3.2828 \(\int \frac{1}{\left (\frac{c}{(a+b x)^3}\right )^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 (a+b x)^4}{11 b c \sqrt{\frac{c}{(a+b x)^3}}} \]

[Out]

(2*(a + b*x)^4)/(11*b*c*Sqrt[c/(a + b*x)^3])

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Rubi [A]  time = 0.0260761, 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 \[ \frac{2 (a+b x)^4}{11 b c \sqrt{\frac{c}{(a+b x)^3}}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^3)^(-3/2),x]

[Out]

(2*(a + b*x)^4)/(11*b*c*Sqrt[c/(a + b*x)^3])

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Rubi in Sympy [A]  time = 2.53199, size = 26, normalized size = 0.87 \[ \frac{2 \sqrt{\frac{c}{\left (a + b x\right )^{3}}} \left (a + b x\right )^{7}}{11 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c/(a + b*x)**3)*(a + b*x)**7/(11*b*c**2)

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Mathematica [A]  time = 0.0326347, size = 25, normalized size = 0.83 \[ \frac{2 (a+b x)}{11 b \left (\frac{c}{(a+b x)^3}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x))/(11*b*(c/(a + b*x)^3)^(3/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{11\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{3}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a)^3)^(3/2),x)

[Out]

2/11*(b*x+a)/b/(c/(b*x+a)^3)^(3/2)

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Maxima [A]  time = 1.43685, size = 36, normalized size = 1.2 \[ \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{9}{2}}}{11 \, b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b*sqrt(c)*x + a*sqrt(c))*(b*x + a)^(9/2)/(b*c^2)

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Fricas [A]  time = 0.21845, size = 151, normalized size = 5.03 \[ \frac{2 \,{\left (b^{7} x^{7} + 7 \, a b^{6} x^{6} + 21 \, a^{2} b^{5} x^{5} + 35 \, a^{3} b^{4} x^{4} + 35 \, a^{4} b^{3} x^{3} + 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{11 \, b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b^7*x^7 + 7*a*b^6*x^6 + 21*a^2*b^5*x^5 + 35*a^3*b^4*x^4 + 35*a^4*b^3*x^3 +
21*a^5*b^2*x^2 + 7*a^6*b*x + a^7)*sqrt(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a
^3))/(b*c^2)

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c/(b*x + a)^3)^(-3/2), x)