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 = 0.46, antiderivative size = 276, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.12, size = 361, normalized size = 1.31 \begin {gather*} \frac {6435 a^4 x \left (7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right )+715 a^3 c x^3 \left (84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right )+117 a^2 c^2 x^5 \left (462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right )+15 a c^3 x^7 \left (1716 d^6+9009 d^5 e x+20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+6006 d e^5 x^5+924 e^6 x^6\right )+c^4 x^9 \left (5005 d^6+27027 d^5 e x+61425 d^4 e^2 x^2+75075 d^3 e^3 x^3+51975 d^2 e^4 x^4+19305 d e^5 x^5+3003 e^6 x^6\right )}{45045} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 472, normalized size = 1.71 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 452, normalized size = 1.64 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 445, normalized size = 1.61 \begin {gather*} \frac {c^{4} e^{6} x^{15}}{15}+\frac {3 c^{4} d \,e^{5} x^{14}}{7}+\frac {\left (4 e^{6} a \,c^{3}+15 d^{2} e^{4} c^{4}\right ) x^{13}}{13}+3 a^{4} d^{5} e \,x^{2}+\frac {\left (24 d \,e^{5} a \,c^{3}+20 d^{3} e^{3} c^{4}\right ) x^{12}}{12}+a^{4} d^{6} x +\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} e a \,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 441, normalized size = 1.60 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 425, normalized size = 1.54 \begin {gather*} 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^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^3\,\left (5\,a^4\,d^4\,e^2+\frac {4\,c\,a^3\,d^6}{3}\right )+x^{13}\,\left (\frac {15\,c^4\,d^2\,e^4}{13}+\frac {4\,a\,c^3\,e^6}{13}\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^{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 )+a^4\,d^6\,x+\frac {c^4\,e^6\,x^{15}}{15}+3\,a^4\,d^5\,e\,x^2+\frac {3\,c^4\,d\,e^5\,x^{14}}{7}+a^3\,d^3\,e\,x^4\,\left (6\,c\,d^2+5\,a\,e^2\right )+\frac {c^3\,d\,e^3\,x^{12}\,\left (5\,c\,d^2+6\,a\,e^2\right )}{3}+\frac {a^2\,d\,e\,x^6\,\left (3\,a^2\,e^4+40\,a\,c\,d^2\,e^2+18\,c^2\,d^4\right )}{3}+\frac {c^2\,d\,e\,x^{10}\,\left (18\,a^2\,e^4+40\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{5}+3\,a\,c\,d\,e\,x^8\,\left (a^2\,e^4+5\,a\,c\,d^2\,e^2+c^2\,d^4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 486, normalized size = 1.76 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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