3.59 \(\int \frac{1}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{128 c^3 (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2}{7 b x^3 \sqrt{b x+c x^2}} \]

[Out]

-2/(7*b*x^3*Sqrt[b*x + c*x^2]) + (16*c)/(35*b^2*x^2*Sqrt[b*x + c*x^2]) - (32*c^2
)/(35*b^3*x*Sqrt[b*x + c*x^2]) + (128*c^3*(b + 2*c*x))/(35*b^5*Sqrt[b*x + c*x^2]
)

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Rubi [A]  time = 0.126569, antiderivative size = 103, 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 \[ \frac{128 c^3 (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2}{7 b x^3 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(b*x + c*x^2)^(3/2)),x]

[Out]

-2/(7*b*x^3*Sqrt[b*x + c*x^2]) + (16*c)/(35*b^2*x^2*Sqrt[b*x + c*x^2]) - (32*c^2
)/(35*b^3*x*Sqrt[b*x + c*x^2]) + (128*c^3*(b + 2*c*x))/(35*b^5*Sqrt[b*x + c*x^2]
)

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Rubi in Sympy [A]  time = 12.9309, size = 97, normalized size = 0.94 \[ - \frac{2}{7 b x^{3} \sqrt{b x + c x^{2}}} + \frac{16 c}{35 b^{2} x^{2} \sqrt{b x + c x^{2}}} - \frac{32 c^{2}}{35 b^{3} x \sqrt{b x + c x^{2}}} + \frac{64 c^{3} \left (2 b + 4 c x\right )}{35 b^{5} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(c*x**2+b*x)**(3/2),x)

[Out]

-2/(7*b*x**3*sqrt(b*x + c*x**2)) + 16*c/(35*b**2*x**2*sqrt(b*x + c*x**2)) - 32*c
**2/(35*b**3*x*sqrt(b*x + c*x**2)) + 64*c**3*(2*b + 4*c*x)/(35*b**5*sqrt(b*x + c
*x**2))

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Mathematica [A]  time = 0.0443151, size = 62, normalized size = 0.6 \[ \frac{2 \left (-5 b^4+8 b^3 c x-16 b^2 c^2 x^2+64 b c^3 x^3+128 c^4 x^4\right )}{35 b^5 x^3 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(b*x + c*x^2)^(3/2)),x]

[Out]

(2*(-5*b^4 + 8*b^3*c*x - 16*b^2*c^2*x^2 + 64*b*c^3*x^3 + 128*c^4*x^4))/(35*b^5*x
^3*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 66, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -128\,{c}^{4}{x}^{4}-64\,{x}^{3}{c}^{3}b+16\,{c}^{2}{x}^{2}{b}^{2}-8\,cx{b}^{3}+5\,{b}^{4} \right ) }{35\,{x}^{2}{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^2+b*x)^(3/2),x)

[Out]

-2/35*(c*x+b)*(-128*c^4*x^4-64*b*c^3*x^3+16*b^2*c^2*x^2-8*b^3*c*x+5*b^4)/x^2/b^5
/(c*x^2+b*x)^(3/2)

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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(1/((c*x^2 + b*x)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236176, size = 81, normalized size = 0.79 \[ \frac{2 \,{\left (128 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 16 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 5 \, b^{4}\right )}}{35 \, \sqrt{c x^{2} + b x} b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*x^3),x, algorithm="fricas")

[Out]

2/35*(128*c^4*x^4 + 64*b*c^3*x^3 - 16*b^2*c^2*x^2 + 8*b^3*c*x - 5*b^4)/(sqrt(c*x
^2 + b*x)*b^5*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(1/(x**3*(x*(b + c*x))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*x^3),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*x^3), x)