Optimal. Leaf size=264 \[ \frac{c^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac{2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac{8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac{8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac{2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}-\frac{8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac{\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}+\frac{c^4 (d+e x)^8}{8 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9} \]
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Rubi [A] time = 0.280778, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{c^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac{2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac{8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac{8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac{2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}-\frac{8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac{\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}+\frac{c^4 (d+e x)^8}{8 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^4}{d+e x} \, dx &=\int \left (-\frac{8 c d \left (c d^2+a e^2\right )^3}{e^8}+\frac{\left (c d^2+a e^2\right )^4}{e^8 (d+e x)}+\frac{4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)}{e^8}+\frac{8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^2}{e^8}+\frac{2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^3}{e^8}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^4}{e^8}+\frac{4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^5}{e^8}-\frac{8 c^4 d (d+e x)^6}{e^8}+\frac{c^4 (d+e x)^7}{e^8}\right ) \, dx\\ &=-\frac{8 c d \left (c d^2+a e^2\right )^3 x}{e^8}+\frac{2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^2}{e^9}-\frac{8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^3}{3 e^9}+\frac{c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^4}{2 e^9}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^5}{5 e^9}+\frac{2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^6}{3 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9}+\frac{c^4 (d+e x)^8}{8 e^9}+\frac{\left (c d^2+a e^2\right )^4 \log (d+e x)}{e^9}\\ \end{align*}
Mathematica [A] time = 0.105041, size = 227, normalized size = 0.86 \[ \frac{c x \left (420 a^2 c e^4 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+1680 a^3 e^6 (e x-2 d)+56 a c^2 e^2 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )+c^3 \left (-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5+420 d^6 e x-840 d^7-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac{\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 358, normalized size = 1.4 \begin{align*} -{\frac{4\,{c}^{3}{x}^{5}ad}{5\,{e}^{2}}}+4\,{\frac{\ln \left ( ex+d \right ){a}^{3}c{d}^{2}}{{e}^{3}}}+{\frac{3\,{c}^{2}{x}^{4}{a}^{2}}{2\,e}}+{\frac{{c}^{4}{x}^{4}{d}^{4}}{4\,{e}^{5}}}-{\frac{{c}^{4}{x}^{3}{d}^{5}}{3\,{e}^{6}}}+{\frac{\ln \left ( ex+d \right ){c}^{4}{d}^{8}}{{e}^{9}}}+2\,{\frac{c{x}^{2}{a}^{3}}{e}}+{\frac{{c}^{4}{x}^{2}{d}^{6}}{2\,{e}^{7}}}-{\frac{{c}^{4}{d}^{7}x}{{e}^{8}}}-{\frac{{c}^{4}d{x}^{7}}{7\,{e}^{2}}}+{\frac{2\,{c}^{3}{x}^{6}a}{3\,e}}+{\frac{{c}^{4}{x}^{6}{d}^{2}}{6\,{e}^{3}}}+{\frac{{c}^{3}{x}^{4}a{d}^{2}}{{e}^{3}}}+{\frac{{c}^{4}{x}^{8}}{8\,e}}+{\frac{\ln \left ( ex+d \right ){a}^{4}}{e}}-4\,{\frac{{d}^{5}a{c}^{3}x}{{e}^{6}}}-2\,{\frac{{c}^{2}{x}^{3}{a}^{2}d}{{e}^{2}}}-{\frac{4\,{x}^{3}{c}^{3}a{d}^{3}}{3\,{e}^{4}}}+3\,{\frac{{c}^{2}{x}^{2}{a}^{2}{d}^{2}}{{e}^{3}}}+2\,{\frac{{c}^{3}{x}^{2}a{d}^{4}}{{e}^{5}}}-4\,{\frac{{a}^{3}cdx}{{e}^{2}}}-6\,{\frac{{a}^{2}{c}^{2}{d}^{3}x}{{e}^{4}}}+6\,{\frac{\ln \left ( ex+d \right ){a}^{2}{c}^{2}{d}^{4}}{{e}^{5}}}+4\,{\frac{\ln \left ( ex+d \right ) a{c}^{3}{d}^{6}}{{e}^{7}}}-{\frac{{c}^{4}{x}^{5}{d}^{3}}{5\,{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18096, size = 431, normalized size = 1.63 \begin{align*} \frac{105 \, c^{4} e^{7} x^{8} - 120 \, c^{4} d e^{6} x^{7} + 140 \,{\left (c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{6} - 168 \,{\left (c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{5} + 210 \,{\left (c^{4} d^{4} e^{3} + 4 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{4} - 280 \,{\left (c^{4} d^{5} e^{2} + 4 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{3} + 420 \,{\left (c^{4} d^{6} e + 4 \, a c^{3} d^{4} e^{3} + 6 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{2} - 840 \,{\left (c^{4} d^{7} + 4 \, a c^{3} d^{5} e^{2} + 6 \, a^{2} c^{2} d^{3} e^{4} + 4 \, a^{3} c d e^{6}\right )} x}{840 \, e^{8}} + \frac{{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (e x + d\right )}{e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88995, size = 662, normalized size = 2.51 \begin{align*} \frac{105 \, c^{4} e^{8} x^{8} - 120 \, c^{4} d e^{7} x^{7} + 140 \,{\left (c^{4} d^{2} e^{6} + 4 \, a c^{3} e^{8}\right )} x^{6} - 168 \,{\left (c^{4} d^{3} e^{5} + 4 \, a c^{3} d e^{7}\right )} x^{5} + 210 \,{\left (c^{4} d^{4} e^{4} + 4 \, a c^{3} d^{2} e^{6} + 6 \, a^{2} c^{2} e^{8}\right )} x^{4} - 280 \,{\left (c^{4} d^{5} e^{3} + 4 \, a c^{3} d^{3} e^{5} + 6 \, a^{2} c^{2} d e^{7}\right )} x^{3} + 420 \,{\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} + 6 \, a^{2} c^{2} d^{2} e^{6} + 4 \, a^{3} c e^{8}\right )} x^{2} - 840 \,{\left (c^{4} d^{7} e + 4 \, a c^{3} d^{5} e^{3} + 6 \, a^{2} c^{2} d^{3} e^{5} + 4 \, a^{3} c d e^{7}\right )} x + 840 \,{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.734259, size = 287, normalized size = 1.09 \begin{align*} - \frac{c^{4} d x^{7}}{7 e^{2}} + \frac{c^{4} x^{8}}{8 e} + \frac{x^{6} \left (4 a c^{3} e^{2} + c^{4} d^{2}\right )}{6 e^{3}} - \frac{x^{5} \left (4 a c^{3} d e^{2} + c^{4} d^{3}\right )}{5 e^{4}} + \frac{x^{4} \left (6 a^{2} c^{2} e^{4} + 4 a c^{3} d^{2} e^{2} + c^{4} d^{4}\right )}{4 e^{5}} - \frac{x^{3} \left (6 a^{2} c^{2} d e^{4} + 4 a c^{3} d^{3} e^{2} + c^{4} d^{5}\right )}{3 e^{6}} + \frac{x^{2} \left (4 a^{3} c e^{6} + 6 a^{2} c^{2} d^{2} e^{4} + 4 a c^{3} d^{4} e^{2} + c^{4} d^{6}\right )}{2 e^{7}} - \frac{x \left (4 a^{3} c d e^{6} + 6 a^{2} c^{2} d^{3} e^{4} + 4 a c^{3} d^{5} e^{2} + c^{4} d^{7}\right )}{e^{8}} + \frac{\left (a e^{2} + c d^{2}\right )^{4} \log{\left (d + e x \right )}}{e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33633, size = 427, normalized size = 1.62 \begin{align*}{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{840} \,{\left (105 \, c^{4} x^{8} e^{7} - 120 \, c^{4} d x^{7} e^{6} + 140 \, c^{4} d^{2} x^{6} e^{5} - 168 \, c^{4} d^{3} x^{5} e^{4} + 210 \, c^{4} d^{4} x^{4} e^{3} - 280 \, c^{4} d^{5} x^{3} e^{2} + 420 \, c^{4} d^{6} x^{2} e - 840 \, c^{4} d^{7} x + 560 \, a c^{3} x^{6} e^{7} - 672 \, a c^{3} d x^{5} e^{6} + 840 \, a c^{3} d^{2} x^{4} e^{5} - 1120 \, a c^{3} d^{3} x^{3} e^{4} + 1680 \, a c^{3} d^{4} x^{2} e^{3} - 3360 \, a c^{3} d^{5} x e^{2} + 1260 \, a^{2} c^{2} x^{4} e^{7} - 1680 \, a^{2} c^{2} d x^{3} e^{6} + 2520 \, a^{2} c^{2} d^{2} x^{2} e^{5} - 5040 \, a^{2} c^{2} d^{3} x e^{4} + 1680 \, a^{3} c x^{2} e^{7} - 3360 \, a^{3} c d x e^{6}\right )} e^{\left (-8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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