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

Optimal. Leaf size=32 \[ \frac{b \sqrt{c x^2} \log (x)}{x}-\frac{a \sqrt{c x^2}}{x^2} \]

[Out]

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

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Rubi [A]  time = 0.0067455, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{b \sqrt{c x^2} \log (x)}{x}-\frac{a \sqrt{c x^2}}{x^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0050649, size = 20, normalized size = 0.62 \[ \frac{c (b x \log (x)-a)}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-a + b*x*Log[x]))/Sqrt[c*x^2]

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Maple [A]  time = 0.005, size = 21, normalized size = 0.7 \begin{align*}{\frac{b\ln \left ( x \right ) x-a}{{x}^{2}}\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^3,x)

[Out]

(c*x^2)^(1/2)*(b*ln(x)*x-a)/x^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((b*x+a)*(c*x^2)^(1/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.52064, size = 46, normalized size = 1.44 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{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^3,x, algorithm="fricas")

[Out]

sqrt(c*x^2)*(b*x*log(x) - a)/x^2

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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 )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05994, size = 27, normalized size = 0.84 \begin{align*}{\left (b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) - \frac{a \mathrm{sgn}\left (x\right )}{x}\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^3,x, algorithm="giac")

[Out]

(b*log(abs(x))*sgn(x) - a*sgn(x)/x)*sqrt(c)