Optimal. Leaf size=103 \[ -\frac{6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac{4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac{2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac{(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac{e^4 x}{b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0884639, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac{6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac{4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac{2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac{(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac{e^4 x}{b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^4}{(a+b x)^4} \, dx\\ &=\int \left (\frac{e^4}{b^4}+\frac{(b d-a e)^4}{b^4 (a+b x)^4}+\frac{4 e (b d-a e)^3}{b^4 (a+b x)^3}+\frac{6 e^2 (b d-a e)^2}{b^4 (a+b x)^2}+\frac{4 e^3 (b d-a e)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{e^4 x}{b^4}-\frac{(b d-a e)^4}{3 b^5 (a+b x)^3}-\frac{2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac{6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac{4 e^3 (b d-a e) \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0606032, size = 166, normalized size = 1.61 \[ \frac{-3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a^3 b e^3 (22 d-27 e x)-13 a^4 e^4+a b^3 e \left (-18 d^2 e x-2 d^3+36 d e^2 x^2+9 e^3 x^3\right )-12 e^3 (a+b x)^3 (a e-b d) \log (a+b x)+b^4 \left (-\left (18 d^2 e^2 x^2+6 d^3 e x+d^4-3 e^4 x^4\right )\right )}{3 b^5 (a+b x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 255, normalized size = 2.5 \begin{align*}{\frac{{e}^{4}x}{{b}^{4}}}+2\,{\frac{{a}^{3}{e}^{4}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{a}^{2}{e}^{3}d}{{b}^{4} \left ( bx+a \right ) ^{2}}}+6\,{\frac{a{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-2\,{\frac{e{d}^{3}}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{4}{e}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{4\,{a}^{3}d{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-2\,{\frac{{d}^{2}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{4\,a{d}^{3}e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{4}}{3\,b \left ( bx+a \right ) ^{3}}}-4\,{\frac{{e}^{4}\ln \left ( bx+a \right ) a}{{b}^{5}}}+4\,{\frac{{e}^{3}\ln \left ( bx+a \right ) d}{{b}^{4}}}-6\,{\frac{{a}^{2}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+12\,{\frac{ad{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-6\,{\frac{{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12441, size = 271, normalized size = 2.63 \begin{align*} \frac{e^{4} x}{b^{4}} - \frac{b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac{4 \,{\left (b d e^{3} - a e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.69989, size = 581, normalized size = 5.64 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} + 9 \, a b^{3} e^{4} x^{3} - b^{4} d^{4} - 2 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} + 22 \, a^{3} b d e^{3} - 13 \, a^{4} e^{4} - 9 \,{\left (2 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} - 3 \,{\left (2 \, b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 18 \, a^{2} b^{2} d e^{3} + 9 \, a^{3} b e^{4}\right )} x + 12 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.46601, size = 209, normalized size = 2.03 \begin{align*} - \frac{13 a^{4} e^{4} - 22 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 2 a b^{3} d^{3} e + b^{4} d^{4} + x^{2} \left (18 a^{2} b^{2} e^{4} - 36 a b^{3} d e^{3} + 18 b^{4} d^{2} e^{2}\right ) + x \left (30 a^{3} b e^{4} - 54 a^{2} b^{2} d e^{3} + 18 a b^{3} d^{2} e^{2} + 6 b^{4} d^{3} e\right )}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac{e^{4} x}{b^{4}} - \frac{4 e^{3} \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14, size = 225, normalized size = 2.18 \begin{align*} \frac{x e^{4}}{b^{4}} + \frac{4 \,{\left (b d e^{3} - a e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \,{\left (b x + a\right )}^{3} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]