3.811 \(\int \frac{\sqrt{c x^2} (a+b x)^2}{x^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0115592, antiderivative size = 54, 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} \[ -\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0092095, size = 36, normalized size = 0.67 \[ \frac{\sqrt{c x^2} \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 34, normalized size = 0.6 \begin{align*}{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2}}{2\,{x}^{3}}\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((b*x+a)^2*(c*x^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.60008, size = 76, normalized size = 1.41 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

1/2*(2*b^2*x^2*log(x) - 4*a*b*x - a^2)*sqrt(c*x^2)/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(c*x**2)**(1/2)/x**4,x)

[Out]

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

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Giac [A]  time = 1.06497, size = 47, normalized size = 0.87 \begin{align*} \frac{1}{2} \,{\left (2 \, b^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) - \frac{4 \, a b x \mathrm{sgn}\left (x\right ) + a^{2} \mathrm{sgn}\left (x\right )}{x^{2}}\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

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