Optimal. Leaf size=109 \[ \frac {e^2}{(d+e x) (b d-a e)^3}+\frac {3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac {3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac {2 b e}{(a+b x) (b d-a e)^3}-\frac {b}{2 (a+b x)^2 (b d-a e)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 44} \begin {gather*} \frac {e^2}{(d+e x) (b d-a e)^3}+\frac {3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac {3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac {2 b e}{(a+b x) (b d-a e)^3}-\frac {b}{2 (a+b x)^2 (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^3 (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^3}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^2}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)}-\frac {e^3}{(b d-a e)^3 (d+e x)^2}-\frac {3 b e^3}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac {b}{2 (b d-a e)^2 (a+b x)^2}+\frac {2 b e}{(b d-a e)^3 (a+b x)}+\frac {e^2}{(b d-a e)^3 (d+e x)}+\frac {3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac {3 b e^2 \log (d+e x)}{(b d-a e)^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 98, normalized size = 0.90 \begin {gather*} \frac {\frac {2 e^2 (b d-a e)}{d+e x}+\frac {4 b e (b d-a e)}{a+b x}-\frac {b (b d-a e)^2}{(a+b x)^2}+6 b e^2 \log (a+b x)-6 b e^2 \log (d+e x)}{2 (b d-a e)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 494, normalized size = 4.53 \begin {gather*} -\frac {b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} + {\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} + {\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} + {\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 212, normalized size = 1.94 \begin {gather*} \frac {3 \, b e^{3} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac {e^{5}}{{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} {\left (x e + d\right )}} + \frac {5 \, b^{3} e^{2} - \frac {6 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \, {\left (b d - a e\right )}^{4} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 109, normalized size = 1.00 \begin {gather*} \frac {3 b \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {3 b \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {2 b e}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {e^{2}}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {b}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.66, size = 386, normalized size = 3.54 \begin {gather*} \frac {3 \, b e^{2} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {3 \, b e^{2} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {6 \, b^{2} e^{2} x^{2} - b^{2} d^{2} + 5 \, a b d e + 2 \, a^{2} e^{2} + 3 \, {\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{2 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.21, size = 330, normalized size = 3.03 \begin {gather*} \frac {6\,b\,e^2\,\mathrm {atanh}\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,e\,x\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {2\,a^2\,e^2+5\,a\,b\,d\,e-b^2\,d^2}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {3\,e\,x\,\left (d\,b^2+3\,a\,e\,b\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {3\,b^2\,e^2\,x^2}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{x\,\left (e\,a^2+2\,b\,d\,a\right )+a^2\,d+x^2\,\left (d\,b^2+2\,a\,e\,b\right )+b^2\,e\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.73, size = 634, normalized size = 5.82 \begin {gather*} - \frac {3 b e^{2} \log {\left (x + \frac {- \frac {3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} + \frac {15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} - \frac {30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} + \frac {30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} - \frac {15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} + \frac {3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac {3 b e^{2} \log {\left (x + \frac {\frac {3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} - \frac {15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} + \frac {30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} - \frac {30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} + \frac {15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} - \frac {3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 2 a^{2} e^{2} - 5 a b d e + b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (- 9 a b e^{2} - 3 b^{2} d e\right )}{2 a^{5} d e^{3} - 6 a^{4} b d^{2} e^{2} + 6 a^{3} b^{2} d^{3} e - 2 a^{2} b^{3} d^{4} + x^{3} \left (2 a^{3} b^{2} e^{4} - 6 a^{2} b^{3} d e^{3} + 6 a b^{4} d^{2} e^{2} - 2 b^{5} d^{3} e\right ) + x^{2} \left (4 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 6 a^{2} b^{3} d^{2} e^{2} + 2 a b^{4} d^{3} e - 2 b^{5} d^{4}\right ) + x \left (2 a^{5} e^{4} - 2 a^{4} b d e^{3} - 6 a^{3} b^{2} d^{2} e^{2} + 10 a^{2} b^{3} d^{3} e - 4 a b^{4} d^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________