3.8.88 \(\int \frac {x^3 (a+b x)}{(c x^2)^{3/2}} \, dx\) [788]

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

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Rubi [A]
time = 0.00, 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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Maple [A]
time = 0.02, size = 20, normalized size = 0.53

method	result	size
gosper	\(\frac {x^{4} \left (b x +2 a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(20\)
default	\(\frac {x^{4} \left (b x +2 a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(20\)
trager	\(\frac {\left (b x +2 a +b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{2 c^{2} x}\)	\(27\)
risch	\(\frac {a \,x^{2}}{c \sqrt {c \,x^{2}}}+\frac {b \,x^{3}}{2 c \sqrt {c \,x^{2}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)/(c*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, 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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Fricas [A]
time = 1.19, 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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Sympy [A]
time = 0.23, size = 27, normalized size = 0.71 \begin {gather*} \frac {a x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b x^{5}}{2 \left (c 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*x**2)**(3/2) + b*x**5/(2*(c*x**2)**(3/2))

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Giac [A]
time = 2.42, size = 19, normalized size = 0.50 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \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*(b*x^2 + 2*a*x)/(c^(3/2)*sgn(x))

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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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