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

Optimal. Leaf size=249 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3)) + (5*a^3*b^2*x^2*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6])/(a + b*x^3) + (2*a^2*b^3*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^
3) + (5*a*b^4*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (b^5*x^11*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a + b*x^3))

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Rubi [A]  time = 0.158662, antiderivative size = 249, 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 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3)) + (5*a^3*b^2*x^2*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6])/(a + b*x^3) + (2*a^2*b^3*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^
3) + (5*a*b^4*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (b^5*x^11*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a + b*x^3))

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Rubi in Sympy [A]  time = 25.9729, size = 216, normalized size = 0.87 \[ \frac{729 a^{3} b^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88 \left (a + b x^{3}\right )} + \frac{243 a^{2} b^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{44} + \frac{405 a b^{2} x^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{4 x^{4}} + \frac{45 b^{2} x^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{11} - \frac{4 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{x^{4}} \]

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**5,x)

[Out]

729*a**3*b**2*x**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(88*(a + b*x**3)) + 243*a
**2*b**2*x**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/44 + 405*a*b**2*x**2*(a + b*x*
*3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/88 + 15*a*(a + b*x**3)*(a**2 + 2*a*b*x**
3 + b**2*x**6)**(3/2)/(4*x**4) + 45*b**2*x**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(
3/2)/11 - 4*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/x**4

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Mathematica [A]  time = 0.0497039, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-22 a^5-440 a^4 b x^3+440 a^3 b^2 x^6+176 a^2 b^3 x^9+55 a b^4 x^{12}+8 b^5 x^{15}\right )}{88 x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-22*a^5 - 440*a^4*b*x^3 + 440*a^3*b^2*x^6 + 176*a^2*b^3*x^
9 + 55*a*b^4*x^12 + 8*b^5*x^15))/(88*x^4*(a + b*x^3))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.3 \[ -{\frac{-8\,{b}^{5}{x}^{15}-55\,a{b}^{4}{x}^{12}-176\,{a}^{2}{b}^{3}{x}^{9}-440\,{a}^{3}{b}^{2}{x}^{6}+440\,{a}^{4}b{x}^{3}+22\,{a}^{5}}{88\,{x}^{4} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^5,x)

[Out]

-1/88*(-8*b^5*x^15-55*a*b^4*x^12-176*a^2*b^3*x^9-440*a^3*b^2*x^6+440*a^4*b*x^3+2
2*a^5)*((b*x^3+a)^2)^(5/2)/x^4/(b*x^3+a)^5

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Maxima [A]  time = 0.764393, size = 80, normalized size = 0.32 \[ \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \]

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^5,x, algorithm="maxima")

[Out]

1/88*(8*b^5*x^15 + 55*a*b^4*x^12 + 176*a^2*b^3*x^9 + 440*a^3*b^2*x^6 - 440*a^4*b
*x^3 - 22*a^5)/x^4

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Fricas [A]  time = 0.270441, size = 80, normalized size = 0.32 \[ \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \]

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^5,x, algorithm="fricas")

[Out]

1/88*(8*b^5*x^15 + 55*a*b^4*x^12 + 176*a^2*b^3*x^9 + 440*a^3*b^2*x^6 - 440*a^4*b
*x^3 - 22*a^5)/x^4

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

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**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.299673, size = 144, normalized size = 0.58 \[ \frac{1}{11} \, b^{5} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{8} \, a b^{4} x^{8}{\rm sign}\left (b x^{3} + a\right ) + 2 \, a^{2} b^{3} x^{5}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x^{3} + a\right ) - \frac{20 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm sign}\left (b x^{3} + a\right )}{4 \, x^{4}} \]

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^5,x, algorithm="giac")

[Out]

1/11*b^5*x^11*sign(b*x^3 + a) + 5/8*a*b^4*x^8*sign(b*x^3 + a) + 2*a^2*b^3*x^5*si
gn(b*x^3 + a) + 5*a^3*b^2*x^2*sign(b*x^3 + a) - 1/4*(20*a^4*b*x^3*sign(b*x^3 + a
) + a^5*sign(b*x^3 + a))/x^4