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

Optimal. Leaf size=77 \[ \frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}+\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0536256, antiderivative size = 77, 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 \[ \frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}+\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.203, size = 82, normalized size = 1.06 \[ - \frac{\sqrt{c x^{2}}}{2 a c x^{3}} + \frac{b \sqrt{c x^{2}}}{a^{2} c x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{a^{3} c x} - \frac{b^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{3} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)/(c*x**2)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0237383, size = 52, normalized size = 0.68 \[ \frac{c x \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 51, normalized size = 0.7 \[{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2}}{2\,x{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.36181, size = 74, normalized size = 0.96 \[ -\frac{b^{2} \log \left (b x + a\right )}{a^{3} \sqrt{c}} + \frac{b^{2} \log \left (x\right )}{a^{3} \sqrt{c}} + \frac{2 \, b \sqrt{c} x - a \sqrt{c}}{2 \, a^{2} c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.219372, size = 63, normalized size = 0.82 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, a^{3} c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{c x^{2}} \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.517077, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x