3.2833 \(\int \frac{1}{\left (\frac{c}{(a+b x)^{2/3}}\right )^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ \frac{(a+b x)^{5/3}}{2 b c \sqrt{\frac{c}{(a+b x)^{2/3}}}} \]

[Out]

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

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Rubi [A]  time = 0.0299885, antiderivative size = 34, 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 \[ \frac{(a+b x)^{5/3}}{2 b c \sqrt{\frac{c}{(a+b x)^{2/3}}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0340955, size = 34, normalized size = 1. \[ \frac{x (2 a+b x)}{2 (a+b x) \left (\frac{c}{(a+b x)^{2/3}}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 29, normalized size = 0.9 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ({c \left ( bx+a \right ) ^{-{\frac{2}{3}}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34058, size = 20, normalized size = 0.59 \[ \frac{b x^{2} + 2 \, a x}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)/c^(3/2)

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Fricas [A]  time = 0.210852, size = 20, normalized size = 0.59 \[ \frac{b x^{2} + 2 \, a x}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)/c^(3/2)

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Sympy [A]  time = 9.37012, size = 134, normalized size = 3.94 \[ \begin{cases} \frac{x}{\left (\tilde{\infty } c\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{x}{\left (\frac{c}{a^{\frac{2}{3}}}\right )^{\frac{3}{2}}} & \text{for}\: b = 0 \\\frac{2 a^{2}}{\frac{2 a b c^{\frac{3}{2}}}{a + b x} + \frac{2 b^{2} c^{\frac{3}{2}} x}{a + b x}} + \frac{2 a b x}{\frac{2 a b c^{\frac{3}{2}}}{a + b x} + \frac{2 b^{2} c^{\frac{3}{2}} x}{a + b x}} + \frac{b^{2} x^{2}}{\frac{2 a b c^{\frac{3}{2}}}{a + b x} + \frac{2 b^{2} c^{\frac{3}{2}} x}{a + b x}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/(zoo*c)**(3/2), Eq(a, 0) & Eq(b, 0)), (x/(c/a**(2/3))**(3/2), Eq(b,
0)), (2*a**2/(2*a*b*c**(3/2)/(a + b*x) + 2*b**2*c**(3/2)*x/(a + b*x)) + 2*a*b*x
/(2*a*b*c**(3/2)/(a + b*x) + 2*b**2*c**(3/2)*x/(a + b*x)) + b**2*x**2/(2*a*b*c**
(3/2)/(a + b*x) + 2*b**2*c**(3/2)*x/(a + b*x)), True))

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GIAC/XCAS [A]  time = 0.213428, size = 20, normalized size = 0.59 \[ \frac{b x^{2} + 2 \, a x}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)/c^(3/2)