Optimal. Leaf size=59 \[ -\frac {(a e+c d) \log (a-c x)}{2 a^3}+\frac {(c d-a e) \log (a+c x)}{2 a^3}-\frac {d}{a^2 x}+\frac {e \log (x)}{a^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {801} \begin {gather*} -\frac {(a e+c d) \log (a-c x)}{2 a^3}+\frac {(c d-a e) \log (a+c x)}{2 a^3}-\frac {d}{a^2 x}+\frac {e \log (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 801
Rubi steps
\begin {align*} \int \frac {d+e x}{x^2 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac {d}{a^2 x^2}+\frac {e}{a^2 x}+\frac {c (c d+a e)}{2 a^3 (a-c x)}-\frac {c (-c d+a e)}{2 a^3 (a+c x)}\right ) \, dx\\ &=-\frac {d}{a^2 x}+\frac {e \log (x)}{a^2}-\frac {(c d+a e) \log (a-c x)}{2 a^3}+\frac {(c d-a e) \log (a+c x)}{2 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 51, normalized size = 0.86 \begin {gather*} \frac {c d \tanh ^{-1}\left (\frac {c x}{a}\right )}{a^3}-\frac {e \log \left (a^2-c^2 x^2\right )}{2 a^2}-\frac {d}{a^2 x}+\frac {e \log (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x^2 \left (a^2-c^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 54, normalized size = 0.92 \begin {gather*} \frac {2 \, a e x \log \relax (x) + {\left (c d - a e\right )} x \log \left (c x + a\right ) - {\left (c d + a e\right )} x \log \left (c x - a\right ) - 2 \, a d}{2 \, a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 74, normalized size = 1.25 \begin {gather*} \frac {e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {d}{a^{2} x} + \frac {{\left (c^{2} d - a c e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{3} c} - \frac {{\left (c^{2} d + a c e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 72, normalized size = 1.22 \begin {gather*} \frac {e \ln \relax (x )}{a^{2}}-\frac {e \ln \left (c x -a \right )}{2 a^{2}}-\frac {e \ln \left (c x +a \right )}{2 a^{2}}-\frac {c d \ln \left (c x -a \right )}{2 a^{3}}+\frac {c d \ln \left (c x +a \right )}{2 a^{3}}-\frac {d}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 56, normalized size = 0.95 \begin {gather*} \frac {e \log \relax (x)}{a^{2}} + \frac {{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{3}} - \frac {{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{3}} - \frac {d}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 55, normalized size = 0.93 \begin {gather*} \frac {e\,\ln \relax (x)}{a^2}-\frac {\ln \left (a-c\,x\right )\,\left (a\,e+c\,d\right )}{2\,a^3}-\frac {d}{a^2\,x}-\frac {\ln \left (a+c\,x\right )\,\left (a\,e-c\,d\right )}{2\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.70, size = 221, normalized size = 3.75 \begin {gather*} - \frac {d}{a^{2} x} + \frac {e \log {\relax (x )}}{a^{2}} - \frac {\left (a e - c d\right ) \log {\left (x + \frac {6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e - c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e - c d\right )^{2} + a c^{2} d^{2} \left (a e - c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} - \frac {\left (a e + c d\right ) \log {\left (x + \frac {6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e + c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e + c d\right )^{2} + a c^{2} d^{2} \left (a e + c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________