Optimal. Leaf size=39 \[ \frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 39, 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, 43} \[ \frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {c x^2} (a+b x)} \, dx &=\frac {x \int \frac {x}{a+b x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 0.69 \[ \frac {x (b x-a \log (a+b x))}{b^2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 30, normalized size = 0.77 \[ \frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.19, size = 51, normalized size = 1.31 \[ \frac {a \log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b^{2} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 27, normalized size = 0.69 \[ -\frac {\left (a \ln \left (b x +a \right )-b x \right ) x}{\sqrt {c \,x^{2}}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.47, size = 64, normalized size = 1.64 \[ -\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} \sqrt {c}} - \frac {a \log \left (b x\right )}{b^{2} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________