3.37 \(\int \frac{\left (a x+b x^2\right )^{5/2}}{x^{11}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac{128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(15*a*x^11) + (16*b*(a*x + b*x^2)^(7/2))/(195*a^2*x^10)
- (32*b^2*(a*x + b*x^2)^(7/2))/(715*a^3*x^9) + (128*b^3*(a*x + b*x^2)^(7/2))/(6
435*a^4*x^8) - (256*b^4*(a*x + b*x^2)^(7/2))/(45045*a^5*x^7)

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Rubi [A]  time = 0.178883, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac{128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^11,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(15*a*x^11) + (16*b*(a*x + b*x^2)^(7/2))/(195*a^2*x^10)
- (32*b^2*(a*x + b*x^2)^(7/2))/(715*a^3*x^9) + (128*b^3*(a*x + b*x^2)^(7/2))/(6
435*a^4*x^8) - (256*b^4*(a*x + b*x^2)^(7/2))/(45045*a^5*x^7)

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Rubi in Sympy [A]  time = 18.5336, size = 119, normalized size = 0.94 \[ - \frac{2 \left (a x + b x^{2}\right )^{\frac{7}{2}}}{15 a x^{11}} + \frac{16 b \left (a x + b x^{2}\right )^{\frac{7}{2}}}{195 a^{2} x^{10}} - \frac{32 b^{2} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{715 a^{3} x^{9}} + \frac{128 b^{3} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{6435 a^{4} x^{8}} - \frac{256 b^{4} \left (a x + b x^{2}\right )^{\frac{7}{2}}}{45045 a^{5} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a*x)**(5/2)/x**11,x)

[Out]

-2*(a*x + b*x**2)**(7/2)/(15*a*x**11) + 16*b*(a*x + b*x**2)**(7/2)/(195*a**2*x**
10) - 32*b**2*(a*x + b*x**2)**(7/2)/(715*a**3*x**9) + 128*b**3*(a*x + b*x**2)**(
7/2)/(6435*a**4*x**8) - 256*b**4*(a*x + b*x**2)**(7/2)/(45045*a**5*x**7)

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Mathematica [A]  time = 0.0455707, size = 69, normalized size = 0.55 \[ -\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (3003 a^4-1848 a^3 b x+1008 a^2 b^2 x^2-448 a b^3 x^3+128 b^4 x^4\right )}{45045 a^5 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^2)^(5/2)/x^11,x]

[Out]

(-2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(3003*a^4 - 1848*a^3*b*x + 1008*a^2*b^2*x^2 -
448*a*b^3*x^3 + 128*b^4*x^4))/(45045*a^5*x^8)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 128\,{b}^{4}{x}^{4}-448\,a{b}^{3}{x}^{3}+1008\,{b}^{2}{x}^{2}{a}^{2}-1848\,bx{a}^{3}+3003\,{a}^{4} \right ) }{45045\,{x}^{10}{a}^{5}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^11,x)

[Out]

-2/45045*(b*x+a)*(128*b^4*x^4-448*a*b^3*x^3+1008*a^2*b^2*x^2-1848*a^3*b*x+3003*a
^4)*(b*x^2+a*x)^(5/2)/x^10/a^5

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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^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218195, size = 126, normalized size = 1. \[ -\frac{2 \,{\left (128 \, b^{7} x^{7} - 64 \, a b^{6} x^{6} + 48 \, a^{2} b^{5} x^{5} - 40 \, a^{3} b^{4} x^{4} + 35 \, a^{4} b^{3} x^{3} + 4473 \, a^{5} b^{2} x^{2} + 7161 \, a^{6} b x + 3003 \, a^{7}\right )} \sqrt{b x^{2} + a x}}{45045 \, a^{5} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^11,x, algorithm="fricas")

[Out]

-2/45045*(128*b^7*x^7 - 64*a*b^6*x^6 + 48*a^2*b^5*x^5 - 40*a^3*b^4*x^4 + 35*a^4*
b^3*x^3 + 4473*a^5*b^2*x^2 + 7161*a^6*b*x + 3003*a^7)*sqrt(b*x^2 + a*x)/(a^5*x^8
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a*x)**(5/2)/x**11,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**11, x)

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GIAC/XCAS [A]  time = 0.219299, size = 419, normalized size = 3.33 \[ \frac{2 \,{\left (144144 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{10} b^{5} + 960960 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} a b^{\frac{9}{2}} + 2934360 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a^{2} b^{4} + 5360355 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{3} b^{\frac{7}{2}} + 6451445 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{4} b^{3} + 5324319 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{5} b^{\frac{5}{2}} + 3042585 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{6} b^{2} + 1186185 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{7} b^{\frac{3}{2}} + 301455 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{8} b + 45045 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{9} \sqrt{b} + 3003 \, a^{10}\right )}}{45045 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^11,x, algorithm="giac")

[Out]

2/45045*(144144*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*b^5 + 960960*(sqrt(b)*x - sqr
t(b*x^2 + a*x))^9*a*b^(9/2) + 2934360*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^2*b^4
+ 5360355*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^3*b^(7/2) + 6451445*(sqrt(b)*x - s
qrt(b*x^2 + a*x))^6*a^4*b^3 + 5324319*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^5*b^(5
/2) + 3042585*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^6*b^2 + 1186185*(sqrt(b)*x - s
qrt(b*x^2 + a*x))^3*a^7*b^(3/2) + 301455*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^8*b
+ 45045*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^9*sqrt(b) + 3003*a^10)/(sqrt(b)*x - s
qrt(b*x^2 + a*x))^15