3.2825 \(\int \left (\frac{c}{(a+b x)^3}\right )^{3/2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{2 c \sqrt{\frac{c}{(a+b x)^3}}}{7 b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0248742, antiderivative size = 28, 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 \[ -\frac{2 c \sqrt{\frac{c}{(a+b x)^3}}}{7 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^3)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.09856, size = 26, normalized size = 0.93 \[ - \frac{2 c \sqrt{\frac{c}{\left (a + b x\right )^{3}}}}{7 b \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0145915, size = 25, normalized size = 0.89 \[ -\frac{2 (a+b x) \left (\frac{c}{(a+b x)^3}\right )^{3/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 22, normalized size = 0.8 \[ -{\frac{2\,bx+2\,a}{7\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{3}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a)^3)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.47643, size = 32, normalized size = 1.14 \[ -\frac{2 \,{\left (b c^{\frac{3}{2}} x + a c^{\frac{3}{2}}\right )}}{7 \,{\left (b x + a\right )}^{\frac{9}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*(b*c^(3/2)*x + a*c^(3/2))/((b*x + a)^(9/2)*b)

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Fricas [A]  time = 0.218368, size = 78, normalized size = 2.79 \[ -\frac{2 \, c \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{7 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.221549, size = 70, normalized size = 2.5 \[ -\frac{2 \, c^{5}{\rm sign}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ){\rm sign}\left (b x + a\right )}{7 \,{\left (b c x + a c\right )}^{\frac{7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*c^5*sign(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sign(b*x + a)/((b*c*x + a
*c)^(7/2)*b)