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

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

[Out]

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

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Rubi [A]  time = 0.0043358, antiderivative size = 29, 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} \[ a c \sqrt{c x^2}+\frac{1}{2} b c x \sqrt{c x^2} \]

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*}

Mathematica [A]  time = 0.0028447, size = 21, normalized size = 0.72 \[ \frac{1}{2} c \sqrt{c x^2} (2 a+b x) \]

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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Maple [A]  time = 0.002, size = 20, normalized size = 0.7 \begin{align*}{\frac{bx+2\,a}{2\,{x}^{2}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [A]  time = 1.49662, size = 45, normalized size = 1.55 \begin{align*} \frac{1}{2} \,{\left (b c x + 2 \, a c\right )} \sqrt{c x^{2}} \end{align*}

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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Sympy [A]  time = 0.695981, size = 32, normalized size = 1.1 \begin{align*} \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{align*}

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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Giac [A]  time = 1.06579, size = 23, normalized size = 0.79 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) \end{align*}

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)