3.78 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{15}} \, dx\)

Optimal. Leaf size=248 \[ \frac{b^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*x^14*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(4*x^8*(a + b*x^3)) - (2*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(
x^5*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3))
+ (b^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3)

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Rubi [A]  time = 0.156539, antiderivative size = 248, 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 \[ \frac{b^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^15,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*x^14*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(4*x^8*(a + b*x^3)) - (2*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(
x^5*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3))
+ (b^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3)

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Rubi in Sympy [A]  time = 26.6954, size = 211, normalized size = 0.85 \[ - \frac{729 a b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{308 x^{2} \left (a + b x^{3}\right )} + \frac{81 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{308 x^{8}} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{154 x^{14}} + \frac{243 b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{154 x^{2}} - \frac{9 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{22 x^{8}} - \frac{13 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{77 x^{14}} \]

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**15,x)

[Out]

-729*a*b**4*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(308*x**2*(a + b*x**3)) + 81*a*b
**2*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(308*x**8) + 15*a*(a + b*x*
*3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(154*x**14) + 243*b**4*sqrt(a**2 + 2*
a*b*x**3 + b**2*x**6)/(154*x**2) - 9*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)
/(22*x**8) - 13*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(77*x**14)

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Mathematica [A]  time = 0.0328613, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (22 a^5+140 a^4 b x^3+385 a^3 b^2 x^6+616 a^2 b^3 x^9+770 a b^4 x^{12}-308 b^5 x^{15}\right )}{308 x^{14} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^15,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(22*a^5 + 140*a^4*b*x^3 + 385*a^3*b^2*x^6 + 616*a^2*b^3*x^
9 + 770*a*b^4*x^12 - 308*b^5*x^15))/(308*x^14*(a + b*x^3))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.3 \[ -{\frac{-308\,{b}^{5}{x}^{15}+770\,a{b}^{4}{x}^{12}+616\,{a}^{2}{b}^{3}{x}^{9}+385\,{a}^{3}{b}^{2}{x}^{6}+140\,{a}^{4}b{x}^{3}+22\,{a}^{5}}{308\,{x}^{14} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^15,x)

[Out]

-1/308*(-308*b^5*x^15+770*a*b^4*x^12+616*a^2*b^3*x^9+385*a^3*b^2*x^6+140*a^4*b*x
^3+22*a^5)*((b*x^3+a)^2)^(5/2)/x^14/(b*x^3+a)^5

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Maxima [A]  time = 0.803092, size = 80, normalized size = 0.32 \[ \frac{308 \, b^{5} x^{15} - 770 \, a b^{4} x^{12} - 616 \, a^{2} b^{3} x^{9} - 385 \, a^{3} b^{2} x^{6} - 140 \, a^{4} b x^{3} - 22 \, a^{5}}{308 \, x^{14}} \]

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^15,x, algorithm="maxima")

[Out]

1/308*(308*b^5*x^15 - 770*a*b^4*x^12 - 616*a^2*b^3*x^9 - 385*a^3*b^2*x^6 - 140*a
^4*b*x^3 - 22*a^5)/x^14

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Fricas [A]  time = 0.255989, size = 80, normalized size = 0.32 \[ \frac{308 \, b^{5} x^{15} - 770 \, a b^{4} x^{12} - 616 \, a^{2} b^{3} x^{9} - 385 \, a^{3} b^{2} x^{6} - 140 \, a^{4} b x^{3} - 22 \, a^{5}}{308 \, x^{14}} \]

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^15,x, algorithm="fricas")

[Out]

1/308*(308*b^5*x^15 - 770*a*b^4*x^12 - 616*a^2*b^3*x^9 - 385*a^3*b^2*x^6 - 140*a
^4*b*x^3 - 22*a^5)/x^14

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{15}}\, dx \]

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**15,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**15, x)

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GIAC/XCAS [A]  time = 0.293629, size = 142, normalized size = 0.57 \[ b^{5} x{\rm sign}\left (b x^{3} + a\right ) - \frac{770 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 616 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 385 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 140 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 22 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{308 \, x^{14}} \]

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^15,x, algorithm="giac")

[Out]

b^5*x*sign(b*x^3 + a) - 1/308*(770*a*b^4*x^12*sign(b*x^3 + a) + 616*a^2*b^3*x^9*
sign(b*x^3 + a) + 385*a^3*b^2*x^6*sign(b*x^3 + a) + 140*a^4*b*x^3*sign(b*x^3 + a
) + 22*a^5*sign(b*x^3 + a))/x^14