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

Optimal. Leaf size=78 \[ -\frac{b x}{a^2 \sqrt{c x^2} (a+b x)}-\frac{2 b x \log (x)}{a^3 \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 \sqrt{c x^2}}-\frac{1}{a^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0224452, antiderivative size = 78, 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} \[ -\frac{b x}{a^2 \sqrt{c x^2} (a+b x)}-\frac{2 b x \log (x)}{a^3 \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 \sqrt{c x^2}}-\frac{1}{a^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

-(1/(a^2*Sqrt[c*x^2])) - (b*x)/(a^2*Sqrt[c*x^2]*(a + b*x)) - (2*b*x*Log[x])/(a^3*Sqrt[c*x^2]) + (2*b*x*Log[a +
b*x])/(a^3*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)^2} \, dx &=\frac{x \int \frac{1}{x^2 (a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{1}{a^2 \sqrt{c x^2}}-\frac{b x}{a^2 \sqrt{c x^2} (a+b x)}-\frac{2 b x \log (x)}{a^3 \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0247229, size = 60, normalized size = 0.77 \[ \frac{c x^2 (-a (a+2 b x)-2 b x \log (x) (a+b x)+2 b x (a+b x) \log (a+b x))}{a^3 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 71, normalized size = 0.9 \begin{align*} -{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,ab\ln \left ( x \right ) x-2\,\ln \left ( bx+a \right ) xab+2\,abx+{a}^{2}}{{a}^{3} \left ( bx+a \right ) }{\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)^2/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.06927, size = 77, normalized size = 0.99 \begin{align*} -\frac{2 \, b x + a}{a^{2} b \sqrt{c} x^{2} + a^{3} \sqrt{c} x} + \frac{2 \, b \log \left (b x + a\right )}{a^{3} \sqrt{c}} - \frac{2 \, b \log \left (x\right )}{a^{3} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3451, size = 128, normalized size = 1.64 \begin{align*} -\frac{{\left (2 \, a b x + a^{2} - 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (\frac{b x + a}{x}\right )\right )} \sqrt{c x^{2}}}{a^{3} b c x^{3} + a^{4} c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*b*x + a^2 - 2*(b^2*x^2 + a*b*x)*log((b*x + a)/x))*sqrt(c*x^2)/(a^3*b*c*x^3 + a^4*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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: RuntimeError