Optimal. Leaf size=81 \[ -\frac {a^2 e x}{c^4}-\frac {d x^2}{2 c^2}-\frac {e x^3}{3 c^2}-\frac {a^2 (c d+a e) \log (a-c x)}{2 c^5}-\frac {a^2 (c d-a e) \log (a+c x)}{2 c^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {815, 647, 31}
\begin {gather*} -\frac {a^2 (a e+c d) \log (a-c x)}{2 c^5}-\frac {a^2 (c d-a e) \log (a+c x)}{2 c^5}-\frac {a^2 e x}{c^4}-\frac {d x^2}{2 c^2}-\frac {e x^3}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 647
Rule 815
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)}{a^2-c^2 x^2} \, dx &=\int \left (-\frac {a^2 e}{c^4}-\frac {d x}{c^2}-\frac {e x^2}{c^2}+\frac {a^4 e+a^2 c^2 d x}{c^4 \left (a^2-c^2 x^2\right )}\right ) \, dx\\ &=-\frac {a^2 e x}{c^4}-\frac {d x^2}{2 c^2}-\frac {e x^3}{3 c^2}+\frac {\int \frac {a^4 e+a^2 c^2 d x}{a^2-c^2 x^2} \, dx}{c^4}\\ &=-\frac {a^2 e x}{c^4}-\frac {d x^2}{2 c^2}-\frac {e x^3}{3 c^2}+\frac {\left (a^2 (c d-a e)\right ) \int \frac {1}{-a c-c^2 x} \, dx}{2 c^3}+\frac {\left (a^2 (c d+a e)\right ) \int \frac {1}{a c-c^2 x} \, dx}{2 c^3}\\ &=-\frac {a^2 e x}{c^4}-\frac {d x^2}{2 c^2}-\frac {e x^3}{3 c^2}-\frac {a^2 (c d+a e) \log (a-c x)}{2 c^5}-\frac {a^2 (c d-a e) \log (a+c x)}{2 c^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 72, normalized size = 0.89 \begin {gather*} -\frac {a^2 e x}{c^4}-\frac {d x^2}{2 c^2}-\frac {e x^3}{3 c^2}+\frac {a^3 e \tanh ^{-1}\left (\frac {c x}{a}\right )}{c^5}-\frac {a^2 d \log \left (a^2-c^2 x^2\right )}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.58, size = 76, normalized size = 0.94
method | result | size |
norman | \(-\frac {d \,x^{2}}{2 c^{2}}-\frac {e \,x^{3}}{3 c^{2}}-\frac {a^{2} e x}{c^{4}}+\frac {a^{2} \left (a e -c d \right ) \ln \left (c x +a \right )}{2 c^{5}}-\frac {a^{2} \left (a e +c d \right ) \ln \left (-c x +a \right )}{2 c^{5}}\) | \(74\) |
default | \(-\frac {\frac {1}{3} c^{2} e \,x^{3}+\frac {1}{2} c^{2} d \,x^{2}+a^{2} e x}{c^{4}}-\frac {a^{2} \left (a e +c d \right ) \ln \left (-c x +a \right )}{2 c^{5}}+\frac {a^{2} \left (a e -c d \right ) \ln \left (c x +a \right )}{2 c^{5}}\) | \(76\) |
risch | \(-\frac {d \,x^{2}}{2 c^{2}}-\frac {e \,x^{3}}{3 c^{2}}-\frac {a^{2} e x}{c^{4}}-\frac {a^{3} \ln \left (c x -a \right ) e}{2 c^{5}}-\frac {a^{2} \ln \left (c x -a \right ) d}{2 c^{4}}+\frac {a^{3} \ln \left (-c x -a \right ) e}{2 c^{5}}-\frac {a^{2} \ln \left (-c x -a \right ) d}{2 c^{4}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 85, normalized size = 1.05 \begin {gather*} -\frac {2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} + 6 \, a^{2} x e}{6 \, c^{4}} - \frac {{\left (a^{2} c d - a^{3} e\right )} \log \left (c x + a\right )}{2 \, c^{5}} - \frac {{\left (a^{2} c d + a^{3} e\right )} \log \left (c x - a\right )}{2 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.07, size = 79, normalized size = 0.98 \begin {gather*} -\frac {3 \, c^{3} d x^{2} + 2 \, {\left (c^{3} x^{3} + 3 \, a^{2} c x\right )} e + 3 \, {\left (a^{2} c d - a^{3} e\right )} \log \left (c x + a\right ) + 3 \, {\left (a^{2} c d + a^{3} e\right )} \log \left (c x - a\right )}{6 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.20, size = 110, normalized size = 1.36 \begin {gather*} - \frac {a^{2} e x}{c^{4}} + \frac {a^{2} \left (a e - c d\right ) \log {\left (x + \frac {a^{2} d + \frac {a^{2} \left (a e - c d\right )}{c}}{a^{2} e} \right )}}{2 c^{5}} - \frac {a^{2} \left (a e + c d\right ) \log {\left (x + \frac {a^{2} d - \frac {a^{2} \left (a e + c d\right )}{c}}{a^{2} e} \right )}}{2 c^{5}} - \frac {d x^{2}}{2 c^{2}} - \frac {e x^{3}}{3 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.06, size = 90, normalized size = 1.11 \begin {gather*} -\frac {{\left (a^{2} c d - a^{3} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{5}} - \frac {{\left (a^{2} c d + a^{3} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{5}} - \frac {2 \, c^{4} x^{3} e + 3 \, c^{4} d x^{2} + 6 \, a^{2} c^{2} x e}{6 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.09, size = 77, normalized size = 0.95 \begin {gather*} \frac {\ln \left (a+c\,x\right )\,\left (a^3\,e-a^2\,c\,d\right )}{2\,c^5}-\frac {\ln \left (a-c\,x\right )\,\left (e\,a^3+c\,d\,a^2\right )}{2\,c^5}-\frac {d\,x^2}{2\,c^2}-\frac {e\,x^3}{3\,c^2}-\frac {a^2\,e\,x}{c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________