Optimal. Leaf size=278 \[ \frac{c^2 (d+e x)^{12} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{6 e^9}+\frac{2 c^3 (d+e x)^{14} \left (a e^2+7 c d^2\right )}{7 e^9}-\frac{8 c^3 d (d+e x)^{13} \left (3 a e^2+7 c d^2\right )}{13 e^9}-\frac{8 c^2 d (d+e x)^{11} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{11 e^9}+\frac{2 c (d+e x)^{10} \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{5 e^9}-\frac{8 c d (d+e x)^9 \left (a e^2+c d^2\right )^3}{9 e^9}+\frac{(d+e x)^8 \left (a e^2+c d^2\right )^4}{8 e^9}+\frac{c^4 (d+e x)^{16}}{16 e^9}-\frac{8 c^4 d (d+e x)^{15}}{15 e^9} \]
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Rubi [A] time = 0.515855, antiderivative size = 278, 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} \[ \frac{c^2 (d+e x)^{12} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{6 e^9}+\frac{2 c^3 (d+e x)^{14} \left (a e^2+7 c d^2\right )}{7 e^9}-\frac{8 c^3 d (d+e x)^{13} \left (3 a e^2+7 c d^2\right )}{13 e^9}-\frac{8 c^2 d (d+e x)^{11} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{11 e^9}+\frac{2 c (d+e x)^{10} \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{5 e^9}-\frac{8 c d (d+e x)^9 \left (a e^2+c d^2\right )^3}{9 e^9}+\frac{(d+e x)^8 \left (a e^2+c d^2\right )^4}{8 e^9}+\frac{c^4 (d+e x)^{16}}{16 e^9}-\frac{8 c^4 d (d+e x)^{15}}{15 e^9} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^4 (d+e x)^7}{e^8}-\frac{8 c d \left (c d^2+a e^2\right )^3 (d+e x)^8}{e^8}+\frac{4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^9}{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)^{10}}{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)^{11}}{e^8}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{12}}{e^8}+\frac{4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{13}}{e^8}-\frac{8 c^4 d (d+e x)^{14}}{e^8}+\frac{c^4 (d+e x)^{15}}{e^8}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^4 (d+e x)^8}{8 e^9}-\frac{8 c d \left (c d^2+a e^2\right )^3 (d+e x)^9}{9 e^9}+\frac{2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^{10}}{5 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)^{11}}{11 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)^{12}}{6 e^9}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{13}}{13 e^9}+\frac{2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{14}}{7 e^9}-\frac{8 c^4 d (d+e x)^{15}}{15 e^9}+\frac{c^4 (d+e x)^{16}}{16 e^9}\\ \end{align*}
Mathematica [A] time = 0.083014, size = 423, normalized size = 1.52 \[ \frac{1}{660} a^2 c^2 x^5 \left (11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+4620 d^6 e x+792 d^7+2520 d e^6 x^6+330 e^7 x^7\right )+\frac{1}{90} a^3 c x^3 \left (1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+630 d^6 e x+120 d^7+280 d e^6 x^6+36 e^7 x^7\right )+\frac{1}{8} a^4 x \left (56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+28 d^6 e x+8 d^7+8 d e^6 x^6+e^7 x^7\right )+\frac{a c^3 x^7 \left (56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+21021 d^6 e x+3432 d^7+12936 d e^6 x^6+1716 e^7 x^7\right )}{6006}+\frac{c^4 x^9 \left (196560 d^5 e^2 x^2+300300 d^4 e^3 x^3+277200 d^3 e^4 x^4+154440 d^2 e^5 x^5+72072 d^6 e x+11440 d^7+48048 d e^6 x^6+6435 e^7 x^7\right )}{102960} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 511, normalized size = 1.8 \begin{align*}{\frac{{e}^{7}{c}^{4}{x}^{16}}{16}}+{\frac{7\,d{e}^{6}{c}^{4}{x}^{15}}{15}}+{\frac{ \left ( 4\,{e}^{7}a{c}^{3}+21\,{d}^{2}{e}^{5}{c}^{4} \right ){x}^{14}}{14}}+{\frac{ \left ( 28\,d{e}^{6}a{c}^{3}+35\,{d}^{3}{e}^{4}{c}^{4} \right ){x}^{13}}{13}}+{\frac{ \left ( 6\,{e}^{7}{a}^{2}{c}^{2}+84\,{d}^{2}{e}^{5}a{c}^{3}+35\,{d}^{4}{e}^{3}{c}^{4} \right ){x}^{12}}{12}}+{\frac{ \left ( 42\,d{e}^{6}{a}^{2}{c}^{2}+140\,{d}^{3}{e}^{4}a{c}^{3}+21\,{d}^{5}{e}^{2}{c}^{4} \right ){x}^{11}}{11}}+{\frac{ \left ( 4\,{e}^{7}{a}^{3}c+126\,{d}^{2}{e}^{5}{a}^{2}{c}^{2}+140\,{d}^{4}{e}^{3}a{c}^{3}+7\,{d}^{6}e{c}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 28\,d{e}^{6}{a}^{3}c+210\,{d}^{3}{e}^{4}{a}^{2}{c}^{2}+84\,{d}^{5}{e}^{2}a{c}^{3}+{d}^{7}{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ({e}^{7}{a}^{4}+84\,{d}^{2}{e}^{5}{a}^{3}c+210\,{d}^{4}{e}^{3}{a}^{2}{c}^{2}+28\,{d}^{6}ea{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,d{e}^{6}{a}^{4}+140\,{d}^{3}{e}^{4}{a}^{3}c+126\,{d}^{5}{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{7}a{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{d}^{2}{e}^{5}{a}^{4}+140\,{d}^{4}{e}^{3}{a}^{3}c+42\,{d}^{6}e{a}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{d}^{3}{e}^{4}{a}^{4}+84\,{d}^{5}{e}^{2}{a}^{3}c+6\,{d}^{7}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{d}^{4}{e}^{3}{a}^{4}+28\,{d}^{6}e{a}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{d}^{5}{e}^{2}{a}^{4}+4\,{d}^{7}{a}^{3}c \right ){x}^{3}}{3}}+{\frac{7\,{d}^{6}e{a}^{4}{x}^{2}}{2}}+{d}^{7}{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1466, size = 689, normalized size = 2.48 \begin{align*} \frac{1}{16} \, c^{4} e^{7} x^{16} + \frac{7}{15} \, c^{4} d e^{6} x^{15} + \frac{1}{14} \,{\left (21 \, c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{14} + \frac{7}{13} \,{\left (5 \, c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{13} + \frac{7}{2} \, a^{4} d^{6} e x^{2} + \frac{1}{12} \,{\left (35 \, c^{4} d^{4} e^{3} + 84 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{12} + a^{4} d^{7} x + \frac{7}{11} \,{\left (3 \, c^{4} d^{5} e^{2} + 20 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{11} + \frac{1}{10} \,{\left (7 \, c^{4} d^{6} e + 140 \, a c^{3} d^{4} e^{3} + 126 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{10} + \frac{1}{9} \,{\left (c^{4} d^{7} + 84 \, a c^{3} d^{5} e^{2} + 210 \, a^{2} c^{2} d^{3} e^{4} + 28 \, a^{3} c d e^{6}\right )} x^{9} + \frac{1}{8} \,{\left (28 \, a c^{3} d^{6} e + 210 \, a^{2} c^{2} d^{4} e^{3} + 84 \, a^{3} c d^{2} e^{5} + a^{4} e^{7}\right )} x^{8} + \frac{1}{7} \,{\left (4 \, a c^{3} d^{7} + 126 \, a^{2} c^{2} d^{5} e^{2} + 140 \, a^{3} c d^{3} e^{4} + 7 \, a^{4} d e^{6}\right )} x^{7} + \frac{7}{6} \,{\left (6 \, a^{2} c^{2} d^{6} e + 20 \, a^{3} c d^{4} e^{3} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, a^{2} c^{2} d^{7} + 84 \, a^{3} c d^{5} e^{2} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac{7}{4} \,{\left (4 \, a^{3} c d^{6} e + 5 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (4 \, a^{3} c d^{7} + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86475, size = 1218, normalized size = 4.38 \begin{align*} \frac{1}{16} x^{16} e^{7} c^{4} + \frac{7}{15} x^{15} e^{6} d c^{4} + \frac{3}{2} x^{14} e^{5} d^{2} c^{4} + \frac{2}{7} x^{14} e^{7} c^{3} a + \frac{35}{13} x^{13} e^{4} d^{3} c^{4} + \frac{28}{13} x^{13} e^{6} d c^{3} a + \frac{35}{12} x^{12} e^{3} d^{4} c^{4} + 7 x^{12} e^{5} d^{2} c^{3} a + \frac{1}{2} x^{12} e^{7} c^{2} a^{2} + \frac{21}{11} x^{11} e^{2} d^{5} c^{4} + \frac{140}{11} x^{11} e^{4} d^{3} c^{3} a + \frac{42}{11} x^{11} e^{6} d c^{2} a^{2} + \frac{7}{10} x^{10} e d^{6} c^{4} + 14 x^{10} e^{3} d^{4} c^{3} a + \frac{63}{5} x^{10} e^{5} d^{2} c^{2} a^{2} + \frac{2}{5} x^{10} e^{7} c a^{3} + \frac{1}{9} x^{9} d^{7} c^{4} + \frac{28}{3} x^{9} e^{2} d^{5} c^{3} a + \frac{70}{3} x^{9} e^{4} d^{3} c^{2} a^{2} + \frac{28}{9} x^{9} e^{6} d c a^{3} + \frac{7}{2} x^{8} e d^{6} c^{3} a + \frac{105}{4} x^{8} e^{3} d^{4} c^{2} a^{2} + \frac{21}{2} x^{8} e^{5} d^{2} c a^{3} + \frac{1}{8} x^{8} e^{7} a^{4} + \frac{4}{7} x^{7} d^{7} c^{3} a + 18 x^{7} e^{2} d^{5} c^{2} a^{2} + 20 x^{7} e^{4} d^{3} c a^{3} + x^{7} e^{6} d a^{4} + 7 x^{6} e d^{6} c^{2} a^{2} + \frac{70}{3} x^{6} e^{3} d^{4} c a^{3} + \frac{7}{2} x^{6} e^{5} d^{2} a^{4} + \frac{6}{5} x^{5} d^{7} c^{2} a^{2} + \frac{84}{5} x^{5} e^{2} d^{5} c a^{3} + 7 x^{5} e^{4} d^{3} a^{4} + 7 x^{4} e d^{6} c a^{3} + \frac{35}{4} x^{4} e^{3} d^{4} a^{4} + \frac{4}{3} x^{3} d^{7} c a^{3} + 7 x^{3} e^{2} d^{5} a^{4} + \frac{7}{2} x^{2} e d^{6} a^{4} + x d^{7} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.137736, size = 571, normalized size = 2.05 \begin{align*} a^{4} d^{7} x + \frac{7 a^{4} d^{6} e x^{2}}{2} + \frac{7 c^{4} d e^{6} x^{15}}{15} + \frac{c^{4} e^{7} x^{16}}{16} + x^{14} \left (\frac{2 a c^{3} e^{7}}{7} + \frac{3 c^{4} d^{2} e^{5}}{2}\right ) + x^{13} \left (\frac{28 a c^{3} d e^{6}}{13} + \frac{35 c^{4} d^{3} e^{4}}{13}\right ) + x^{12} \left (\frac{a^{2} c^{2} e^{7}}{2} + 7 a c^{3} d^{2} e^{5} + \frac{35 c^{4} d^{4} e^{3}}{12}\right ) + x^{11} \left (\frac{42 a^{2} c^{2} d e^{6}}{11} + \frac{140 a c^{3} d^{3} e^{4}}{11} + \frac{21 c^{4} d^{5} e^{2}}{11}\right ) + x^{10} \left (\frac{2 a^{3} c e^{7}}{5} + \frac{63 a^{2} c^{2} d^{2} e^{5}}{5} + 14 a c^{3} d^{4} e^{3} + \frac{7 c^{4} d^{6} e}{10}\right ) + x^{9} \left (\frac{28 a^{3} c d e^{6}}{9} + \frac{70 a^{2} c^{2} d^{3} e^{4}}{3} + \frac{28 a c^{3} d^{5} e^{2}}{3} + \frac{c^{4} d^{7}}{9}\right ) + x^{8} \left (\frac{a^{4} e^{7}}{8} + \frac{21 a^{3} c d^{2} e^{5}}{2} + \frac{105 a^{2} c^{2} d^{4} e^{3}}{4} + \frac{7 a c^{3} d^{6} e}{2}\right ) + x^{7} \left (a^{4} d e^{6} + 20 a^{3} c d^{3} e^{4} + 18 a^{2} c^{2} d^{5} e^{2} + \frac{4 a c^{3} d^{7}}{7}\right ) + x^{6} \left (\frac{7 a^{4} d^{2} e^{5}}{2} + \frac{70 a^{3} c d^{4} e^{3}}{3} + 7 a^{2} c^{2} d^{6} e\right ) + x^{5} \left (7 a^{4} d^{3} e^{4} + \frac{84 a^{3} c d^{5} e^{2}}{5} + \frac{6 a^{2} c^{2} d^{7}}{5}\right ) + x^{4} \left (\frac{35 a^{4} d^{4} e^{3}}{4} + 7 a^{3} c d^{6} e\right ) + x^{3} \left (7 a^{4} d^{5} e^{2} + \frac{4 a^{3} c d^{7}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33366, size = 705, normalized size = 2.54 \begin{align*} \frac{1}{16} \, c^{4} x^{16} e^{7} + \frac{7}{15} \, c^{4} d x^{15} e^{6} + \frac{3}{2} \, c^{4} d^{2} x^{14} e^{5} + \frac{35}{13} \, c^{4} d^{3} x^{13} e^{4} + \frac{35}{12} \, c^{4} d^{4} x^{12} e^{3} + \frac{21}{11} \, c^{4} d^{5} x^{11} e^{2} + \frac{7}{10} \, c^{4} d^{6} x^{10} e + \frac{1}{9} \, c^{4} d^{7} x^{9} + \frac{2}{7} \, a c^{3} x^{14} e^{7} + \frac{28}{13} \, a c^{3} d x^{13} e^{6} + 7 \, a c^{3} d^{2} x^{12} e^{5} + \frac{140}{11} \, a c^{3} d^{3} x^{11} e^{4} + 14 \, a c^{3} d^{4} x^{10} e^{3} + \frac{28}{3} \, a c^{3} d^{5} x^{9} e^{2} + \frac{7}{2} \, a c^{3} d^{6} x^{8} e + \frac{4}{7} \, a c^{3} d^{7} x^{7} + \frac{1}{2} \, a^{2} c^{2} x^{12} e^{7} + \frac{42}{11} \, a^{2} c^{2} d x^{11} e^{6} + \frac{63}{5} \, a^{2} c^{2} d^{2} x^{10} e^{5} + \frac{70}{3} \, a^{2} c^{2} d^{3} x^{9} e^{4} + \frac{105}{4} \, a^{2} c^{2} d^{4} x^{8} e^{3} + 18 \, a^{2} c^{2} d^{5} x^{7} e^{2} + 7 \, a^{2} c^{2} d^{6} x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{7} x^{5} + \frac{2}{5} \, a^{3} c x^{10} e^{7} + \frac{28}{9} \, a^{3} c d x^{9} e^{6} + \frac{21}{2} \, a^{3} c d^{2} x^{8} e^{5} + 20 \, a^{3} c d^{3} x^{7} e^{4} + \frac{70}{3} \, a^{3} c d^{4} x^{6} e^{3} + \frac{84}{5} \, a^{3} c d^{5} x^{5} e^{2} + 7 \, a^{3} c d^{6} x^{4} e + \frac{4}{3} \, a^{3} c d^{7} x^{3} + \frac{1}{8} \, a^{4} x^{8} e^{7} + a^{4} d x^{7} e^{6} + \frac{7}{2} \, a^{4} d^{2} x^{6} e^{5} + 7 \, a^{4} d^{3} x^{5} e^{4} + \frac{35}{4} \, a^{4} d^{4} x^{4} e^{3} + 7 \, a^{4} d^{5} x^{3} e^{2} + \frac{7}{2} \, a^{4} d^{6} x^{2} e + a^{4} d^{7} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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