*a^2*b^3*x^9+180*a^3*b^2*ln
(x)*x^6-30*a^4*b*x^3-3*a^5)/(b*x^3+a)^5/x^6

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258562, size = 82, normalized size = 0.33 \[ \frac{2 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 180 \, a^{3} b^{2} x^{6} \log \left (x\right ) - 30 \, a^{4} b x^{3} - 3 \, a^{5}}{18 \, x^{6}} \]

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^7,x, algorithm="fricas")

[Out]

1/18*(2*b^5*x^15 + 15*a*b^4*x^12 + 60*a^2*b^3*x^9 + 180*a^3*b^2*x^6*log(x) - 30*
a^4*b*x^3 - 3*a^5)/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{7}}\, dx \]

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**7,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.28916, size = 170, normalized size = 0.67 \[ \frac{1}{9} \, b^{5} x^{9}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{6} \, a b^{4} x^{6}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{3} \, a^{2} b^{3} x^{3}{\rm sign}\left (b x^{3} + a\right ) + 10 \, a^{3} b^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) - \frac{30 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 10 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm sign}\left (b x^{3} + a\right )}{6 \, x^{6}} \]

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^7,x, algorithm="giac")

[Out]

1/9*b^5*x^9*sign(b*x^3 + a) + 5/6*a*b^4*x^6*sign(b*x^3 + a) + 10/3*a^2*b^3*x^3*s
ign(b*x^3 + a) + 10*a^3*b^2*ln(abs(x))*sign(b*x^3 + a) - 1/6*(30*a^3*b^2*x^6*sig
n(b*x^3 + a) + 10*a^4*b*x^3*sign(b*x^3 + a) + a^5*sign(b*x^3 + a))/x^6