Optimal. Leaf size=64 \[ -\frac {b c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {c \sqrt {c x^2}}{a x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 64, 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} \[ -\frac {b c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {c \sqrt {c x^2}}{a x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^2 (a+b x)} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {c \sqrt {c x^2}}{a x^2}-\frac {b c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c \sqrt {c x^2} \log (a+b x)}{a^2 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 34, normalized size = 0.53 \[ -\frac {c^2 (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 33, normalized size = 0.52 \[ \frac {{\left (b c x \log \left (\frac {b x + a}{x}\right ) - a c\right )} \sqrt {c x^{2}}}{a^{2} x^{2}} \]
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 = 33, normalized size = 0.52 \[ -\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \right )}{a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 37, normalized size = 0.58 \[ \frac {b c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{2}} - \frac {b c^{\frac {3}{2}} \log \relax (x)}{a^{2}} - \frac {c^{\frac {3}{2}}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x^2\right )}^{3/2}}{x^5\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{5} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________