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

Optimal. Leaf size=29 \[ a c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 29, 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*} a c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(3/2)*(a + b*x))/x^3,x]

[Out]

a*c*Sqrt[c*x^2] + (b*c*x*Sqrt[c*x^2])/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 {\left (c x^2\right )^{3/2} (a+b x)}{x^3} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int (a+b x) \, dx}{x}\\ &=a c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.72 \begin {gather*} \frac {1}{2} c \sqrt {c x^2} (2 a+b x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c*x^2]*(2*a + b*x))/2

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IntegrateAlgebraic [A]  time = 0.02, size = 23, normalized size = 0.79 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (2 a+b x)}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*x^2)^(3/2)*(2*a + b*x))/(2*x^2)

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fricas [A]  time = 1.04, size = 18, normalized size = 0.62 \begin {gather*} \frac {1}{2} \, {\left (b c x + 2 \, a c\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*c*x + 2*a*c)*sqrt(c*x^2)

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giac [A]  time = 0.99, size = 17, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c^(3/2)*sgn(x)

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maple [A]  time = 0.00, size = 20, normalized size = 0.69 \begin {gather*} \frac {\left (b x +2 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.22, size = 14, normalized size = 0.48 \begin {gather*} \frac {c^{3/2}\,\relax |x|\,\left (2\,a+b\,x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(3/2)*(a + b*x))/x^3,x)

[Out]

(c^(3/2)*abs(x)*(2*a + b*x))/2

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sympy [A]  time = 0.74, size = 32, normalized size = 1.10 \begin {gather*} \frac {a c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{x^{2}} + \frac {b c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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