3.74 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{11}} \, dx\)

Optimal. Leaf size=253 \[ \frac{b^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*x^10*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(2*x^4*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x
*(a + b*x^3)) + (5*a*b^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) +
(b^5*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3))

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Rubi [A]  time = 0.161025, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^11,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*x^10*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(2*x^4*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x
*(a + b*x^3)) + (5*a*b^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) +
(b^5*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3))

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Rubi in Sympy [A]  time = 26.3638, size = 212, normalized size = 0.84 \[ \frac{729 a b^{4} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{70 \left (a + b x^{3}\right )} + \frac{81 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{14 x^{4}} + \frac{3 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{14 x^{10}} + \frac{243 b^{4} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{35} - \frac{45 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{7 x^{4}} - \frac{11 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{35 x^{10}} \]

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**11,x)

[Out]

729*a*b**4*x**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(70*(a + b*x**3)) + 81*a*b**
2*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(14*x**4) + 3*a*(a + b*x**3)*
(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(14*x**10) + 243*b**4*x**2*sqrt(a**2 + 2*
a*b*x**3 + b**2*x**6)/35 - 45*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(7*x**
4) - 11*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(35*x**10)

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Mathematica [A]  time = 0.0332968, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (7 a^5+50 a^4 b x^3+175 a^3 b^2 x^6+700 a^2 b^3 x^9-175 a b^4 x^{12}-14 b^5 x^{15}\right )}{70 x^{10} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^11,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(7*a^5 + 50*a^4*b*x^3 + 175*a^3*b^2*x^6 + 700*a^2*b^3*x^9
- 175*a*b^4*x^12 - 14*b^5*x^15))/(70*x^10*(a + b*x^3))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[ -{\frac{-14\,{b}^{5}{x}^{15}-175\,a{b}^{4}{x}^{12}+700\,{a}^{2}{b}^{3}{x}^{9}+175\,{a}^{3}{b}^{2}{x}^{6}+50\,{a}^{4}b{x}^{3}+7\,{a}^{5}}{70\,{x}^{10} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^11,x)

[Out]

-1/70*(-14*b^5*x^15-175*a*b^4*x^12+700*a^2*b^3*x^9+175*a^3*b^2*x^6+50*a^4*b*x^3+
7*a^5)*((b*x^3+a)^2)^(5/2)/x^10/(b*x^3+a)^5

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Maxima [A]  time = 0.816737, size = 80, normalized size = 0.32 \[ \frac{14 \, b^{5} x^{15} + 175 \, a b^{4} x^{12} - 700 \, a^{2} b^{3} x^{9} - 175 \, a^{3} b^{2} x^{6} - 50 \, a^{4} b x^{3} - 7 \, a^{5}}{70 \, x^{10}} \]

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^11,x, algorithm="maxima")

[Out]

1/70*(14*b^5*x^15 + 175*a*b^4*x^12 - 700*a^2*b^3*x^9 - 175*a^3*b^2*x^6 - 50*a^4*
b*x^3 - 7*a^5)/x^10

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Fricas [A]  time = 0.252904, size = 80, normalized size = 0.32 \[ \frac{14 \, b^{5} x^{15} + 175 \, a b^{4} x^{12} - 700 \, a^{2} b^{3} x^{9} - 175 \, a^{3} b^{2} x^{6} - 50 \, a^{4} b x^{3} - 7 \, a^{5}}{70 \, x^{10}} \]

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^11,x, algorithm="fricas")

[Out]

1/70*(14*b^5*x^15 + 175*a*b^4*x^12 - 700*a^2*b^3*x^9 - 175*a^3*b^2*x^6 - 50*a^4*
b*x^3 - 7*a^5)/x^10

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{11}}\, dx \]

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**11,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**11, x)

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GIAC/XCAS [A]  time = 0.259142, size = 146, normalized size = 0.58 \[ \frac{1}{5} \, b^{5} x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{2} \, a b^{4} x^{2}{\rm sign}\left (b x^{3} + a\right ) - \frac{700 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 175 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 50 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 7 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{70 \, x^{10}} \]

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^11,x, algorithm="giac")

[Out]

1/5*b^5*x^5*sign(b*x^3 + a) + 5/2*a*b^4*x^2*sign(b*x^3 + a) - 1/70*(700*a^2*b^3*
x^9*sign(b*x^3 + a) + 175*a^3*b^2*x^6*sign(b*x^3 + a) + 50*a^4*b*x^3*sign(b*x^3
+ a) + 7*a^5*sign(b*x^3 + a))/x^10