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

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])

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Rubi [A]  time = 0.0056526, 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, Rules used = {15} \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]

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])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int (a+b x) \, dx}{c \sqrt{c x^2}}\\ &=\frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0040282, 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))

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*}{\frac{{x}^{4} \left ( bx+2\,a \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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)

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Maxima [A]  time = 1.0237, size = 43, normalized size = 1.13 \begin{align*} \frac{b x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{a x^{2}}{\sqrt{c x^{2}} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)/(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)

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Fricas [A]  time = 1.54675, size = 45, normalized size = 1.18 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + 2 \, a\right )}}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2)*(b*x + 2*a)/c^2

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Sympy [A]  time = 0.597934, size = 34, normalized size = 0.89 \begin{align*} \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}}} \end{align*}

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))

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Giac [A]  time = 1.07361, size = 34, normalized size = 0.89 \begin{align*} \frac{\sqrt{c x^{2}}{\left (\frac{b x}{c} + \frac{2 \, a}{c}\right )}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2)*(b*x/c + 2*a/c)/c