3.87 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{24}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 x^{23} \left (a+b x^3\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^{20} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*x^23*(a + b*x^3)) - (a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(4*x^20*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(17*x^17*(a + b*x^3)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(
7*x^14*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*
x^3)) - (b^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3))

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Rubi [A]  time = 0.160998, antiderivative size = 255, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 x^{23} \left (a+b x^3\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^{20} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^24,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*x^23*(a + b*x^3)) - (a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(4*x^20*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(17*x^17*(a + b*x^3)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(
7*x^14*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*
x^3)) - (b^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3))

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Rubi in Sympy [A]  time = 26.8114, size = 211, normalized size = 0.83 \[ \frac{729 a b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{240856 x^{11} \left (a + b x^{3}\right )} + \frac{81 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{5474 x^{17}} + \frac{3 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{92 x^{23}} - \frac{243 b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{21896 x^{11}} - \frac{9 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{238 x^{17}} - \frac{7 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{92 x^{23}} \]

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**24,x)

[Out]

729*a*b**4*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(240856*x**11*(a + b*x**3)) + 81*
a*b**2*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(5474*x**17) + 3*a*(a +
b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(92*x**23) - 243*b**4*sqrt(a**2 +
2*a*b*x**3 + b**2*x**6)/(21896*x**11) - 9*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)*
*(3/2)/(238*x**17) - 7*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(92*x**23)

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Mathematica [A]  time = 0.0366637, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (10472 a^5+60214 a^4 b x^3+141680 a^3 b^2 x^6+172040 a^2 b^3 x^9+109480 a b^4 x^{12}+30107 b^5 x^{15}\right )}{240856 x^{23} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^24,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(10472*a^5 + 60214*a^4*b*x^3 + 141680*a^3*b^2*x^6 + 172040
*a^2*b^3*x^9 + 109480*a*b^4*x^12 + 30107*b^5*x^15))/(240856*x^23*(a + b*x^3))

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Maple [A]  time = 0.012, size = 80, normalized size = 0.3 \[ -{\frac{30107\,{b}^{5}{x}^{15}+109480\,a{b}^{4}{x}^{12}+172040\,{a}^{2}{b}^{3}{x}^{9}+141680\,{a}^{3}{b}^{2}{x}^{6}+60214\,{a}^{4}b{x}^{3}+10472\,{a}^{5}}{240856\,{x}^{23} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^24,x)

[Out]

-1/240856*(30107*b^5*x^15+109480*a*b^4*x^12+172040*a^2*b^3*x^9+141680*a^3*b^2*x^
6+60214*a^4*b*x^3+10472*a^5)*((b*x^3+a)^2)^(5/2)/x^23/(b*x^3+a)^5

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Maxima [A]  time = 0.783491, size = 80, normalized size = 0.31 \[ -\frac{30107 \, b^{5} x^{15} + 109480 \, a b^{4} x^{12} + 172040 \, a^{2} b^{3} x^{9} + 141680 \, a^{3} b^{2} x^{6} + 60214 \, a^{4} b x^{3} + 10472 \, a^{5}}{240856 \, x^{23}} \]

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^24,x, algorithm="maxima")

[Out]

-1/240856*(30107*b^5*x^15 + 109480*a*b^4*x^12 + 172040*a^2*b^3*x^9 + 141680*a^3*
b^2*x^6 + 60214*a^4*b*x^3 + 10472*a^5)/x^23

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Fricas [A]  time = 0.253849, size = 80, normalized size = 0.31 \[ -\frac{30107 \, b^{5} x^{15} + 109480 \, a b^{4} x^{12} + 172040 \, a^{2} b^{3} x^{9} + 141680 \, a^{3} b^{2} x^{6} + 60214 \, a^{4} b x^{3} + 10472 \, a^{5}}{240856 \, x^{23}} \]

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^24,x, algorithm="fricas")

[Out]

-1/240856*(30107*b^5*x^15 + 109480*a*b^4*x^12 + 172040*a^2*b^3*x^9 + 141680*a^3*
b^2*x^6 + 60214*a^4*b*x^3 + 10472*a^5)/x^23

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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**24,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272021, size = 144, normalized size = 0.56 \[ -\frac{30107 \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + 109480 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 172040 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 141680 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 60214 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 10472 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{240856 \, x^{23}} \]

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^24,x, algorithm="giac")

[Out]

-1/240856*(30107*b^5*x^15*sign(b*x^3 + a) + 109480*a*b^4*x^12*sign(b*x^3 + a) +
172040*a^2*b^3*x^9*sign(b*x^3 + a) + 141680*a^3*b^2*x^6*sign(b*x^3 + a) + 60214*
a^4*b*x^3*sign(b*x^3 + a) + 10472*a^5*sign(b*x^3 + a))/x^23