Optimal. Leaf size=77 \[ \frac {x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {(2 a e+c d) \log (a-c x)}{4 a c^4}-\frac {(c d-2 a e) \log (a+c x)}{4 a c^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {819, 633, 31} \begin {gather*} \frac {x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {(2 a e+c d) \log (a-c x)}{4 a c^4}-\frac {(c d-2 a e) \log (a+c x)}{4 a c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 633
Rule 819
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac {x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \frac {a^2 d+2 a^2 e x}{a^2-c^2 x^2} \, dx}{2 a^2 c^2}\\ &=\frac {x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {(c d-2 a e) \int \frac {1}{-a c-c^2 x} \, dx}{4 a c^2}-\frac {(c d+2 a e) \int \frac {1}{a c-c^2 x} \, dx}{4 a c^2}\\ &=\frac {x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {(c d+2 a e) \log (a-c x)}{4 a c^4}-\frac {(c d-2 a e) \log (a+c x)}{4 a c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 64, normalized size = 0.83 \begin {gather*} \frac {\frac {a^2 e+c^2 d x}{a^2-c^2 x^2}+e \log \left (a^2-c^2 x^2\right )-\frac {c d \tanh ^{-1}\left (\frac {c x}{a}\right )}{a}}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 115, normalized size = 1.49 \begin {gather*} -\frac {2 \, a c^{2} d x + 2 \, a^{3} e - {\left (a^{2} c d - 2 \, a^{3} e - {\left (c^{3} d - 2 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) + {\left (a^{2} c d + 2 \, a^{3} e - {\left (c^{3} d + 2 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \, {\left (a c^{6} x^{2} - a^{3} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 85, normalized size = 1.10 \begin {gather*} -\frac {d x + \frac {a^{2} e}{c^{2}}}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} c^{2}} - \frac {{\left (c d - 2 \, a e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a c^{4}} + \frac {{\left (c d + 2 \, a e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 118, normalized size = 1.53 \begin {gather*} \frac {a e}{4 \left (c x +a \right ) c^{4}}-\frac {a e}{4 \left (c x -a \right ) c^{4}}+\frac {d \ln \left (c x -a \right )}{4 a \,c^{3}}-\frac {d \ln \left (c x +a \right )}{4 a \,c^{3}}-\frac {d}{4 \left (c x +a \right ) c^{3}}-\frac {d}{4 \left (c x -a \right ) c^{3}}+\frac {e \ln \left (c x -a \right )}{2 c^{4}}+\frac {e \ln \left (c x +a \right )}{2 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.58, size = 79, normalized size = 1.03 \begin {gather*} -\frac {c^{2} d x + a^{2} e}{2 \, {\left (c^{6} x^{2} - a^{2} c^{4}\right )}} - \frac {{\left (c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a c^{4}} + \frac {{\left (c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 103, normalized size = 1.34 \begin {gather*} \frac {a^2\,e}{2\,\left (a^2\,c^4-c^6\,x^2\right )}+\frac {d\,x}{2\,\left (a^2\,c^2-c^4\,x^2\right )}+\frac {e\,\ln \left (a+c\,x\right )}{2\,c^4}+\frac {e\,\ln \left (a-c\,x\right )}{2\,c^4}-\frac {d\,\ln \left (a+c\,x\right )}{4\,a\,c^3}+\frac {d\,\ln \left (a-c\,x\right )}{4\,a\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.63, size = 110, normalized size = 1.43 \begin {gather*} \frac {- a^{2} e - c^{2} d x}{- 2 a^{2} c^{4} + 2 c^{6} x^{2}} + \frac {\left (2 a e - c d\right ) \log {\left (x + \frac {2 a^{2} e - a \left (2 a e - c d\right )}{c^{2} d} \right )}}{4 a c^{4}} + \frac {\left (2 a e + c d\right ) \log {\left (x + \frac {2 a^{2} e - a \left (2 a e + c d\right )}{c^{2} d} \right )}}{4 a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________