3.2824 \(\int \left (\frac{c}{(a+b x)^3}\right )^{5/2} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 c^2 \sqrt{\frac{c}{(a+b x)^3}}}{13 b (a+b x)^5} \]

[Out]

(-2*c^2*Sqrt[c/(a + b*x)^3])/(13*b*(a + b*x)^5)

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Rubi [A]  time = 0.027195, 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 \[ -\frac{2 c^2 \sqrt{\frac{c}{(a+b x)^3}}}{13 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^3)^(5/2),x]

[Out]

(-2*c^2*Sqrt[c/(a + b*x)^3])/(13*b*(a + b*x)^5)

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Rubi in Sympy [A]  time = 2.58562, size = 27, normalized size = 0.9 \[ - \frac{2 c^{2} \sqrt{\frac{c}{\left (a + b x\right )^{3}}}}{13 b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c/(b*x+a)**3)**(5/2),x)

[Out]

-2*c**2*sqrt(c/(a + b*x)**3)/(13*b*(a + b*x)**5)

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Mathematica [A]  time = 0.0209611, size = 25, normalized size = 0.83 \[ -\frac{2 (a+b x) \left (\frac{c}{(a+b x)^3}\right )^{5/2}}{13 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c/(a + b*x)^3)^(5/2),x]

[Out]

(-2*(c/(a + b*x)^3)^(5/2)*(a + b*x))/(13*b)

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Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{13\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{3}}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a)^3)^(5/2),x)

[Out]

-2/13*(b*x+a)*(c/(b*x+a)^3)^(5/2)/b

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Maxima [A]  time = 1.54379, size = 32, normalized size = 1.07 \[ -\frac{2 \,{\left (b c^{\frac{5}{2}} x + a c^{\frac{5}{2}}\right )}}{13 \,{\left (b x + a\right )}^{\frac{15}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^3)^(5/2),x, algorithm="maxima")

[Out]

-2/13*(b*c^(5/2)*x + a*c^(5/2))/((b*x + a)^(15/2)*b)

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Fricas [A]  time = 0.220673, size = 126, normalized size = 4.2 \[ -\frac{2 \, c^{2} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{13 \,{\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^3)^(5/2),x, algorithm="fricas")

[Out]

-2/13*c^2*sqrt(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))/(b^6*x^5 + 5*a*b^5*x
^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a)**3)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223281, size = 70, normalized size = 2.33 \[ -\frac{2 \, c^{9}{\rm sign}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ){\rm sign}\left (b x + a\right )}{13 \,{\left (b c x + a c\right )}^{\frac{13}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^3)^(5/2),x, algorithm="giac")

[Out]

-2/13*c^9*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)^(13/2)*b)