Optimal. Leaf size=94 \[ \frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (3 c d+4 a e) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {833, 815, 647,
31} \begin {gather*} \frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 647
Rule 815
Rule 833
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \frac {x^2 \left (3 a^2 d+4 a^2 e x\right )}{a^2-c^2 x^2} \, dx}{2 a^2 c^2}\\ &=\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \left (-\frac {3 a^2 d}{c^2}-\frac {4 a^2 e x}{c^2}+\frac {3 a^4 d+4 a^4 e x}{c^2 \left (a^2-c^2 x^2\right )}\right ) \, dx}{2 a^2 c^2}\\ &=\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \frac {3 a^4 d+4 a^4 e x}{a^2-c^2 x^2} \, dx}{2 a^2 c^4}\\ &=\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {(a (3 c d-4 a e)) \int \frac {1}{-a c-c^2 x} \, dx}{4 c^4}-\frac {(a (3 c d+4 a e)) \int \frac {1}{a c-c^2 x} \, dx}{4 c^4}\\ &=\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (3 c d+4 a e) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 84, normalized size = 0.89 \begin {gather*} \frac {2 c^2 d x+c^2 e x^2+\frac {a^4 e+a^2 c^2 d x}{a^2-c^2 x^2}-3 a c d \tanh ^{-1}\left (\frac {c x}{a}\right )+2 a^2 e \log \left (a^2-c^2 x^2\right )}{2 c^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.55, size = 105, normalized size = 1.12
method | result | size |
norman | \(\frac {\frac {a^{4} e}{c^{6}}-\frac {d \,x^{3}}{c^{2}}-\frac {e \,x^{4}}{2 c^{2}}+\frac {3 a^{2} d x}{2 c^{4}}}{-c^{2} x^{2}+a^{2}}+\frac {a \left (4 a e -3 c d \right ) \ln \left (c x +a \right )}{4 c^{6}}+\frac {a \left (4 a e +3 c d \right ) \ln \left (-c x +a \right )}{4 c^{6}}\) | \(97\) |
default | \(\frac {\frac {1}{2} e \,x^{2}+d x}{c^{4}}+\frac {a \left (4 a e +3 c d \right ) \ln \left (-c x +a \right )}{4 c^{6}}+\frac {a^{2} \left (a e +c d \right )}{4 c^{6} \left (-c x +a \right )}+\frac {a \left (4 a e -3 c d \right ) \ln \left (c x +a \right )}{4 c^{6}}+\frac {a^{2} \left (a e -c d \right )}{4 c^{6} \left (c x +a \right )}\) | \(105\) |
risch | \(\frac {e \,x^{2}}{2 c^{4}}+\frac {d x}{c^{4}}+\frac {\frac {a^{2} d x}{2}+\frac {a^{4} e}{2 c^{2}}}{c^{4} \left (-c^{2} x^{2}+a^{2}\right )}+\frac {a^{2} \ln \left (-c x -a \right ) e}{c^{6}}-\frac {3 a \ln \left (-c x -a \right ) d}{4 c^{5}}+\frac {a^{2} \ln \left (c x -a \right ) e}{c^{6}}+\frac {3 a \ln \left (c x -a \right ) d}{4 c^{5}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 103, normalized size = 1.10 \begin {gather*} -\frac {a^{2} c^{2} d x + a^{4} e}{2 \, {\left (c^{8} x^{2} - a^{2} c^{6}\right )}} + \frac {x^{2} e + 2 \, d x}{2 \, c^{4}} - \frac {{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left (c x + a\right )}{4 \, c^{6}} + \frac {{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left (c x - a\right )}{4 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 5.16, size = 161, normalized size = 1.71 \begin {gather*} \frac {4 \, c^{4} d x^{3} - 6 \, a^{2} c^{2} d x + 2 \, {\left (c^{4} x^{4} - a^{2} c^{2} x^{2} - a^{4}\right )} e - {\left (3 \, a c^{3} d x^{2} - 3 \, a^{3} c d - 4 \, {\left (a^{2} c^{2} x^{2} - a^{4}\right )} e\right )} \log \left (c x + a\right ) + {\left (3 \, a c^{3} d x^{2} - 3 \, a^{3} c d + 4 \, {\left (a^{2} c^{2} x^{2} - a^{4}\right )} e\right )} \log \left (c x - a\right )}{4 \, {\left (c^{8} x^{2} - a^{2} c^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.49, size = 141, normalized size = 1.50 \begin {gather*} \frac {a \left (4 a e - 3 c d\right ) \log {\left (x + \frac {4 a^{2} e - a \left (4 a e - 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac {a \left (4 a e + 3 c d\right ) \log {\left (x + \frac {4 a^{2} e - a \left (4 a e + 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac {- a^{4} e - a^{2} c^{2} d x}{- 2 a^{2} c^{6} + 2 c^{8} x^{2}} + \frac {d x}{c^{4}} + \frac {e x^{2}}{2 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.81, size = 112, normalized size = 1.19 \begin {gather*} -\frac {{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, c^{6}} + \frac {{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, c^{6}} + \frac {c^{4} x^{2} e + 2 \, c^{4} d x}{2 \, c^{8}} - \frac {a^{2} c^{2} d x + a^{4} e}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.09, size = 99, normalized size = 1.05 \begin {gather*} \frac {\frac {a^4\,e}{2\,c^2}+\frac {a^2\,d\,x}{2}}{a^2\,c^4-c^6\,x^2}+\frac {e\,x^2}{2\,c^4}+\frac {\ln \left (a+c\,x\right )\,\left (4\,a^2\,e-3\,a\,c\,d\right )}{4\,c^6}+\frac {\ln \left (a-c\,x\right )\,\left (4\,e\,a^2+3\,c\,d\,a\right )}{4\,c^6}+\frac {d\,x}{c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________