Optimal. Leaf size=276 \[ \frac{2 c^2 (d+e x)^{11} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{11 e^9}+\frac{4 c^3 (d+e x)^{13} \left (a e^2+7 c d^2\right )}{13 e^9}-\frac{2 c^3 d (d+e x)^{12} \left (3 a e^2+7 c d^2\right )}{3 e^9}-\frac{4 c^2 d (d+e x)^{10} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{5 e^9}+\frac{4 c (d+e x)^9 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{9 e^9}-\frac{c d (d+e x)^8 \left (a e^2+c d^2\right )^3}{e^9}+\frac{(d+e x)^7 \left (a e^2+c d^2\right )^4}{7 e^9}+\frac{c^4 (d+e x)^{15}}{15 e^9}-\frac{4 c^4 d (d+e x)^{14}}{7 e^9} \]
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Rubi [A] time = 0.458363, antiderivative size = 276, 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} \[ \frac{2 c^2 (d+e x)^{11} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{11 e^9}+\frac{4 c^3 (d+e x)^{13} \left (a e^2+7 c d^2\right )}{13 e^9}-\frac{2 c^3 d (d+e x)^{12} \left (3 a e^2+7 c d^2\right )}{3 e^9}-\frac{4 c^2 d (d+e x)^{10} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{5 e^9}+\frac{4 c (d+e x)^9 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{9 e^9}-\frac{c d (d+e x)^8 \left (a e^2+c d^2\right )^3}{e^9}+\frac{(d+e x)^7 \left (a e^2+c d^2\right )^4}{7 e^9}+\frac{c^4 (d+e x)^{15}}{15 e^9}-\frac{4 c^4 d (d+e x)^{14}}{7 e^9} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^4 (d+e x)^6}{e^8}-\frac{8 c d \left (c d^2+a e^2\right )^3 (d+e x)^7}{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)^8}{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)^9}{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)^{10}}{e^8}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{11}}{e^8}+\frac{4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{12}}{e^8}-\frac{8 c^4 d (d+e x)^{13}}{e^8}+\frac{c^4 (d+e x)^{14}}{e^8}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^4 (d+e x)^7}{7 e^9}-\frac{c d \left (c d^2+a e^2\right )^3 (d+e x)^8}{e^9}+\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}{9 e^9}-\frac{4 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^{10}}{5 e^9}+\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}}{11 e^9}-\frac{2 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{12}}{3 e^9}+\frac{4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{13}}{13 e^9}-\frac{4 c^4 d (d+e x)^{14}}{7 e^9}+\frac{c^4 (d+e x)^{15}}{15 e^9}\\ \end{align*}
Mathematica [A] time = 0.122988, size = 361, normalized size = 1.31 \[ \frac{117 a^2 c^2 x^5 \left (4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+2310 d^5 e x+462 d^6+1386 d e^5 x^5+210 e^6 x^6\right )+715 a^3 c x^3 \left (756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+378 d^5 e x+84 d^6+189 d e^5 x^5+28 e^6 x^6\right )+6435 a^4 x \left (35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+21 d^5 e x+7 d^6+7 d e^5 x^5+e^6 x^6\right )+15 a c^3 x^7 \left (20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+9009 d^5 e x+1716 d^6+6006 d e^5 x^5+924 e^6 x^6\right )+c^4 x^9 \left (61425 d^4 e^2 x^2+75075 d^3 e^3 x^3+51975 d^2 e^4 x^4+27027 d^5 e x+5005 d^6+19305 d e^5 x^5+3003 e^6 x^6\right )}{45045} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 445, normalized size = 1.6 \begin{align*}{\frac{{e}^{6}{c}^{4}{x}^{15}}{15}}+{\frac{3\,d{e}^{5}{c}^{4}{x}^{14}}{7}}+{\frac{ \left ( 4\,{e}^{6}a{c}^{3}+15\,{d}^{2}{e}^{4}{c}^{4} \right ){x}^{13}}{13}}+{\frac{ \left ( 24\,d{e}^{5}a{c}^{3}+20\,{d}^{3}{e}^{3}{c}^{4} \right ){x}^{12}}{12}}+{\frac{ \left ( 6\,{e}^{6}{a}^{2}{c}^{2}+60\,{d}^{2}{e}^{4}a{c}^{3}+15\,{d}^{4}{e}^{2}{c}^{4} \right ){x}^{11}}{11}}+{\frac{ \left ( 36\,d{e}^{5}{a}^{2}{c}^{2}+80\,{d}^{3}{e}^{3}a{c}^{3}+6\,{d}^{5}e{c}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 4\,{e}^{6}{a}^{3}c+90\,{d}^{2}{e}^{4}{a}^{2}{c}^{2}+60\,{d}^{4}{e}^{2}a{c}^{3}+{d}^{6}{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 24\,d{e}^{5}{a}^{3}c+120\,{d}^{3}{e}^{3}{a}^{2}{c}^{2}+24\,{d}^{5}ea{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{6}{a}^{4}+60\,{d}^{2}{e}^{4}{a}^{3}c+90\,{d}^{4}{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{6}a{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,d{e}^{5}{a}^{4}+80\,{d}^{3}{e}^{3}{a}^{3}c+36\,{d}^{5}e{a}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{d}^{2}{e}^{4}{a}^{4}+60\,{d}^{4}{e}^{2}{a}^{3}c+6\,{d}^{6}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{d}^{3}{e}^{3}{a}^{4}+24\,{d}^{5}e{a}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{d}^{4}{e}^{2}{a}^{4}+4\,{d}^{6}{a}^{3}c \right ){x}^{3}}{3}}+3\,{d}^{5}e{a}^{4}{x}^{2}+{d}^{6}{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19209, size = 595, normalized size = 2.16 \begin{align*} \frac{1}{15} \, c^{4} e^{6} x^{15} + \frac{3}{7} \, c^{4} d e^{5} x^{14} + \frac{1}{13} \,{\left (15 \, c^{4} d^{2} e^{4} + 4 \, a c^{3} e^{6}\right )} x^{13} + \frac{1}{3} \,{\left (5 \, c^{4} d^{3} e^{3} + 6 \, a c^{3} d e^{5}\right )} x^{12} + 3 \, a^{4} d^{5} e x^{2} + \frac{3}{11} \,{\left (5 \, c^{4} d^{4} e^{2} + 20 \, a c^{3} d^{2} e^{4} + 2 \, a^{2} c^{2} e^{6}\right )} x^{11} + a^{4} d^{6} x + \frac{1}{5} \,{\left (3 \, c^{4} d^{5} e + 40 \, a c^{3} d^{3} e^{3} + 18 \, a^{2} c^{2} d e^{5}\right )} x^{10} + \frac{1}{9} \,{\left (c^{4} d^{6} + 60 \, a c^{3} d^{4} e^{2} + 90 \, a^{2} c^{2} d^{2} e^{4} + 4 \, a^{3} c e^{6}\right )} x^{9} + 3 \,{\left (a c^{3} d^{5} e + 5 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (4 \, a c^{3} d^{6} + 90 \, a^{2} c^{2} d^{4} e^{2} + 60 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} x^{7} + \frac{1}{3} \,{\left (18 \, a^{2} c^{2} d^{5} e + 40 \, a^{3} c d^{3} e^{3} + 3 \, a^{4} d e^{5}\right )} x^{6} + \frac{3}{5} \,{\left (2 \, a^{2} c^{2} d^{6} + 20 \, a^{3} c d^{4} e^{2} + 5 \, a^{4} d^{2} e^{4}\right )} x^{5} +{\left (6 \, a^{3} c d^{5} e + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (4 \, a^{3} c d^{6} + 15 \, a^{4} d^{4} e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87541, size = 1023, normalized size = 3.71 \begin{align*} \frac{1}{15} x^{15} e^{6} c^{4} + \frac{3}{7} x^{14} e^{5} d c^{4} + \frac{15}{13} x^{13} e^{4} d^{2} c^{4} + \frac{4}{13} x^{13} e^{6} c^{3} a + \frac{5}{3} x^{12} e^{3} d^{3} c^{4} + 2 x^{12} e^{5} d c^{3} a + \frac{15}{11} x^{11} e^{2} d^{4} c^{4} + \frac{60}{11} x^{11} e^{4} d^{2} c^{3} a + \frac{6}{11} x^{11} e^{6} c^{2} a^{2} + \frac{3}{5} x^{10} e d^{5} c^{4} + 8 x^{10} e^{3} d^{3} c^{3} a + \frac{18}{5} x^{10} e^{5} d c^{2} a^{2} + \frac{1}{9} x^{9} d^{6} c^{4} + \frac{20}{3} x^{9} e^{2} d^{4} c^{3} a + 10 x^{9} e^{4} d^{2} c^{2} a^{2} + \frac{4}{9} x^{9} e^{6} c a^{3} + 3 x^{8} e d^{5} c^{3} a + 15 x^{8} e^{3} d^{3} c^{2} a^{2} + 3 x^{8} e^{5} d c a^{3} + \frac{4}{7} x^{7} d^{6} c^{3} a + \frac{90}{7} x^{7} e^{2} d^{4} c^{2} a^{2} + \frac{60}{7} x^{7} e^{4} d^{2} c a^{3} + \frac{1}{7} x^{7} e^{6} a^{4} + 6 x^{6} e d^{5} c^{2} a^{2} + \frac{40}{3} x^{6} e^{3} d^{3} c a^{3} + x^{6} e^{5} d a^{4} + \frac{6}{5} x^{5} d^{6} c^{2} a^{2} + 12 x^{5} e^{2} d^{4} c a^{3} + 3 x^{5} e^{4} d^{2} a^{4} + 6 x^{4} e d^{5} c a^{3} + 5 x^{4} e^{3} d^{3} a^{4} + \frac{4}{3} x^{3} d^{6} c a^{3} + 5 x^{3} e^{2} d^{4} a^{4} + 3 x^{2} e d^{5} a^{4} + x d^{6} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.12627, size = 486, normalized size = 1.76 \begin{align*} a^{4} d^{6} x + 3 a^{4} d^{5} e x^{2} + \frac{3 c^{4} d e^{5} x^{14}}{7} + \frac{c^{4} e^{6} x^{15}}{15} + x^{13} \left (\frac{4 a c^{3} e^{6}}{13} + \frac{15 c^{4} d^{2} e^{4}}{13}\right ) + x^{12} \left (2 a c^{3} d e^{5} + \frac{5 c^{4} d^{3} e^{3}}{3}\right ) + x^{11} \left (\frac{6 a^{2} c^{2} e^{6}}{11} + \frac{60 a c^{3} d^{2} e^{4}}{11} + \frac{15 c^{4} d^{4} e^{2}}{11}\right ) + x^{10} \left (\frac{18 a^{2} c^{2} d e^{5}}{5} + 8 a c^{3} d^{3} e^{3} + \frac{3 c^{4} d^{5} e}{5}\right ) + x^{9} \left (\frac{4 a^{3} c e^{6}}{9} + 10 a^{2} c^{2} d^{2} e^{4} + \frac{20 a c^{3} d^{4} e^{2}}{3} + \frac{c^{4} d^{6}}{9}\right ) + x^{8} \left (3 a^{3} c d e^{5} + 15 a^{2} c^{2} d^{3} e^{3} + 3 a c^{3} d^{5} e\right ) + x^{7} \left (\frac{a^{4} e^{6}}{7} + \frac{60 a^{3} c d^{2} e^{4}}{7} + \frac{90 a^{2} c^{2} d^{4} e^{2}}{7} + \frac{4 a c^{3} d^{6}}{7}\right ) + x^{6} \left (a^{4} d e^{5} + \frac{40 a^{3} c d^{3} e^{3}}{3} + 6 a^{2} c^{2} d^{5} e\right ) + x^{5} \left (3 a^{4} d^{2} e^{4} + 12 a^{3} c d^{4} e^{2} + \frac{6 a^{2} c^{2} d^{6}}{5}\right ) + x^{4} \left (5 a^{4} d^{3} e^{3} + 6 a^{3} c d^{5} e\right ) + x^{3} \left (5 a^{4} d^{4} e^{2} + \frac{4 a^{3} c d^{6}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28685, size = 610, normalized size = 2.21 \begin{align*} \frac{1}{15} \, c^{4} x^{15} e^{6} + \frac{3}{7} \, c^{4} d x^{14} e^{5} + \frac{15}{13} \, c^{4} d^{2} x^{13} e^{4} + \frac{5}{3} \, c^{4} d^{3} x^{12} e^{3} + \frac{15}{11} \, c^{4} d^{4} x^{11} e^{2} + \frac{3}{5} \, c^{4} d^{5} x^{10} e + \frac{1}{9} \, c^{4} d^{6} x^{9} + \frac{4}{13} \, a c^{3} x^{13} e^{6} + 2 \, a c^{3} d x^{12} e^{5} + \frac{60}{11} \, a c^{3} d^{2} x^{11} e^{4} + 8 \, a c^{3} d^{3} x^{10} e^{3} + \frac{20}{3} \, a c^{3} d^{4} x^{9} e^{2} + 3 \, a c^{3} d^{5} x^{8} e + \frac{4}{7} \, a c^{3} d^{6} x^{7} + \frac{6}{11} \, a^{2} c^{2} x^{11} e^{6} + \frac{18}{5} \, a^{2} c^{2} d x^{10} e^{5} + 10 \, a^{2} c^{2} d^{2} x^{9} e^{4} + 15 \, a^{2} c^{2} d^{3} x^{8} e^{3} + \frac{90}{7} \, a^{2} c^{2} d^{4} x^{7} e^{2} + 6 \, a^{2} c^{2} d^{5} x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{6} x^{5} + \frac{4}{9} \, a^{3} c x^{9} e^{6} + 3 \, a^{3} c d x^{8} e^{5} + \frac{60}{7} \, a^{3} c d^{2} x^{7} e^{4} + \frac{40}{3} \, a^{3} c d^{3} x^{6} e^{3} + 12 \, a^{3} c d^{4} x^{5} e^{2} + 6 \, a^{3} c d^{5} x^{4} e + \frac{4}{3} \, a^{3} c d^{6} x^{3} + \frac{1}{7} \, a^{4} x^{7} e^{6} + a^{4} d x^{6} e^{5} + 3 \, a^{4} d^{2} x^{5} e^{4} + 5 \, a^{4} d^{3} x^{4} e^{3} + 5 \, a^{4} d^{4} x^{3} e^{2} + 3 \, a^{4} d^{5} x^{2} e + a^{4} d^{6} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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