Optimal. Leaf size=107 \[ -\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac {e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 46}
\begin {gather*} \frac {e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac {e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3 (d+e x)} \, dx\\ &=\int \left (\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac {c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac {e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 83, normalized size = 0.78 \begin {gather*} -\frac {\frac {\left (c d^2-a e^2\right ) \left (-3 a e^2+c d (d-2 e x)\right )}{(a e+c d x)^2}-2 e^2 \log (a e+c d x)+2 e^2 \log (d+e x)}{2 \left (c d^2-a e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.81, size = 106, normalized size = 0.99
method | result | size |
default | \(\frac {e^{2} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}+\frac {1}{2 \left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right )^{2}}+\frac {e}{\left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )}-\frac {e^{2} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}\) | \(106\) |
risch | \(\frac {\frac {c d e x}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {3 e^{2} a -c \,d^{2}}{2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}}}{\left (c d x +a e \right )^{2}}+\frac {e^{2} \ln \left (-e x -d \right )}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}-\frac {e^{2} \ln \left (c d x +a e \right )}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}\) | \(199\) |
norman | \(\frac {\frac {c d \,e^{3} x^{3}}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {3 a d \,e^{3} x}{a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {3 d^{2} e^{2} c^{2} a -d^{4} c^{3}}{2 c^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {\left (3 c^{2} d^{2} a \,e^{6}+3 c^{3} d^{4} e^{4}\right ) x^{2}}{2 e^{2} d^{2} c^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}+\frac {e^{2} \ln \left (e x +d \right )}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}-\frac {e^{2} \ln \left (c d x +a e \right )}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}\) | \(318\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs.
\(2 (104) = 208\).
time = 0.29, size = 226, normalized size = 2.11 \begin {gather*} \frac {e^{2} \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac {e^{2} \log \left (x e + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac {2 \, c d x e - c d^{2} + 3 \, a e^{2}}{2 \, {\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs.
\(2 (104) = 208\).
time = 3.84, size = 275, normalized size = 2.57 \begin {gather*} \frac {2 \, c^{2} d^{3} x e - c^{2} d^{4} - 2 \, a c d x e^{3} + 4 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 2 \, {\left (c^{2} d^{2} x^{2} e^{2} + 2 \, a c d x e^{3} + a^{2} e^{4}\right )} \log \left (c d x + a e\right ) - 2 \, {\left (c^{2} d^{2} x^{2} e^{2} + 2 \, a c d x e^{3} + a^{2} e^{4}\right )} \log \left (x e + d\right )}{2 \, {\left (c^{5} d^{8} x^{2} + 2 \, a c^{4} d^{7} x e - 6 \, a^{2} c^{3} d^{5} x e^{3} + 6 \, a^{3} c^{2} d^{3} x e^{5} - 2 \, a^{4} c d x e^{7} - a^{5} e^{8} - {\left (a^{3} c^{2} d^{2} x^{2} - 3 \, a^{4} c d^{2}\right )} e^{6} + 3 \, {\left (a^{2} c^{3} d^{4} x^{2} - a^{3} c^{2} d^{4}\right )} e^{4} - {\left (3 \, a c^{4} d^{6} x^{2} - a^{2} c^{3} d^{6}\right )} e^{2}\right )}} \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. 457 vs.
\(2 (92) = 184\).
time = 0.66, size = 457, normalized size = 4.27 \begin {gather*} \frac {e^{2} \log {\left (x + \frac {- \frac {a^{4} e^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {4 a^{3} c d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {6 a^{2} c^{2} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {4 a c^{3} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + a e^{4} - \frac {c^{4} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + c d^{2} e^{2}}{2 c d e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {e^{2} \log {\left (x + \frac {\frac {a^{4} e^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {4 a^{3} c d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {6 a^{2} c^{2} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {4 a c^{3} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + a e^{4} + \frac {c^{4} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + c d^{2} e^{2}}{2 c d e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {3 a e^{2} - c d^{2} + 2 c d e x}{2 a^{4} e^{6} - 4 a^{3} c d^{2} e^{4} + 2 a^{2} c^{2} d^{4} e^{2} + x^{2} \cdot \left (2 a^{2} c^{2} d^{2} e^{4} - 4 a c^{3} d^{4} e^{2} + 2 c^{4} d^{6}\right ) + x \left (4 a^{3} c d e^{5} - 8 a^{2} c^{2} d^{3} e^{3} + 4 a c^{3} d^{5} e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.16, size = 183, normalized size = 1.71 \begin {gather*} \frac {c d e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}} - \frac {e^{3} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}} - \frac {c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 2 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{2 \, {\left (c d^{2} - a e^{2}\right )}^{3} {\left (c d x + a e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.69, size = 225, normalized size = 2.10 \begin {gather*} \frac {\frac {3\,a\,e^2-c\,d^2}{2\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {c\,d\,e\,x}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}}{a^2\,e^2+2\,a\,c\,d\,e\,x+c^2\,d^2\,x^2}-\frac {2\,e^2\,\mathrm {atanh}\left (\frac {a^3\,e^6-a^2\,c\,d^2\,e^4-a\,c^2\,d^4\,e^2+c^3\,d^6}{{\left (a\,e^2-c\,d^2\right )}^3}+\frac {2\,c\,d\,e\,x\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3}\right )}{{\left (a\,e^2-c\,d^2\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________