3.23 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=126 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{5/2}}{15015 b^5 x^5}+\frac{128 c^3 \left (b x+c x^2\right )^{5/2}}{3003 b^4 x^6}-\frac{32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}+\frac{16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac{2 \left (b x+c x^2\right )^{5/2}}{13 b x^9} \]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(13*b*x^9) + (16*c*(b*x + c*x^2)^(5/2))/(143*b^2*x^8) -
(32*c^2*(b*x + c*x^2)^(5/2))/(429*b^3*x^7) + (128*c^3*(b*x + c*x^2)^(5/2))/(300
3*b^4*x^6) - (256*c^4*(b*x + c*x^2)^(5/2))/(15015*b^5*x^5)

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Rubi [A]  time = 0.175897, 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 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{5/2}}{15015 b^5 x^5}+\frac{128 c^3 \left (b x+c x^2\right )^{5/2}}{3003 b^4 x^6}-\frac{32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}+\frac{16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac{2 \left (b x+c x^2\right )^{5/2}}{13 b x^9} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^9,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(13*b*x^9) + (16*c*(b*x + c*x^2)^(5/2))/(143*b^2*x^8) -
(32*c^2*(b*x + c*x^2)^(5/2))/(429*b^3*x^7) + (128*c^3*(b*x + c*x^2)^(5/2))/(300
3*b^4*x^6) - (256*c^4*(b*x + c*x^2)^(5/2))/(15015*b^5*x^5)

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Rubi in Sympy [A]  time = 18.9522, size = 119, normalized size = 0.94 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{13 b x^{9}} + \frac{16 c \left (b x + c x^{2}\right )^{\frac{5}{2}}}{143 b^{2} x^{8}} - \frac{32 c^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{429 b^{3} x^{7}} + \frac{128 c^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3003 b^{4} x^{6}} - \frac{256 c^{4} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15015 b^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**9,x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(13*b*x**9) + 16*c*(b*x + c*x**2)**(5/2)/(143*b**2*x**8
) - 32*c**2*(b*x + c*x**2)**(5/2)/(429*b**3*x**7) + 128*c**3*(b*x + c*x**2)**(5/
2)/(3003*b**4*x**6) - 256*c**4*(b*x + c*x**2)**(5/2)/(15015*b**5*x**5)

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Mathematica [A]  time = 0.046993, size = 62, normalized size = 0.49 \[ -\frac{2 (x (b+c x))^{5/2} \left (1155 b^4-840 b^3 c x+560 b^2 c^2 x^2-320 b c^3 x^3+128 c^4 x^4\right )}{15015 b^5 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^9,x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(1155*b^4 - 840*b^3*c*x + 560*b^2*c^2*x^2 - 320*b*c^3*x^
3 + 128*c^4*x^4))/(15015*b^5*x^9)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,{c}^{4}{x}^{4}-320\,{x}^{3}{c}^{3}b+560\,{c}^{2}{x}^{2}{b}^{2}-840\,cx{b}^{3}+1155\,{b}^{4} \right ) }{15015\,{x}^{8}{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^9,x)

[Out]

-2/15015*(c*x+b)*(128*c^4*x^4-320*b*c^3*x^3+560*b^2*c^2*x^2-840*b^3*c*x+1155*b^4
)*(c*x^2+b*x)^(3/2)/x^8/b^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((c*x^2 + b*x)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.214518, size = 111, normalized size = 0.88 \[ -\frac{2 \,{\left (128 \, c^{6} x^{6} - 64 \, b c^{5} x^{5} + 48 \, b^{2} c^{4} x^{4} - 40 \, b^{3} c^{3} x^{3} + 35 \, b^{4} c^{2} x^{2} + 1470 \, b^{5} c x + 1155 \, b^{6}\right )} \sqrt{c x^{2} + b x}}{15015 \, b^{5} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^9,x, algorithm="fricas")

[Out]

-2/15015*(128*c^6*x^6 - 64*b*c^5*x^5 + 48*b^2*c^4*x^4 - 40*b^3*c^3*x^3 + 35*b^4*
c^2*x^2 + 1470*b^5*c*x + 1155*b^6)*sqrt(c*x^2 + b*x)/(b^5*x^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**9,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.218903, size = 340, normalized size = 2.7 \[ \frac{2 \,{\left (48048 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} c^{4} + 240240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} b c^{\frac{7}{2}} + 531960 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} b^{2} c^{3} + 675675 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} b^{3} c^{\frac{5}{2}} + 535535 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b^{4} c^{2} + 270270 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{5} c^{\frac{3}{2}} + 84630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{6} c + 15015 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{7} \sqrt{c} + 1155 \, b^{8}\right )}}{15015 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^9,x, algorithm="giac")

[Out]

2/15015*(48048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*c^4 + 240240*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^7*b*c^(7/2) + 531960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^2*c^3 + 6
75675*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^3*c^(5/2) + 535535*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^4*b^4*c^2 + 270270*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^5*c^(3/2) +
84630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^6*c + 15015*(sqrt(c)*x - sqrt(c*x^2 +
b*x))*b^7*sqrt(c) + 1155*b^8)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^13