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

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

[Out]

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

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Rubi [A]  time = 0.0343412, 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 \[ \frac{16 b (a+2 b x)}{3 a^4 \sqrt{a x+b x^2}}-\frac{2 (a+2 b x)}{3 a^2 \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(-5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.77902, size = 53, normalized size = 0.98 \[ - \frac{2 \left (a + 2 b x\right )}{3 a^{2} \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{8 b \left (2 a + 4 b x\right )}{3 a^{4} \sqrt{a x + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0415415, size = 48, normalized size = 0.89 \[ \frac{-2 a^3+12 a^2 b x+48 a b^2 x^2+32 b^3 x^3}{3 a^4 (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a*x)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.699501, size = 97, normalized size = 1.8 \[ -\frac{4 \, b x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a^{2}} + \frac{32 \, b^{2} x}{3 \, \sqrt{b x^{2} + a x} a^{4}} - \frac{2}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a} + \frac{16 \, b}{3 \, \sqrt{b x^{2} + a x} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217639, size = 80, normalized size = 1.48 \[ \frac{2 \,{\left (16 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} + 6 \, a^{2} b x - a^{3}\right )}}{3 \,{\left (a^{4} b x^{2} + a^{5} x\right )} \sqrt{b x^{2} + a x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x + b*x**2)**(-5/2), x)

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GIAC/XCAS [A]  time = 0.218681, size = 68, normalized size = 1.26 \[ \frac{2 \,{\left (2 \,{\left (4 \, x{\left (\frac{2 \, b^{3} x}{a^{4}} + \frac{3 \, b^{2}}{a^{3}}\right )} + \frac{3 \, b}{a^{2}}\right )} x - \frac{1}{a}\right )}}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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