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

Optimal. Leaf size=26 \[ -\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3} \]

[Out]

-(Sqrt[c*x^2]*(a + b*x)^2)/(2*a*x^3)

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Rubi [A]  time = 0.0040421, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 37} \[ -\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x^4,x]

[Out]

-(Sqrt[c*x^2]*(a + b*x)^2)/(2*a*x^3)

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0042912, size = 22, normalized size = 0.85 \[ -\frac{\sqrt{c x^2} (a+2 b x)}{2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[c*x^2]*(a + 2*b*x))/(2*x^3)

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Maple [A]  time = 0.004, size = 19, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+a}{2\,{x}^{3}}\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.45878, size = 46, normalized size = 1.77 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/x^3

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Sympy [A]  time = 0.482106, size = 36, normalized size = 1.38 \begin{align*} - \frac{a \sqrt{c} \sqrt{x^{2}}}{2 x^{3}} - \frac{b \sqrt{c} \sqrt{x^{2}}}{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*sqrt(c)*sqrt(x**2)/(2*x**3) - b*sqrt(c)*sqrt(x**2)/x**2

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Giac [A]  time = 1.06677, size = 26, normalized size = 1. \begin{align*} -\frac{{\left (2 \, b x \mathrm{sgn}\left (x\right ) + a \mathrm{sgn}\left (x\right )\right )} \sqrt{c}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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