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

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

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 0.61 \begin {gather*} \frac {x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 23, normalized size = 0.61 \begin {gather*} \frac {x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [A]  time = 1.05, size = 19, normalized size = 0.50 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + 2 \, a\right )}}{2 \, c^{2}} \end {gather*}

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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giac [A]  time = 1.01, size = 25, normalized size = 0.66 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (\frac {b x}{c} + \frac {2 \, a}{c}\right )}}{2 \, c} \end {gather*}

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

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maple [A]  time = 0.00, size = 20, normalized size = 0.53 \begin {gather*} \frac {\left (b x +2 a \right ) x^{4}}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}

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 = 1.31, size = 32, normalized size = 0.84 \begin {gather*} \frac {b x^{3}}{2 \, \sqrt {c x^{2}} c} + \frac {a x^{2}}{\sqrt {c x^{2}} c} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^3\,\left (a+b\,x\right )}{{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*x))/(c*x^2)^(3/2),x)

[Out]

int((x^3*(a + b*x))/(c*x^2)^(3/2), x)

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sympy [A]  time = 0.64, size = 34, normalized size = 0.89 \begin {gather*} \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 {gather*}

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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