3.819 \(\int \frac{(c x^2)^{3/2} (a+b x)^2}{x^4} \, dx\)

Optimal. Leaf size=52 \[ \frac{a^2 c \sqrt{c x^2} \log (x)}{x}+2 a b c \sqrt{c x^2}+\frac{1}{2} b^2 c x \sqrt{c x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0097471, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ \frac{a^2 c \sqrt{c x^2} \log (x)}{x}+2 a b c \sqrt{c x^2}+\frac{1}{2} b^2 c x \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(3/2)*(a + b*x)^2)/x^4,x]

[Out]

2*a*b*c*Sqrt[c*x^2] + (b^2*c*x*Sqrt[c*x^2])/2 + (a^2*c*Sqrt[c*x^2]*Log[x])/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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)^2}{x^4} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{(a+b x)^2}{x} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx}{x}\\ &=2 a b c \sqrt{c x^2}+\frac{1}{2} b^2 c x \sqrt{c x^2}+\frac{a^2 c \sqrt{c x^2} \log (x)}{x}\\ \end{align*}

Mathematica [A]  time = 0.0081543, size = 34, normalized size = 0.65 \[ \frac{\left (c x^2\right )^{3/2} \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*x^2)^(3/2)*(a + b*x)^2)/x^4,x]

[Out]

((c*x^2)^(3/2)*(b*x*(4*a + b*x) + 2*a^2*Log[x]))/(2*x^3)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 33, normalized size = 0.6 \begin{align*}{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx}{2\,{x}^{3}} \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^4,x)

[Out]

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

________________________________________________________________________________________

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)^2/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.50387, size = 81, normalized size = 1.56 \begin{align*} \frac{{\left (b^{2} c x^{2} + 4 \, a b c x + 2 \, a^{2} c \log \left (x\right )\right )} \sqrt{c x^{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)^2/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{2}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)*(b*x+a)**2/x**4,x)

[Out]

Integral((c*x**2)**(3/2)*(a + b*x)**2/x**4, x)

________________________________________________________________________________________

Giac [A]  time = 1.05001, size = 43, normalized size = 0.83 \begin{align*} \frac{1}{2} \,{\left (b^{2} x^{2} \mathrm{sgn}\left (x\right ) + 4 \, a b x \mathrm{sgn}\left (x\right ) + 2 \, a^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right )\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^4,x, algorithm="giac")

[Out]

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