3.316 \(\int \frac{1}{x^2 \sqrt{a x^3+b x^4}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{16 b^2 \sqrt{a x^3+b x^4}}{15 a^3 x^2}+\frac{8 b \sqrt{a x^3+b x^4}}{15 a^2 x^3}-\frac{2 \sqrt{a x^3+b x^4}}{5 a x^4} \]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(5*a*x^4) + (8*b*Sqrt[a*x^3 + b*x^4])/(15*a^2*x^3) - (1
6*b^2*Sqrt[a*x^3 + b*x^4])/(15*a^3*x^2)

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Rubi [A]  time = 0.156459, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{16 b^2 \sqrt{a x^3+b x^4}}{15 a^3 x^2}+\frac{8 b \sqrt{a x^3+b x^4}}{15 a^2 x^3}-\frac{2 \sqrt{a x^3+b x^4}}{5 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[a*x^3 + b*x^4]),x]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(5*a*x^4) + (8*b*Sqrt[a*x^3 + b*x^4])/(15*a^2*x^3) - (1
6*b^2*Sqrt[a*x^3 + b*x^4])/(15*a^3*x^2)

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Rubi in Sympy [A]  time = 14.0659, size = 73, normalized size = 0.91 \[ - \frac{2 \sqrt{a x^{3} + b x^{4}}}{5 a x^{4}} + \frac{8 b \sqrt{a x^{3} + b x^{4}}}{15 a^{2} x^{3}} - \frac{16 b^{2} \sqrt{a x^{3} + b x^{4}}}{15 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**4+a*x**3)**(1/2),x)

[Out]

-2*sqrt(a*x**3 + b*x**4)/(5*a*x**4) + 8*b*sqrt(a*x**3 + b*x**4)/(15*a**2*x**3) -
16*b**2*sqrt(a*x**3 + b*x**4)/(15*a**3*x**2)

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Mathematica [A]  time = 0.030552, size = 42, normalized size = 0.52 \[ -\frac{2 \sqrt{x^3 (a+b x)} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*Sqrt[a*x^3 + b*x^4]),x]

[Out]

(-2*Sqrt[x^3*(a + b*x)]*(3*a^2 - 4*a*b*x + 8*b^2*x^2))/(15*a^3*x^4)

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Maple [A]  time = 0.006, size = 46, normalized size = 0.6 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 8\,{b}^{2}{x}^{2}-4\,abx+3\,{a}^{2} \right ) }{15\,x{a}^{3}}{\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^4+a*x^3)^(1/2),x)

[Out]

-2/15*(b*x+a)*(8*b^2*x^2-4*a*b*x+3*a^2)/x/a^3/(b*x^4+a*x^3)^(1/2)

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Maxima [A]  time = 1.43855, size = 62, normalized size = 0.78 \[ -\frac{2 \,{\left (\frac{15 \, \sqrt{b x + a} b^{2}}{\sqrt{x}} - \frac{10 \,{\left (b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x^3)*x^2),x, algorithm="maxima")

[Out]

-2/15*(15*sqrt(b*x + a)*b^2/sqrt(x) - 10*(b*x + a)^(3/2)*b/x^(3/2) + 3*(b*x + a)
^(5/2)/x^(5/2))/a^3

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Fricas [A]  time = 0.233186, size = 54, normalized size = 0.68 \[ -\frac{2 \, \sqrt{b x^{4} + a x^{3}}{\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x^3)*x^2),x, algorithm="fricas")

[Out]

-2/15*sqrt(b*x^4 + a*x^3)*(8*b^2*x^2 - 4*a*b*x + 3*a^2)/(a^3*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**4+a*x**3)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.236588, size = 70, normalized size = 0.88 \[ -\frac{2 \,{\left (3 \, a^{12}{\left (b + \frac{a}{x}\right )}^{\frac{5}{2}} - 10 \, a^{12}{\left (b + \frac{a}{x}\right )}^{\frac{3}{2}} b + 15 \, a^{12} \sqrt{b + \frac{a}{x}} b^{2}\right )}}{15 \, a^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x^3)*x^2),x, algorithm="giac")

[Out]

-2/15*(3*a^12*(b + a/x)^(5/2) - 10*a^12*(b + a/x)^(3/2)*b + 15*a^12*sqrt(b + a/x
)*b^2)/a^15