Optimal. Leaf size=84 \[ \frac {b^2 \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b \sqrt {c x^2}}{a^2 x^2}-\frac {\sqrt {c x^2}}{2 a x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 44} \[ \frac {b^2 \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b \sqrt {c x^2}}{a^2 x^2}-\frac {\sqrt {c x^2}}{2 a x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^4 (a+b x)} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x^3 (a+b x)} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {\sqrt {c x^2}}{2 a x^3}+\frac {b \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 \sqrt {c x^2} \log (a+b x)}{a^3 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 53, normalized size = 0.63 \[ \frac {\sqrt {c x^2} \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 44, normalized size = 0.52 \[ \frac {{\left (2 \, b^{2} x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 51, normalized size = 0.61 \[ \frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} x^{2} \ln \relax (x )-2 b^{2} x^{2} \ln \left (b x +a \right )+2 a b x -a^{2}\right )}{2 a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 52, normalized size = 0.62 \[ -\frac {b^{2} \sqrt {c} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \sqrt {c} \log \relax (x)}{a^{3}} + \frac {2 \, b \sqrt {c} x - a \sqrt {c}}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,x^2}}{x^4\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2}}}{x^{4} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________