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

Optimal. Leaf size=20 \[ \frac{x \log (a+b x)}{b \sqrt{c x^2}} \]

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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Rubi [A]  time = 0.0037139, antiderivative size = 20, 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, 31} \[ \frac{x \log (a+b x)}{b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0021436, size = 20, normalized size = 1. \[ \frac{x \log (a+b x)}{b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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Maple [A]  time = 0.003, size = 19, normalized size = 1. \begin{align*}{\frac{x\ln \left ( bx+a \right ) }{b}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81381, size = 46, normalized size = 2.3 \begin{align*} \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{b c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08111, size = 47, normalized size = 2.35 \begin{align*} -\frac{\log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-(sqrt(c)*x - sqrt(c*x^2))*b - 2*a*sqrt(c)))/(b*sqrt(c))