3.36 \(\int \frac{\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=100 \[ \frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(13*a*x^10) + (12*b*(a*x + b*x^2)^(7/2))/(143*a^2*x^9)
- (16*b^2*(a*x + b*x^2)^(7/2))/(429*a^3*x^8) + (32*b^3*(a*x + b*x^2)^(7/2))/(300
3*a^4*x^7)

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Rubi [A]  time = 0.136434, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^10,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(13*a*x^10) + (12*b*(a*x + b*x^2)^(7/2))/(143*a^2*x^9)
- (16*b^2*(a*x + b*x^2)^(7/2))/(429*a^3*x^8) + (32*b^3*(a*x + b*x^2)^(7/2))/(300
3*a^4*x^7)

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Rubi in Sympy [A]  time = 13.5397, size = 94, normalized size = 0.94 \[ - \frac{2 \left (a x + b x^{2}\right )^{\frac{7}{2}}}{13 a x^{10}} + \frac{12 b \left (a x + b x^{2}\right )^{\frac{7}{2}}}{143 a^{2} x^{9}} - \frac{16 b^{2} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{429 a^{3} x^{8}} + \frac{32 b^{3} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{3003 a^{4} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a*x)**(5/2)/x**10,x)

[Out]

-2*(a*x + b*x**2)**(7/2)/(13*a*x**10) + 12*b*(a*x + b*x**2)**(7/2)/(143*a**2*x**
9) - 16*b**2*(a*x + b*x**2)**(7/2)/(429*a**3*x**8) + 32*b**3*(a*x + b*x**2)**(7/
2)/(3003*a**4*x**7)

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Mathematica [A]  time = 0.0424585, size = 58, normalized size = 0.58 \[ \frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (-231 a^3+126 a^2 b x-56 a b^2 x^2+16 b^3 x^3\right )}{3003 a^4 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^2)^(5/2)/x^10,x]

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-231*a^3 + 126*a^2*b*x - 56*a*b^2*x^2 + 16*b^3
*x^3))/(3003*a^4*x^7)

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Maple [A]  time = 0.007, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -16\,{b}^{3}{x}^{3}+56\,a{b}^{2}{x}^{2}-126\,bx{a}^{2}+231\,{a}^{3} \right ) }{3003\,{x}^{9}{a}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^10,x)

[Out]

-2/3003*(b*x+a)*(-16*b^3*x^3+56*a*b^2*x^2-126*a^2*b*x+231*a^3)*(b*x^2+a*x)^(5/2)
/x^9/a^4

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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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217928, size = 111, normalized size = 1.11 \[ \frac{2 \,{\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt{b x^{2} + a x}}{3003 \, a^{4} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^10,x, algorithm="fricas")

[Out]

2/3003*(16*b^6*x^6 - 8*a*b^5*x^5 + 6*a^2*b^4*x^4 - 5*a^3*b^3*x^3 - 371*a^4*b^2*x
^2 - 567*a^5*b*x - 231*a^6)*sqrt(b*x^2 + a*x)/(a^4*x^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a*x)**(5/2)/x**10,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**10, x)

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GIAC/XCAS [A]  time = 0.219698, size = 379, normalized size = 3.79 \[ \frac{2 \,{\left (6006 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} b^{\frac{9}{2}} + 36036 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{2} b^{\frac{7}{2}} + 161733 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{4} b^{\frac{5}{2}} + 121121 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{6} b^{\frac{3}{2}} + 17199 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{8} \sqrt{b} + 231 \, a^{9}\right )}}{3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^10,x, algorithm="giac")

[Out]

2/3003*(6006*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*b^(9/2) + 36036*(sqrt(b)*x - sqrt
(b*x^2 + a*x))^8*a*b^4 + 99099*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^2*b^(7/2) + 1
61733*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*b^3 + 171171*(sqrt(b)*x - sqrt(b*x^2
+ a*x))^5*a^4*b^(5/2) + 121121*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^5*b^2 + 5705
7*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^6*b^(3/2) + 17199*(sqrt(b)*x - sqrt(b*x^2
+ a*x))^2*a^7*b + 3003*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^8*sqrt(b) + 231*a^9)/(s
qrt(b)*x - sqrt(b*x^2 + a*x))^13