3.882 \(\int \frac{1}{x \sqrt{c x^2} (a+b x)} \, dx\)

Optimal. Leaf size=54 \[ -\frac{b x \log (x)}{a^2 \sqrt{c x^2}}+\frac{b x \log (a+b x)}{a^2 \sqrt{c x^2}}-\frac{1}{a \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0155541, 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, 44} \[ -\frac{b x \log (x)}{a^2 \sqrt{c x^2}}+\frac{b x \log (a+b x)}{a^2 \sqrt{c x^2}}-\frac{1}{a \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

-(1/(a*Sqrt[c*x^2])) - (b*x*Log[x])/(a^2*Sqrt[c*x^2]) + (b*x*Log[a + b*x])/(a^2*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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0108824, size = 36, normalized size = 0.67 \[ \frac{c x^2 (b x \log (a+b x)-a-b x \log (x))}{a^2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 0.6 \begin{align*} -{\frac{b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a}{{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02622, size = 50, normalized size = 0.93 \begin{align*} \frac{b \log \left (b x + a\right )}{a^{2} \sqrt{c}} - \frac{b \log \left (x\right )}{a^{2} \sqrt{c}} - \frac{1}{a \sqrt{c} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66856, size = 70, normalized size = 1.3 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (\frac{b x + a}{x}\right ) - a\right )}}{a^{2} c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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(1/x/(b*x+a)/(c*x**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.08413, size = 123, normalized size = 2.28 \begin{align*} -\sqrt{c}{\left (\frac{b \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{a^{2} c} - \frac{b \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{a^{2} c} - \frac{2}{{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} a \sqrt{c}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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