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

Optimal. Leaf size=103 \[ \frac{b^2 x}{a^3 \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 \sqrt{c x^2}}+\frac{2 b}{a^3 \sqrt{c x^2}}-\frac{1}{2 a^2 x \sqrt{c x^2}} \]

[Out]

(2*b)/(a^3*Sqrt[c*x^2]) - 1/(2*a^2*x*Sqrt[c*x^2]) + (b^2*x)/(a^3*Sqrt[c*x^2]*(a + b*x)) + (3*b^2*x*Log[x])/(a^
4*Sqrt[c*x^2]) - (3*b^2*x*Log[a + b*x])/(a^4*Sqrt[c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0316542, antiderivative size = 103, 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} \[ \frac{b^2 x}{a^3 \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 \sqrt{c x^2}}+\frac{2 b}{a^3 \sqrt{c x^2}}-\frac{1}{2 a^2 x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b)/(a^3*Sqrt[c*x^2]) - 1/(2*a^2*x*Sqrt[c*x^2]) + (b^2*x)/(a^3*Sqrt[c*x^2]*(a + b*x)) + (3*b^2*x*Log[x])/(a^
4*Sqrt[c*x^2]) - (3*b^2*x*Log[a + b*x])/(a^4*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^2 \sqrt{c x^2} (a+b x)^2} \, dx &=\frac{x \int \frac{1}{x^3 (a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{2 b}{a^3 \sqrt{c x^2}}-\frac{1}{2 a^2 x \sqrt{c x^2}}+\frac{b^2 x}{a^3 \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0170383, size = 81, normalized size = 0.79 \[ \frac{c x \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(a*(-a^2 + 3*a*b*x + 6*b^2*x^2) + 6*b^2*x^2*(a + b*x)*Log[x] - 6*b^2*x^2*(a + b*x)*Log[a + b*x]))/(2*a^4*
(c*x^2)^(3/2)*(a + b*x))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 95, normalized size = 0.9 \begin{align*}{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,\ln \left ( x \right ){x}^{2}a{b}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx-{a}^{3}}{2\,x{a}^{4} \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^2/(b*x+a)^2/(c*x^2)^(1/2),x)

[Out]

1/2/x*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*ln(x)*x^2*a*b^2-6*ln(b*x+a)*x^2*a*b^2+6*a*b^2*x^2+3*a^2*b*x-a^3)/
(c*x^2)^(1/2)/a^4/(b*x+a)

________________________________________________________________________________________

Maxima [A]  time = 1.05652, size = 103, normalized size = 1. \begin{align*} \frac{6 \, b^{2} x^{2} + 3 \, a b x - a^{2}}{2 \,{\left (a^{3} b \sqrt{c} x^{3} + a^{4} \sqrt{c} x^{2}\right )}} - \frac{3 \, b^{2} \log \left (b x + a\right )}{a^{4} \sqrt{c}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*b^2*x^2 + 3*a*b*x - a^2)/(a^3*b*sqrt(c)*x^3 + a^4*sqrt(c)*x^2) - 3*b^2*log(b*x + a)/(a^4*sqrt(c)) + 3*b
^2*log(x)/(a^4*sqrt(c))

________________________________________________________________________________________

Fricas [A]  time = 1.3217, size = 159, normalized size = 1.54 \begin{align*} \frac{{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac{x}{b x + a}\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (a^{4} b c x^{4} + a^{5} c x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*a*b^2*x^2 + 3*a^2*b*x - a^3 + 6*(b^3*x^3 + a*b^2*x^2)*log(x/(b*x + a)))*sqrt(c*x^2)/(a^4*b*c*x^4 + a^5*
c*x^3)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError