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

Optimal. Leaf size=27 \[ \frac{c \sqrt{c x^2} (a+b x)^3}{3 b x} \]

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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Rubi [A]  time = 0.0044034, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{c \sqrt{c x^2} (a+b x)^3}{3 b x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0052405, size = 26, normalized size = 0.96 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^3}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*x^2)^(3/2)*(a + b*x)^3)/(3*b*x^3)

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Maple [A]  time = 0.002, size = 31, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}{x}^{2}+3\,abx+3\,{a}^{2}}{3\,{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)^2/x^3,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43776, size = 69, normalized size = 2.56 \begin{align*} \frac{1}{3} \,{\left (b^{2} c x^{2} + 3 \, a b c x + 3 \, a^{2} 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)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.854937, size = 51, normalized size = 1.89 \begin{align*} \frac{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{2}} + \frac{a b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x} + \frac{b^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06025, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{3} \,{\left (\frac{{\left (b x + a\right )}^{3} \mathrm{sgn}\left (x\right )}{b} - \frac{a^{3} \mathrm{sgn}\left (x\right )}{b}\right )} c^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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