3.2832 \(\int \frac{1}{\left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \, dx\)

Optimal. Leaf size=27 \[ \frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0266699, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c/(a + b*x)**(3/2))**(1/3)*sqrt(a + b*x)*Integral(x, (x, a + b*x))/(b*c)

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Mathematica [A]  time = 0.0345671, size = 34, normalized size = 1.26 \[ \frac{x (2 a+b x)}{2 (a+b x) \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 29, normalized size = 1.1 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ({c \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(b*x+2*a)/(b*x+a)/(c/(b*x+a)^(3/2))^(2/3)

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Maxima [A]  time = 1.33623, size = 28, normalized size = 1.04 \[ \frac{b x + a}{2 \, b \left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x + a)/(b*(c/(b*x + a)^(3/2))^(2/3))

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Fricas [A]  time = 0.336873, size = 59, normalized size = 2.19 \[ \frac{{\left (b^{2} x^{3} + 3 \, a b x^{2} + 2 \, a^{2} x\right )} \left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{1}{3}}}{2 \, \sqrt{b x + a} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*x^3 + 3*a*b*x^2 + 2*a^2*x)*(c/(b*x + a)^(3/2))^(1/3)/(sqrt(b*x + a)*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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