3.38 \(\int \frac{\left (a x+b x^2\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=152 \[ \frac{512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(17*a*x^12) + (4*b*(a*x + b*x^2)^(7/2))/(51*a^2*x^11) -
(32*b^2*(a*x + b*x^2)^(7/2))/(663*a^3*x^10) + (64*b^3*(a*x + b*x^2)^(7/2))/(243
1*a^4*x^9) - (256*b^4*(a*x + b*x^2)^(7/2))/(21879*a^5*x^8) + (512*b^5*(a*x + b*x
^2)^(7/2))/(153153*a^6*x^7)

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Rubi [A]  time = 0.226303, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^12,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(17*a*x^12) + (4*b*(a*x + b*x^2)^(7/2))/(51*a^2*x^11) -
(32*b^2*(a*x + b*x^2)^(7/2))/(663*a^3*x^10) + (64*b^3*(a*x + b*x^2)^(7/2))/(243
1*a^4*x^9) - (256*b^4*(a*x + b*x^2)^(7/2))/(21879*a^5*x^8) + (512*b^5*(a*x + b*x
^2)^(7/2))/(153153*a^6*x^7)

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Rubi in Sympy [A]  time = 24.2457, size = 144, normalized size = 0.95 \[ - \frac{2 \left (a x + b x^{2}\right )^{\frac{7}{2}}}{17 a x^{12}} + \frac{4 b \left (a x + b x^{2}\right )^{\frac{7}{2}}}{51 a^{2} x^{11}} - \frac{32 b^{2} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{663 a^{3} x^{10}} + \frac{64 b^{3} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{2431 a^{4} x^{9}} - \frac{256 b^{4} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{21879 a^{5} x^{8}} + \frac{512 b^{5} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{153153 a^{6} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a*x)**(5/2)/x**12,x)

[Out]

-2*(a*x + b*x**2)**(7/2)/(17*a*x**12) + 4*b*(a*x + b*x**2)**(7/2)/(51*a**2*x**11
) - 32*b**2*(a*x + b*x**2)**(7/2)/(663*a**3*x**10) + 64*b**3*(a*x + b*x**2)**(7/
2)/(2431*a**4*x**9) - 256*b**4*(a*x + b*x**2)**(7/2)/(21879*a**5*x**8) + 512*b**
5*(a*x + b*x**2)**(7/2)/(153153*a**6*x**7)

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Mathematica [A]  time = 0.0504098, size = 80, normalized size = 0.53 \[ \frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (-9009 a^5+6006 a^4 b x-3696 a^3 b^2 x^2+2016 a^2 b^3 x^3-896 a b^4 x^4+256 b^5 x^5\right )}{153153 a^6 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^2)^(5/2)/x^12,x]

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-9009*a^5 + 6006*a^4*b*x - 3696*a^3*b^2*x^2 +
2016*a^2*b^3*x^3 - 896*a*b^4*x^4 + 256*b^5*x^5))/(153153*a^6*x^9)

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Maple [A]  time = 0.008, size = 77, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -256\,{b}^{5}{x}^{5}+896\,{b}^{4}{x}^{4}a-2016\,{b}^{3}{x}^{3}{a}^{2}+3696\,{b}^{2}{x}^{2}{a}^{3}-6006\,bx{a}^{4}+9009\,{a}^{5} \right ) }{153153\,{x}^{11}{a}^{6}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^12,x)

[Out]

-2/153153*(b*x+a)*(-256*b^5*x^5+896*a*b^4*x^4-2016*a^2*b^3*x^3+3696*a^3*b^2*x^2-
6006*a^4*b*x+9009*a^5)*(b*x^2+a*x)^(5/2)/x^11/a^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*x^2 + a*x)^(5/2)/x^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213395, size = 140, normalized size = 0.92 \[ \frac{2 \,{\left (256 \, b^{8} x^{8} - 128 \, a b^{7} x^{7} + 96 \, a^{2} b^{6} x^{6} - 80 \, a^{3} b^{5} x^{5} + 70 \, a^{4} b^{4} x^{4} - 63 \, a^{5} b^{3} x^{3} - 12705 \, a^{6} b^{2} x^{2} - 21021 \, a^{7} b x - 9009 \, a^{8}\right )} \sqrt{b x^{2} + a x}}{153153 \, a^{6} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^12,x, algorithm="fricas")

[Out]

2/153153*(256*b^8*x^8 - 128*a*b^7*x^7 + 96*a^2*b^6*x^6 - 80*a^3*b^5*x^5 + 70*a^4
*b^4*x^4 - 63*a^5*b^3*x^3 - 12705*a^6*b^2*x^2 - 21021*a^7*b*x - 9009*a^8)*sqrt(b
*x^2 + a*x)/(a^6*x^9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{12}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a*x)**(5/2)/x**12,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.223372, size = 458, normalized size = 3.01 \[ \frac{2 \,{\left (816816 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{11} b^{\frac{11}{2}} + 5951088 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{10} a b^{5} + 19909890 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} a^{2} b^{\frac{9}{2}} + 40160120 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a^{3} b^{4} + 54063009 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{4} b^{\frac{7}{2}} + 50860719 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{5} b^{3} + 34051017 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{6} b^{\frac{5}{2}} + 16198875 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{7} b^{2} + 5360355 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{8} b^{\frac{3}{2}} + 1174173 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{9} b + 153153 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{10} \sqrt{b} + 9009 \, a^{11}\right )}}{153153 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^12,x, algorithm="giac")

[Out]

2/153153*(816816*(sqrt(b)*x - sqrt(b*x^2 + a*x))^11*b^(11/2) + 5951088*(sqrt(b)*
x - sqrt(b*x^2 + a*x))^10*a*b^5 + 19909890*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a^2
*b^(9/2) + 40160120*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^3*b^4 + 54063009*(sqrt(b
)*x - sqrt(b*x^2 + a*x))^7*a^4*b^(7/2) + 50860719*(sqrt(b)*x - sqrt(b*x^2 + a*x)
)^6*a^5*b^3 + 34051017*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^6*b^(5/2) + 16198875*
(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^7*b^2 + 5360355*(sqrt(b)*x - sqrt(b*x^2 + a*
x))^3*a^8*b^(3/2) + 1174173*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^9*b + 153153*(sq
rt(b)*x - sqrt(b*x^2 + a*x))*a^10*sqrt(b) + 9009*a^11)/(sqrt(b)*x - sqrt(b*x^2 +
a*x))^17