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

Optimal. Leaf size=108 \[ \frac{32 b^3 \sqrt{a x^3+b x^4}}{35 a^4 x^2}-\frac{16 b^2 \sqrt{a x^3+b x^4}}{35 a^3 x^3}+\frac{12 b \sqrt{a x^3+b x^4}}{35 a^2 x^4}-\frac{2 \sqrt{a x^3+b x^4}}{7 a x^5} \]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(7*a*x^5) + (12*b*Sqrt[a*x^3 + b*x^4])/(35*a^2*x^4) - (
16*b^2*Sqrt[a*x^3 + b*x^4])/(35*a^3*x^3) + (32*b^3*Sqrt[a*x^3 + b*x^4])/(35*a^4*
x^2)

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Rubi [A]  time = 0.236629, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{32 b^3 \sqrt{a x^3+b x^4}}{35 a^4 x^2}-\frac{16 b^2 \sqrt{a x^3+b x^4}}{35 a^3 x^3}+\frac{12 b \sqrt{a x^3+b x^4}}{35 a^2 x^4}-\frac{2 \sqrt{a x^3+b x^4}}{7 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(7*a*x^5) + (12*b*Sqrt[a*x^3 + b*x^4])/(35*a^2*x^4) - (
16*b^2*Sqrt[a*x^3 + b*x^4])/(35*a^3*x^3) + (32*b^3*Sqrt[a*x^3 + b*x^4])/(35*a^4*
x^2)

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Rubi in Sympy [A]  time = 20.5225, size = 100, normalized size = 0.93 \[ - \frac{2 \sqrt{a x^{3} + b x^{4}}}{7 a x^{5}} + \frac{12 b \sqrt{a x^{3} + b x^{4}}}{35 a^{2} x^{4}} - \frac{16 b^{2} \sqrt{a x^{3} + b x^{4}}}{35 a^{3} x^{3}} + \frac{32 b^{3} \sqrt{a x^{3} + b x^{4}}}{35 a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a*x**3 + b*x**4)/(7*a*x**5) + 12*b*sqrt(a*x**3 + b*x**4)/(35*a**2*x**4)
- 16*b**2*sqrt(a*x**3 + b*x**4)/(35*a**3*x**3) + 32*b**3*sqrt(a*x**3 + b*x**4)/(
35*a**4*x**2)

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Mathematica [A]  time = 0.042248, size = 53, normalized size = 0.49 \[ \frac{2 \sqrt{x^3 (a+b x)} \left (-5 a^3+6 a^2 b x-8 a b^2 x^2+16 b^3 x^3\right )}{35 a^4 x^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x^3*(a + b*x)]*(-5*a^3 + 6*a^2*b*x - 8*a*b^2*x^2 + 16*b^3*x^3))/(35*a^4*
x^5)

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Maple [A]  time = 0.006, size = 57, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -16\,{b}^{3}{x}^{3}+8\,a{b}^{2}{x}^{2}-6\,bx{a}^{2}+5\,{a}^{3} \right ) }{35\,{x}^{2}{a}^{4}}{\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(b*x+a)*(-16*b^3*x^3+8*a*b^2*x^2-6*a^2*b*x+5*a^3)/x^2/a^4/(b*x^4+a*x^3)^(1
/2)

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Maxima [A]  time = 1.42885, size = 82, normalized size = 0.76 \[ \frac{2 \,{\left (\frac{35 \, \sqrt{b x + a} b^{3}}{\sqrt{x}} - \frac{35 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}}{x^{\frac{3}{2}}} + \frac{21 \,{\left (b x + a\right )}^{\frac{5}{2}} b}{x^{\frac{5}{2}}} - \frac{5 \,{\left (b x + a\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}}\right )}}{35 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(35*sqrt(b*x + a)*b^3/sqrt(x) - 35*(b*x + a)^(3/2)*b^2/x^(3/2) + 21*(b*x +
a)^(5/2)*b/x^(5/2) - 5*(b*x + a)^(7/2)/x^(7/2))/a^4

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Fricas [A]  time = 0.227234, size = 69, normalized size = 0.64 \[ \frac{2 \,{\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt{b x^{4} + a x^{3}}}{35 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(16*b^3*x^3 - 8*a*b^2*x^2 + 6*a^2*b*x - 5*a^3)*sqrt(b*x^4 + a*x^3)/(a^4*x^5
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235022, size = 93, normalized size = 0.86 \[ -\frac{2 \,{\left (5 \, a^{24}{\left (b + \frac{a}{x}\right )}^{\frac{7}{2}} - 21 \, a^{24}{\left (b + \frac{a}{x}\right )}^{\frac{5}{2}} b + 35 \, a^{24}{\left (b + \frac{a}{x}\right )}^{\frac{3}{2}} b^{2} - 35 \, a^{24} \sqrt{b + \frac{a}{x}} b^{3}\right )}}{35 \, a^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(5*a^24*(b + a/x)^(7/2) - 21*a^24*(b + a/x)^(5/2)*b + 35*a^24*(b + a/x)^(3
/2)*b^2 - 35*a^24*sqrt(b + a/x)*b^3)/a^28