Optimal. Leaf size=184 \[ \frac{e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.197296, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 43} \[ \frac{e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{(d+e x)^5}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac{10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac{\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^2}+\frac{5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac{10 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^5 d^5}+\frac{5 \left (c d^2 e^4-a e^6\right ) (a e+c d x)^2}{c^5 d^5}+\frac{e^5 (a e+c d x)^3}{c^5 d^5}\right ) \, dx\\ &=\frac{10 e^2 \left (c d^2-a e^2\right )^3 x}{c^5 d^5}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}\\ \end{align*}
Mathematica [A] time = 0.0858901, size = 263, normalized size = 1.43 \[ \frac{30 a^3 c^2 d^2 e^6 \left (4 d^2+6 d e x-e^2 x^2\right )-10 a^2 c^3 d^3 e^4 \left (24 d^2 e x+12 d^3-12 d e^2 x^2-e^3 x^3\right )-12 a^4 c d e^8 (5 d+4 e x)+12 a^5 e^{10}+5 a c^4 d^4 e^2 \left (-36 d^2 e^2 x^2+24 d^3 e x+12 d^4-8 d e^3 x^3-e^4 x^4\right )+60 e \left (c d^2-a e^2\right )^4 (a e+c d x) \log (a e+c d x)+c^5 d^5 \left (120 d^3 e^2 x^2+60 d^2 e^3 x^3-12 d^5+20 d e^4 x^4+3 e^5 x^5\right )}{12 c^6 d^6 (a e+c d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 378, normalized size = 2.1 \begin{align*}{\frac{{e}^{5}{x}^{4}}{4\,{c}^{2}{d}^{2}}}-{\frac{2\,{e}^{6}{x}^{3}a}{3\,{c}^{3}{d}^{3}}}+{\frac{5\,{e}^{4}{x}^{3}}{3\,{c}^{2}d}}+{\frac{3\,{e}^{7}{x}^{2}{a}^{2}}{2\,{c}^{4}{d}^{4}}}-5\,{\frac{{e}^{5}{x}^{2}a}{{c}^{3}{d}^{2}}}+5\,{\frac{{e}^{3}{x}^{2}}{{c}^{2}}}-4\,{\frac{{a}^{3}{e}^{8}x}{{c}^{5}{d}^{5}}}+15\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{3}}}-20\,{\frac{a{e}^{4}x}{{c}^{3}d}}+10\,{\frac{d{e}^{2}x}{{c}^{2}}}+{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{6} \left ( cdx+ae \right ) }}-5\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{4} \left ( cdx+ae \right ) }}+10\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{2} \left ( cdx+ae \right ) }}-10\,{\frac{{a}^{2}{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) }}+5\,{\frac{a{d}^{2}{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{d}^{4}}{c \left ( cdx+ae \right ) }}+5\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{6}{d}^{6}}}-20\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{4}}}+30\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}{d}^{2}}}-20\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}}}+5\,{\frac{{d}^{2}e\ln \left ( cdx+ae \right ) }{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10539, size = 405, normalized size = 2.2 \begin{align*} -\frac{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{c^{7} d^{7} x + a c^{6} d^{6} e} + \frac{3 \, c^{3} d^{3} e^{5} x^{4} + 4 \,{\left (5 \, c^{3} d^{4} e^{4} - 2 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \,{\left (10 \, c^{3} d^{5} e^{3} - 10 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{2} + 12 \,{\left (10 \, c^{3} d^{6} e^{2} - 20 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} - 4 \, a^{3} e^{8}\right )} x}{12 \, c^{5} d^{5}} + \frac{5 \,{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.96588, size = 841, normalized size = 4.57 \begin{align*} \frac{3 \, c^{5} d^{5} e^{5} x^{5} - 12 \, c^{5} d^{10} + 60 \, a c^{4} d^{8} e^{2} - 120 \, a^{2} c^{3} d^{6} e^{4} + 120 \, a^{3} c^{2} d^{4} e^{6} - 60 \, a^{4} c d^{2} e^{8} + 12 \, a^{5} e^{10} + 5 \,{\left (4 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \,{\left (6 \, c^{5} d^{7} e^{3} - 4 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 30 \,{\left (4 \, c^{5} d^{8} e^{2} - 6 \, a c^{4} d^{6} e^{4} + 4 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 12 \,{\left (10 \, a c^{4} d^{7} e^{3} - 20 \, a^{2} c^{3} d^{5} e^{5} + 15 \, a^{3} c^{2} d^{3} e^{7} - 4 \, a^{4} c d e^{9}\right )} x + 60 \,{\left (a c^{4} d^{8} e^{2} - 4 \, a^{2} c^{3} d^{6} e^{4} + 6 \, a^{3} c^{2} d^{4} e^{6} - 4 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} +{\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{12 \,{\left (c^{7} d^{7} x + a c^{6} d^{6} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.58569, size = 265, normalized size = 1.44 \begin{align*} \frac{a^{5} e^{10} - 5 a^{4} c d^{2} e^{8} + 10 a^{3} c^{2} d^{4} e^{6} - 10 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} - c^{5} d^{10}}{a c^{6} d^{6} e + c^{7} d^{7} x} + \frac{e^{5} x^{4}}{4 c^{2} d^{2}} - \frac{x^{3} \left (2 a e^{6} - 5 c d^{2} e^{4}\right )}{3 c^{3} d^{3}} + \frac{x^{2} \left (3 a^{2} e^{7} - 10 a c d^{2} e^{5} + 10 c^{2} d^{4} e^{3}\right )}{2 c^{4} d^{4}} - \frac{x \left (4 a^{3} e^{8} - 15 a^{2} c d^{2} e^{6} + 20 a c^{2} d^{4} e^{4} - 10 c^{3} d^{6} e^{2}\right )}{c^{5} d^{5}} + \frac{5 e \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.46234, size = 909, normalized size = 4.94 \begin{align*} \frac{5 \,{\left (c^{7} d^{14} e - 7 \, a c^{6} d^{12} e^{3} + 21 \, a^{2} c^{5} d^{10} e^{5} - 35 \, a^{3} c^{4} d^{8} e^{7} + 35 \, a^{4} c^{3} d^{6} e^{9} - 21 \, a^{5} c^{2} d^{4} e^{11} + 7 \, a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{8} d^{10} - 2 \, a c^{7} d^{8} e^{2} + a^{2} c^{6} d^{6} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{5 \,{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{6} d^{6}} - \frac{c^{7} d^{15} - 7 \, a c^{6} d^{13} e^{2} + 21 \, a^{2} c^{5} d^{11} e^{4} - 35 \, a^{3} c^{4} d^{9} e^{6} + 35 \, a^{4} c^{3} d^{7} e^{8} - 21 \, a^{5} c^{2} d^{5} e^{10} + 7 \, a^{6} c d^{3} e^{12} - a^{7} d e^{14} +{\left (c^{7} d^{14} e - 7 \, a c^{6} d^{12} e^{3} + 21 \, a^{2} c^{5} d^{10} e^{5} - 35 \, a^{3} c^{4} d^{8} e^{7} + 35 \, a^{4} c^{3} d^{6} e^{9} - 21 \, a^{5} c^{2} d^{4} e^{11} + 7 \, a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{6} d^{6}} + \frac{{\left (3 \, c^{6} d^{6} x^{4} e^{13} + 20 \, c^{6} d^{7} x^{3} e^{12} + 60 \, c^{6} d^{8} x^{2} e^{11} + 120 \, c^{6} d^{9} x e^{10} - 8 \, a c^{5} d^{5} x^{3} e^{14} - 60 \, a c^{5} d^{6} x^{2} e^{13} - 240 \, a c^{5} d^{7} x e^{12} + 18 \, a^{2} c^{4} d^{4} x^{2} e^{15} + 180 \, a^{2} c^{4} d^{5} x e^{14} - 48 \, a^{3} c^{3} d^{3} x e^{16}\right )} e^{\left (-8\right )}}{12 \, c^{8} d^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]