3.2810 \(\int \frac{1}{\left (c (a+b x)^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2}{7 b c (a+b x)^2 \sqrt{c (a+b x)^3}} \]

[Out]

-2/(7*b*c*(a + b*x)^2*Sqrt[c*(a + b*x)^3])

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Rubi [A]  time = 0.0325733, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2}{7 b c (a+b x)^2 \sqrt{c (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^3)^(-3/2),x]

[Out]

-2/(7*b*c*(a + b*x)^2*Sqrt[c*(a + b*x)^3])

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Rubi in Sympy [A]  time = 9.25948, size = 51, normalized size = 1.7 \[ - \frac{2 \left (3 a + 3 b x\right )}{21 b \left (a^{3} c + 3 a^{2} b c x + 3 a b^{2} c x^{2} + b^{3} c x^{3}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(3*a + 3*b*x)/(21*b*(a**3*c + 3*a**2*b*c*x + 3*a*b**2*c*x**2 + b**3*c*x**3)**
(3/2))

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Mathematica [A]  time = 0.016754, size = 25, normalized size = 0.83 \[ -\frac{2 (a+b x)}{7 b \left (c (a+b x)^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*(a + b*x)^3)^(-3/2),x]

[Out]

(-2*(a + b*x))/(7*b*(c*(a + b*x)^3)^(3/2))

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Maple [A]  time = 0.003, size = 22, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{7\,b} \left ( c \left ( bx+a \right ) ^{3} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(b*x+a)^3)^(3/2),x)

[Out]

-2/7*(b*x+a)/b/(c*(b*x+a)^3)^(3/2)

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Maxima [A]  time = 1.42304, size = 58, normalized size = 1.93 \[ -\frac{2 \, \sqrt{c}}{7 \,{\left (b^{3} c^{2} x^{2} + 2 \, a b^{2} c^{2} x + a^{2} b c^{2}\right )}{\left (b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*sqrt(c)/((b^3*c^2*x^2 + 2*a*b^2*c^2*x + a^2*b*c^2)*(b*x + a)^(3/2))

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Fricas [A]  time = 0.21556, size = 147, normalized size = 4.9 \[ -\frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{7 \,{\left (b^{6} c^{2} x^{5} + 5 \, a b^{5} c^{2} x^{4} + 10 \, a^{2} b^{4} c^{2} x^{3} + 10 \, a^{3} b^{3} c^{2} x^{2} + 5 \, a^{4} b^{2} c^{2} x + a^{5} b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*sqrt(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)/(b^6*c^2*x^5 + 5*a*b^
5*c^2*x^4 + 10*a^2*b^4*c^2*x^3 + 10*a^3*b^3*c^2*x^2 + 5*a^4*b^2*c^2*x + a^5*b*c^
2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*(a + b*x)**3)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.568761, size = 4, normalized size = 0.13 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x