Optimal. Leaf size=75 \[ -\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 46}
\begin {gather*} -\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 640
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 (d+e x)} \, dx\\ &=\int \left (\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {e^2}{\left (c d^2-a e^2\right )^2 (d+e x)}\right ) \, dx\\ &=-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 74, normalized size = 0.99 \begin {gather*} \frac {1}{\left (-c d^2+a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (-c d^2+a e^2\right )^2}+\frac {e \log (d+e x)}{\left (-c d^2+a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.68, size = 75, normalized size = 1.00
method | result | size |
default | \(\frac {e \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}+\frac {1}{\left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right )}-\frac {e \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\) | \(75\) |
risch | \(\frac {1}{\left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right )}-\frac {e \ln \left (c d x +a e \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {e \ln \left (-e x -d \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}\) | \(104\) |
norman | \(\frac {\frac {d}{e^{2} a -c \,d^{2}}+\frac {e x}{e^{2} a -c \,d^{2}}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {e \ln \left (e x +d \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}-\frac {e \ln \left (c d x +a e \right )}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 112, normalized size = 1.49 \begin {gather*} -\frac {e \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {e \log \left (x e + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {1}{a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.06, size = 114, normalized size = 1.52 \begin {gather*} -\frac {c d^{2} - a e^{2} + {\left (c d x e + a e^{2}\right )} \log \left (c d x + a e\right ) - {\left (c d x e + a e^{2}\right )} \log \left (x e + d\right )}{c^{3} d^{5} x - 2 \, a c^{2} d^{3} x e^{2} + a c^{2} d^{4} e + a^{2} c d x e^{4} - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 287 vs.
\(2 (63) = 126\).
time = 0.40, size = 287, normalized size = 3.83 \begin {gather*} \frac {e \log {\left (x + \frac {- \frac {a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} + \frac {c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {e \log {\left (x + \frac {\frac {a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} - \frac {c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {1}{a^{2} e^{3} - a c d^{2} e + x \left (a c d e^{2} - c^{2} d^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.05, size = 111, normalized size = 1.48 \begin {gather*} -\frac {c d e \log \left ({\left | c d x + a e \right |}\right )}{c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}} + \frac {e^{2} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}} - \frac {1}{{\left (c d^{2} - a e^{2}\right )} {\left (c d x + a e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.63, size = 96, normalized size = 1.28 \begin {gather*} \frac {1}{\left (a\,e+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )}-\frac {2\,e\,\mathrm {atanh}\left (\frac {a^2\,e^4-c^2\,d^4}{{\left (a\,e^2-c\,d^2\right )}^2}+\frac {2\,c\,d\,e\,x}{a\,e^2-c\,d^2}\right )}{{\left (a\,e^2-c\,d^2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________