Optimal. Leaf size=26 \[ \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1593, 1584, 43} \[ \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 1584
Rule 1593
Rubi steps
\begin {align*} \int \frac {a x^2+b x^3}{c x^2+d x^3} \, dx &=\int \frac {x^2 (a+b x)}{c x^2+d x^3} \, dx\\ &=\int \frac {a+b x}{c+d x} \, dx\\ &=\int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx\\ &=\frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 25, normalized size = 0.96 \[ \frac {(a d-b c) \log (c+d x)}{d^2}+\frac {b x}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 25, normalized size = 0.96 \[ \frac {b d x - {\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.28, size = 27, normalized size = 1.04 \[ \frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 32, normalized size = 1.23 \[ \frac {a \ln \left (d x +c \right )}{d}-\frac {b c \ln \left (d x +c \right )}{d^{2}}+\frac {b x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.72, size = 26, normalized size = 1.00 \[ \frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.23, size = 25, normalized size = 0.96 \[ \frac {\ln \left (c+d\,x\right )\,\left (a\,d-b\,c\right )}{d^2}+\frac {b\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 20, normalized size = 0.77 \[ \frac {b x}{d} + \frac {\left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________