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

Optimal. Leaf size=38 \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0163118, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00729913, size = 23, normalized size = 0.61 \[ \frac{x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*(2*a + b*x))/(2*(c*x^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 20, normalized size = 0.5 \[{\frac{{x}^{4} \left ( bx+2\,a \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^4*(b*x+2*a)/(c*x^2)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.33841, size = 43, normalized size = 1.13 \[ \frac{b x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{a x^{2}}{\sqrt{c x^{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.206474, size = 26, normalized size = 0.68 \[ \frac{\sqrt{c x^{2}}{\left (b x + 2 \, a\right )}}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(c*x^2)*(b*x + 2*a)/c^2

_______________________________________________________________________________________

Sympy [A]  time = 2.27244, size = 34, normalized size = 0.89 \[ \frac{a x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{b x^{5}}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/(c**(3/2)*(x**2)**(3/2)) + b*x**5/(2*c**(3/2)*(x**2)**(3/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210118, size = 34, normalized size = 0.89 \[ \frac{\sqrt{c x^{2}}{\left (\frac{b x}{c} + \frac{2 \, a}{c}\right )}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(c*x^2)*(b*x/c + 2*a/c)/c