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

Optimal. Leaf size=136 \[ -\frac{256 b^4 \sqrt{a x^3+b x^4}}{315 a^5 x^2}+\frac{128 b^3 \sqrt{a x^3+b x^4}}{315 a^4 x^3}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6} \]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(9*a*x^6) + (16*b*Sqrt[a*x^3 + b*x^4])/(63*a^2*x^5) - (
32*b^2*Sqrt[a*x^3 + b*x^4])/(105*a^3*x^4) + (128*b^3*Sqrt[a*x^3 + b*x^4])/(315*a
^4*x^3) - (256*b^4*Sqrt[a*x^3 + b*x^4])/(315*a^5*x^2)

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Rubi [A]  time = 0.316923, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 b^4 \sqrt{a x^3+b x^4}}{315 a^5 x^2}+\frac{128 b^3 \sqrt{a x^3+b x^4}}{315 a^4 x^3}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(9*a*x^6) + (16*b*Sqrt[a*x^3 + b*x^4])/(63*a^2*x^5) - (
32*b^2*Sqrt[a*x^3 + b*x^4])/(105*a^3*x^4) + (128*b^3*Sqrt[a*x^3 + b*x^4])/(315*a
^4*x^3) - (256*b^4*Sqrt[a*x^3 + b*x^4])/(315*a^5*x^2)

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Rubi in Sympy [A]  time = 27.912, size = 128, normalized size = 0.94 \[ - \frac{2 \sqrt{a x^{3} + b x^{4}}}{9 a x^{6}} + \frac{16 b \sqrt{a x^{3} + b x^{4}}}{63 a^{2} x^{5}} - \frac{32 b^{2} \sqrt{a x^{3} + b x^{4}}}{105 a^{3} x^{4}} + \frac{128 b^{3} \sqrt{a x^{3} + b x^{4}}}{315 a^{4} x^{3}} - \frac{256 b^{4} \sqrt{a x^{3} + b x^{4}}}{315 a^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a*x**3 + b*x**4)/(9*a*x**6) + 16*b*sqrt(a*x**3 + b*x**4)/(63*a**2*x**5)
- 32*b**2*sqrt(a*x**3 + b*x**4)/(105*a**3*x**4) + 128*b**3*sqrt(a*x**3 + b*x**4)
/(315*a**4*x**3) - 256*b**4*sqrt(a*x**3 + b*x**4)/(315*a**5*x**2)

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Mathematica [A]  time = 0.0443768, size = 64, normalized size = 0.47 \[ -\frac{2 \sqrt{x^3 (a+b x)} \left (35 a^4-40 a^3 b x+48 a^2 b^2 x^2-64 a b^3 x^3+128 b^4 x^4\right )}{315 a^5 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x^3*(a + b*x)]*(35*a^4 - 40*a^3*b*x + 48*a^2*b^2*x^2 - 64*a*b^3*x^3 + 1
28*b^4*x^4))/(315*a^5*x^6)

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Maple [A]  time = 0.007, size = 68, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 128\,{b}^{4}{x}^{4}-64\,a{b}^{3}{x}^{3}+48\,{b}^{2}{x}^{2}{a}^{2}-40\,x{a}^{3}b+35\,{a}^{4} \right ) }{315\,{x}^{3}{a}^{5}}{\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(b*x+a)*(128*b^4*x^4-64*a*b^3*x^3+48*a^2*b^2*x^2-40*a^3*b*x+35*a^4)/x^3/a
^5/(b*x^4+a*x^3)^(1/2)

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Maxima [A]  time = 1.41931, size = 103, normalized size = 0.76 \[ -\frac{2 \,{\left (\frac{315 \, \sqrt{b x + a} b^{4}}{\sqrt{x}} - \frac{420 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}}{x^{\frac{3}{2}}} + \frac{378 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{2}}{x^{\frac{5}{2}}} - \frac{180 \,{\left (b x + a\right )}^{\frac{7}{2}} b}{x^{\frac{7}{2}}} + \frac{35 \,{\left (b x + a\right )}^{\frac{9}{2}}}{x^{\frac{9}{2}}}\right )}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(315*sqrt(b*x + a)*b^4/sqrt(x) - 420*(b*x + a)^(3/2)*b^3/x^(3/2) + 378*(b
*x + a)^(5/2)*b^2/x^(5/2) - 180*(b*x + a)^(7/2)*b/x^(7/2) + 35*(b*x + a)^(9/2)/x
^(9/2))/a^5

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Fricas [A]  time = 0.221085, size = 84, normalized size = 0.62 \[ -\frac{2 \,{\left (128 \, b^{4} x^{4} - 64 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} - 40 \, a^{3} b x + 35 \, a^{4}\right )} \sqrt{b x^{4} + a x^{3}}}{315 \, a^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(128*b^4*x^4 - 64*a*b^3*x^3 + 48*a^2*b^2*x^2 - 40*a^3*b*x + 35*a^4)*sqrt(
b*x^4 + a*x^3)/(a^5*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{x^{3} \left (a + b x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*sqrt(x**3*(a + b*x))), x)

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GIAC/XCAS [A]  time = 0.235313, size = 116, normalized size = 0.85 \[ -\frac{2 \,{\left (35 \, a^{40}{\left (b + \frac{a}{x}\right )}^{\frac{9}{2}} - 180 \, a^{40}{\left (b + \frac{a}{x}\right )}^{\frac{7}{2}} b + 378 \, a^{40}{\left (b + \frac{a}{x}\right )}^{\frac{5}{2}} b^{2} - 420 \, a^{40}{\left (b + \frac{a}{x}\right )}^{\frac{3}{2}} b^{3} + 315 \, a^{40} \sqrt{b + \frac{a}{x}} b^{4}\right )}}{315 \, a^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(35*a^40*(b + a/x)^(9/2) - 180*a^40*(b + a/x)^(7/2)*b + 378*a^40*(b + a/x
)^(5/2)*b^2 - 420*a^40*(b + a/x)^(3/2)*b^3 + 315*a^40*sqrt(b + a/x)*b^4)/a^45