3.333 \(\int \frac{1}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{16 c (b+2 c x)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(b + 2*c*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (16*c*(b + 2*c*x))/(3*b^4*Sqrt[b*
x + c*x^2])

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Rubi [A]  time = 0.0349233, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{16 c (b+2 c x)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-5/2),x]

[Out]

(-2*(b + 2*c*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (16*c*(b + 2*c*x))/(3*b^4*Sqrt[b*
x + c*x^2])

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Rubi in Sympy [A]  time = 2.74332, size = 53, normalized size = 0.98 \[ - \frac{2 \left (b + 2 c x\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{8 c \left (2 b + 4 c x\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**(5/2),x)

[Out]

-2*(b + 2*c*x)/(3*b**2*(b*x + c*x**2)**(3/2)) + 8*c*(2*b + 4*c*x)/(3*b**4*sqrt(b
*x + c*x**2))

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Mathematica [A]  time = 0.0352122, size = 48, normalized size = 0.89 \[ \frac{-2 b^3+12 b^2 c x+48 b c^2 x^2+32 c^3 x^3}{3 b^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-5/2),x]

[Out]

(-2*b^3 + 12*b^2*c*x + 48*b*c^2*x^2 + 32*c^3*x^3)/(3*b^4*(x*(b + c*x))^(3/2))

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Maple [A]  time = 0.006, size = 51, normalized size = 0.9 \[ -{\frac{2\,x \left ( cx+b \right ) \left ( -16\,{x}^{3}{c}^{3}-24\,b{x}^{2}{c}^{2}-6\,{b}^{2}xc+{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(5/2),x)

[Out]

-2/3*x*(c*x+b)*(-16*c^3*x^3-24*b*c^2*x^2-6*b^2*c*x+b^3)/b^4/(c*x^2+b*x)^(5/2)

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Maxima [A]  time = 0.686977, size = 97, normalized size = 1.8 \[ -\frac{4 \, c x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, c^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} - \frac{2}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, c}{3 \, \sqrt{c x^{2} + b x} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-5/2),x, algorithm="maxima")

[Out]

-4/3*c*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*x/(sqrt(c*x^2 + b*x)*b^4) - 2/3/((
c*x^2 + b*x)^(3/2)*b) + 16/3*c/(sqrt(c*x^2 + b*x)*b^3)

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Fricas [A]  time = 0.214943, size = 80, normalized size = 1.48 \[ \frac{2 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} + 6 \, b^{2} c x - b^{3}\right )}}{3 \,{\left (b^{4} c x^{2} + b^{5} x\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(16*c^3*x^3 + 24*b*c^2*x^2 + 6*b^2*c*x - b^3)/((b^4*c*x^2 + b^5*x)*sqrt(c*x^
2 + b*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((b*x + c*x**2)**(-5/2), x)

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GIAC/XCAS [A]  time = 0.217061, size = 68, normalized size = 1.26 \[ \frac{2 \,{\left (2 \,{\left (4 \, x{\left (\frac{2 \, c^{3} x}{b^{4}} + \frac{3 \, c^{2}}{b^{3}}\right )} + \frac{3 \, c}{b^{2}}\right )} x - \frac{1}{b}\right )}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(2*(4*x*(2*c^3*x/b^4 + 3*c^2/b^3) + 3*c/b^2)*x - 1/b)/(c*x^2 + b*x)^(3/2)