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

Optimal. Leaf size=87 \[ -\frac{b \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{2 b \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b \sqrt{c x^2} \log (a+b x)}{a^3 x}-\frac{\sqrt{c x^2}}{a^2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0270424, antiderivative size = 87, 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} \[ -\frac{b \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{2 b \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b \sqrt{c x^2} \log (a+b x)}{a^3 x}-\frac{\sqrt{c x^2}}{a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^3*(a + b*x)^2),x]

[Out]

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

Mathematica [A]  time = 0.0268839, size = 57, normalized size = 0.66 \[ -\frac{c (a (a+2 b x)+2 b x \log (x) (a+b x)-2 b x (a+b x) \log (a+b x))}{a^3 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*(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*Sqrt[c*x^2]*(a + b*x)))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 74, 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}}{{x}^{2}{a}^{3} \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02047, size = 78, normalized size = 0.9 \begin{align*} -\frac{2 \, b \sqrt{c} x + a \sqrt{c}}{a^{2} b x^{2} + a^{3} x} + \frac{2 \, b \sqrt{c} \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b \sqrt{c} \log \left (x\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.32187, size = 123, normalized size = 1.41 \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 x^{3} + a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a)^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*x^3 + a^4*x^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2}}}{x^{3} \left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError