3.22 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=100 \[ \frac{32 c^3 \left (b x+c x^2\right )^{5/2}}{1155 b^4 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{231 b^3 x^6}+\frac{4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}-\frac{2 \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(11*b*x^8) + (4*c*(b*x + c*x^2)^(5/2))/(33*b^2*x^7) - (
16*c^2*(b*x + c*x^2)^(5/2))/(231*b^3*x^6) + (32*c^3*(b*x + c*x^2)^(5/2))/(1155*b
^4*x^5)

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Rubi [A]  time = 0.132433, 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 \[ \frac{32 c^3 \left (b x+c x^2\right )^{5/2}}{1155 b^4 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{231 b^3 x^6}+\frac{4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}-\frac{2 \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^8,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(11*b*x^8) + (4*c*(b*x + c*x^2)^(5/2))/(33*b^2*x^7) - (
16*c^2*(b*x + c*x^2)^(5/2))/(231*b^3*x^6) + (32*c^3*(b*x + c*x^2)^(5/2))/(1155*b
^4*x^5)

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Rubi in Sympy [A]  time = 13.6403, size = 94, normalized size = 0.94 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{11 b x^{8}} + \frac{4 c \left (b x + c x^{2}\right )^{\frac{5}{2}}}{33 b^{2} x^{7}} - \frac{16 c^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{231 b^{3} x^{6}} + \frac{32 c^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{1155 b^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**8,x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(11*b*x**8) + 4*c*(b*x + c*x**2)**(5/2)/(33*b**2*x**7)
- 16*c**2*(b*x + c*x**2)**(5/2)/(231*b**3*x**6) + 32*c**3*(b*x + c*x**2)**(5/2)/
(1155*b**4*x**5)

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Mathematica [A]  time = 0.0414852, size = 51, normalized size = 0.51 \[ \frac{2 (x (b+c x))^{5/2} \left (-105 b^3+70 b^2 c x-40 b c^2 x^2+16 c^3 x^3\right )}{1155 b^4 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^8,x]

[Out]

(2*(x*(b + c*x))^(5/2)*(-105*b^3 + 70*b^2*c*x - 40*b*c^2*x^2 + 16*c^3*x^3))/(115
5*b^4*x^8)

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Maple [A]  time = 0.007, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,{x}^{3}{c}^{3}+40\,b{x}^{2}{c}^{2}-70\,{b}^{2}xc+105\,{b}^{3} \right ) }{1155\,{x}^{7}{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^8,x)

[Out]

-2/1155*(c*x+b)*(-16*c^3*x^3+40*b*c^2*x^2-70*b^2*c*x+105*b^3)*(c*x^2+b*x)^(3/2)/
x^7/b^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((c*x^2 + b*x)^(3/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218532, size = 96, normalized size = 0.96 \[ \frac{2 \,{\left (16 \, c^{5} x^{5} - 8 \, b c^{4} x^{4} + 6 \, b^{2} c^{3} x^{3} - 5 \, b^{3} c^{2} x^{2} - 140 \, b^{4} c x - 105 \, b^{5}\right )} \sqrt{c x^{2} + b x}}{1155 \, b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^8,x, algorithm="fricas")

[Out]

2/1155*(16*c^5*x^5 - 8*b*c^4*x^4 + 6*b^2*c^3*x^3 - 5*b^3*c^2*x^2 - 140*b^4*c*x -
105*b^5)*sqrt(c*x^2 + b*x)/(b^4*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**8,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**8, x)

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GIAC/XCAS [A]  time = 0.218543, size = 301, normalized size = 3.01 \[ \frac{2 \,{\left (2310 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} c^{\frac{7}{2}} + 10164 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} b c^{3} + 19635 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} b^{2} c^{\frac{5}{2}} + 21285 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b^{3} c^{2} + 13860 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{4} c^{\frac{3}{2}} + 5390 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{5} c + 1155 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{6} \sqrt{c} + 105 \, b^{7}\right )}}{1155 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^8,x, algorithm="giac")

[Out]

2/1155*(2310*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*c^(7/2) + 10164*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^6*b*c^3 + 19635*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^2*c^(5/2) + 2
1285*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^3*c^2 + 13860*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*b^4*c^(3/2) + 5390*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^5*c + 1155*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*b^6*sqrt(c) + 105*b^7)/(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^11