Optimal. Leaf size=270 \[ a^4 d^4 x+\frac {2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^4 d e^3 x^4+\frac {1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {8}{3} a^3 c d e^3 x^6+\frac {4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+3 a^2 c^2 d e^3 x^8+\frac {1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac {8}{5} a c^3 d e^3 x^{10}+\frac {2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13}+\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c} \]
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Rubi [A]
time = 0.17, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {710, 1824}
\begin {gather*} a^4 d^4 x+a^4 d e^3 x^4+\frac {2}{3} a^3 d^2 x^3 \left (3 a e^2+2 c d^2\right )+\frac {8}{3} a^3 c d e^3 x^6+\frac {1}{9} c^2 x^9 \left (6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right )+\frac {4}{7} a c x^7 \left (a^2 e^4+9 a c d^2 e^2+c^2 d^4\right )+\frac {1}{5} a^2 x^5 \left (a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right )+3 a^2 c^2 d e^3 x^8+\frac {2}{11} c^3 e^2 x^{11} \left (2 a e^2+3 c d^2\right )+\frac {8}{5} a c^3 d e^3 x^{10}+\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 710
Rule 1824
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx &=\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-4 d^3 e x+(d+e x)^4\right ) \, dx\\ &=\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^4+2 a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^2+4 a^4 d e^3 x^3+a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^4+16 a^3 c d e^3 x^5+4 a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^6+24 a^2 c^2 d e^3 x^7+c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^8+16 a c^3 d e^3 x^9+2 c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{10}+4 c^4 d e^3 x^{11}+c^4 e^4 x^{12}\right ) \, dx\\ &=a^4 d^4 x+\frac {2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^4 d e^3 x^4+\frac {1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {8}{3} a^3 c d e^3 x^6+\frac {4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+3 a^2 c^2 d e^3 x^8+\frac {1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac {8}{5} a c^3 d e^3 x^{10}+\frac {2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13}+\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 300, normalized size = 1.11 \begin {gather*} a^4 d^4 x+2 a^4 d^3 e x^2+\frac {2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^3 d e \left (4 c d^2+a e^2\right ) x^4+\frac {1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {4}{3} a^2 c d e \left (3 c d^2+2 a e^2\right ) x^6+\frac {4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+a c^2 d e \left (2 c d^2+3 a e^2\right ) x^8+\frac {1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac {2}{5} c^3 d e \left (c d^2+4 a e^2\right ) x^{10}+\frac {2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 313, normalized size = 1.16
method | result | size |
norman | \(\frac {c^{4} e^{4} x^{13}}{13}+\frac {c^{4} d \,e^{3} x^{12}}{3}+\left (\frac {4}{11} e^{4} c^{3} a +\frac {6}{11} d^{2} e^{2} c^{4}\right ) x^{11}+\left (\frac {8}{5} d \,e^{3} c^{3} a +\frac {2}{5} d^{3} e \,c^{4}\right ) x^{10}+\left (\frac {2}{3} e^{4} a^{2} c^{2}+\frac {8}{3} d^{2} e^{2} c^{3} a +\frac {1}{9} c^{4} d^{4}\right ) x^{9}+\left (3 d \,e^{3} a^{2} c^{2}+2 d^{3} e \,c^{3} a \right ) x^{8}+\left (\frac {4}{7} e^{4} c \,a^{3}+\frac {36}{7} d^{2} e^{2} a^{2} c^{2}+\frac {4}{7} d^{4} c^{3} a \right ) x^{7}+\left (\frac {8}{3} d \,e^{3} c \,a^{3}+4 d^{3} e \,a^{2} c^{2}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{4}+\frac {24}{5} d^{2} e^{2} c \,a^{3}+\frac {6}{5} d^{4} a^{2} c^{2}\right ) x^{5}+\left (d \,e^{3} a^{4}+4 d^{3} e c \,a^{3}\right ) x^{4}+\left (2 d^{2} e^{2} a^{4}+\frac {4}{3} d^{4} c \,a^{3}\right ) x^{3}+2 d^{3} e \,a^{4} x^{2}+a^{4} d^{4} x\) | \(305\) |
default | \(\frac {c^{4} e^{4} x^{13}}{13}+\frac {c^{4} d \,e^{3} x^{12}}{3}+\frac {\left (4 e^{4} c^{3} a +6 d^{2} e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (16 d \,e^{3} c^{3} a +4 d^{3} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (6 e^{4} a^{2} c^{2}+24 d^{2} e^{2} c^{3} a +c^{4} d^{4}\right ) x^{9}}{9}+\frac {\left (24 d \,e^{3} a^{2} c^{2}+16 d^{3} e \,c^{3} a \right ) x^{8}}{8}+\frac {\left (4 e^{4} c \,a^{3}+36 d^{2} e^{2} a^{2} c^{2}+4 d^{4} c^{3} a \right ) x^{7}}{7}+\frac {\left (16 d \,e^{3} c \,a^{3}+24 d^{3} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (e^{4} a^{4}+24 d^{2} e^{2} c \,a^{3}+6 d^{4} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{4}+16 d^{3} e c \,a^{3}\right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{4}+4 d^{4} c \,a^{3}\right ) x^{3}}{3}+2 d^{3} e \,a^{4} x^{2}+a^{4} d^{4} x\) | \(313\) |
gosper | \(2 d^{3} e \,a^{4} x^{2}+\frac {4}{11} x^{11} e^{4} c^{3} a +\frac {36}{7} x^{7} d^{2} e^{2} a^{2} c^{2}+2 a \,c^{3} d^{3} e \,x^{8}+\frac {8}{3} x^{9} d^{2} e^{2} c^{3} a +4 x^{6} d^{3} e \,a^{2} c^{2}+\frac {24}{5} x^{5} d^{2} e^{2} c \,a^{3}+4 a^{3} c \,d^{3} e \,x^{4}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {1}{9} x^{9} c^{4} d^{4}+\frac {4}{7} x^{7} d^{4} c^{3} a +\frac {8}{3} a^{3} c d \,e^{3} x^{6}+3 a^{2} c^{2} d \,e^{3} x^{8}+\frac {8}{5} a \,c^{3} d \,e^{3} x^{10}+\frac {6}{5} x^{5} d^{4} a^{2} c^{2}+2 x^{3} d^{2} e^{2} a^{4}+\frac {4}{3} x^{3} d^{4} c \,a^{3}+\frac {6}{11} x^{11} d^{2} e^{2} c^{4}+\frac {2}{5} x^{10} d^{3} e \,c^{4}+\frac {2}{3} x^{9} e^{4} a^{2} c^{2}+\frac {4}{7} x^{7} e^{4} c \,a^{3}+a^{4} d^{4} x +a^{4} d \,e^{3} x^{4}+\frac {1}{13} c^{4} e^{4} x^{13}+\frac {1}{3} c^{4} d \,e^{3} x^{12}\) | \(323\) |
risch | \(2 d^{3} e \,a^{4} x^{2}+\frac {4}{11} x^{11} e^{4} c^{3} a +\frac {36}{7} x^{7} d^{2} e^{2} a^{2} c^{2}+2 a \,c^{3} d^{3} e \,x^{8}+\frac {8}{3} x^{9} d^{2} e^{2} c^{3} a +4 x^{6} d^{3} e \,a^{2} c^{2}+\frac {24}{5} x^{5} d^{2} e^{2} c \,a^{3}+4 a^{3} c \,d^{3} e \,x^{4}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {1}{9} x^{9} c^{4} d^{4}+\frac {4}{7} x^{7} d^{4} c^{3} a +\frac {8}{3} a^{3} c d \,e^{3} x^{6}+3 a^{2} c^{2} d \,e^{3} x^{8}+\frac {8}{5} a \,c^{3} d \,e^{3} x^{10}+\frac {6}{5} x^{5} d^{4} a^{2} c^{2}+2 x^{3} d^{2} e^{2} a^{4}+\frac {4}{3} x^{3} d^{4} c \,a^{3}+\frac {6}{11} x^{11} d^{2} e^{2} c^{4}+\frac {2}{5} x^{10} d^{3} e \,c^{4}+\frac {2}{3} x^{9} e^{4} a^{2} c^{2}+\frac {4}{7} x^{7} e^{4} c \,a^{3}+a^{4} d^{4} x +a^{4} d \,e^{3} x^{4}+\frac {1}{13} c^{4} e^{4} x^{13}+\frac {1}{3} c^{4} d \,e^{3} x^{12}\) | \(323\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 296, normalized size = 1.10 \begin {gather*} \frac {1}{13} \, c^{4} x^{13} e^{4} + \frac {1}{3} \, c^{4} d x^{12} e^{3} + \frac {2}{11} \, {\left (3 \, c^{4} d^{2} e^{2} + 2 \, a c^{3} e^{4}\right )} x^{11} + \frac {2}{5} \, {\left (c^{4} d^{3} e + 4 \, a c^{3} d e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{4} + 24 \, a c^{3} d^{2} e^{2} + 6 \, a^{2} c^{2} e^{4}\right )} x^{9} + 2 \, a^{4} d^{3} x^{2} e + a^{4} d^{4} x + {\left (2 \, a c^{3} d^{3} e + 3 \, a^{2} c^{2} d e^{3}\right )} x^{8} + \frac {4}{7} \, {\left (a c^{3} d^{4} + 9 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{7} + \frac {4}{3} \, {\left (3 \, a^{2} c^{2} d^{3} e + 2 \, a^{3} c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{4} + 24 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} x^{5} + {\left (4 \, a^{3} c d^{3} e + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} c d^{4} + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.95, size = 292, normalized size = 1.08 \begin {gather*} \frac {1}{9} \, c^{4} d^{4} x^{9} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + \frac {6}{5} \, a^{2} c^{2} d^{4} x^{5} + \frac {4}{3} \, a^{3} c d^{4} x^{3} + a^{4} d^{4} x + \frac {1}{15015} \, {\left (1155 \, c^{4} x^{13} + 5460 \, a c^{3} x^{11} + 10010 \, a^{2} c^{2} x^{9} + 8580 \, a^{3} c x^{7} + 3003 \, a^{4} x^{5}\right )} e^{4} + \frac {1}{15} \, {\left (5 \, c^{4} d x^{12} + 24 \, a c^{3} d x^{10} + 45 \, a^{2} c^{2} d x^{8} + 40 \, a^{3} c d x^{6} + 15 \, a^{4} d x^{4}\right )} e^{3} + \frac {2}{1155} \, {\left (315 \, c^{4} d^{2} x^{11} + 1540 \, a c^{3} d^{2} x^{9} + 2970 \, a^{2} c^{2} d^{2} x^{7} + 2772 \, a^{3} c d^{2} x^{5} + 1155 \, a^{4} d^{2} x^{3}\right )} e^{2} + \frac {2}{5} \, {\left (c^{4} d^{3} x^{10} + 5 \, a c^{3} d^{3} x^{8} + 10 \, a^{2} c^{2} d^{3} x^{6} + 10 \, a^{3} c d^{3} x^{4} + 5 \, a^{4} d^{3} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 340, normalized size = 1.26 \begin {gather*} a^{4} d^{4} x + 2 a^{4} d^{3} e x^{2} + \frac {c^{4} d e^{3} x^{12}}{3} + \frac {c^{4} e^{4} x^{13}}{13} + x^{11} \cdot \left (\frac {4 a c^{3} e^{4}}{11} + \frac {6 c^{4} d^{2} e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {8 a c^{3} d e^{3}}{5} + \frac {2 c^{4} d^{3} e}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2} c^{2} e^{4}}{3} + \frac {8 a c^{3} d^{2} e^{2}}{3} + \frac {c^{4} d^{4}}{9}\right ) + x^{8} \cdot \left (3 a^{2} c^{2} d e^{3} + 2 a c^{3} d^{3} e\right ) + x^{7} \cdot \left (\frac {4 a^{3} c e^{4}}{7} + \frac {36 a^{2} c^{2} d^{2} e^{2}}{7} + \frac {4 a c^{3} d^{4}}{7}\right ) + x^{6} \cdot \left (\frac {8 a^{3} c d e^{3}}{3} + 4 a^{2} c^{2} d^{3} e\right ) + x^{5} \left (\frac {a^{4} e^{4}}{5} + \frac {24 a^{3} c d^{2} e^{2}}{5} + \frac {6 a^{2} c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 4 a^{3} c d^{3} e\right ) + x^{3} \cdot \left (2 a^{4} d^{2} e^{2} + \frac {4 a^{3} c d^{4}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.82, size = 312, normalized size = 1.16 \begin {gather*} \frac {1}{13} \, c^{4} x^{13} e^{4} + \frac {1}{3} \, c^{4} d x^{12} e^{3} + \frac {6}{11} \, c^{4} d^{2} x^{11} e^{2} + \frac {2}{5} \, c^{4} d^{3} x^{10} e + \frac {1}{9} \, c^{4} d^{4} x^{9} + \frac {4}{11} \, a c^{3} x^{11} e^{4} + \frac {8}{5} \, a c^{3} d x^{10} e^{3} + \frac {8}{3} \, a c^{3} d^{2} x^{9} e^{2} + 2 \, a c^{3} d^{3} x^{8} e + \frac {4}{7} \, a c^{3} d^{4} x^{7} + \frac {2}{3} \, a^{2} c^{2} x^{9} e^{4} + 3 \, a^{2} c^{2} d x^{8} e^{3} + \frac {36}{7} \, a^{2} c^{2} d^{2} x^{7} e^{2} + 4 \, a^{2} c^{2} d^{3} x^{6} e + \frac {6}{5} \, a^{2} c^{2} d^{4} x^{5} + \frac {4}{7} \, a^{3} c x^{7} e^{4} + \frac {8}{3} \, a^{3} c d x^{6} e^{3} + \frac {24}{5} \, a^{3} c d^{2} x^{5} e^{2} + 4 \, a^{3} c d^{3} x^{4} e + \frac {4}{3} \, a^{3} c d^{4} x^{3} + \frac {1}{5} \, a^{4} x^{5} e^{4} + a^{4} d x^{4} e^{3} + 2 \, a^{4} d^{2} x^{3} e^{2} + 2 \, a^{4} d^{3} x^{2} e + a^{4} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 288, normalized size = 1.07 \begin {gather*} x^5\,\left (\frac {a^4\,e^4}{5}+\frac {24\,a^3\,c\,d^2\,e^2}{5}+\frac {6\,a^2\,c^2\,d^4}{5}\right )+x^9\,\left (\frac {2\,a^2\,c^2\,e^4}{3}+\frac {8\,a\,c^3\,d^2\,e^2}{3}+\frac {c^4\,d^4}{9}\right )+x^3\,\left (2\,a^4\,d^2\,e^2+\frac {4\,c\,a^3\,d^4}{3}\right )+x^{11}\,\left (\frac {6\,c^4\,d^2\,e^2}{11}+\frac {4\,a\,c^3\,e^4}{11}\right )+a^4\,d^4\,x+\frac {c^4\,e^4\,x^{13}}{13}+2\,a^4\,d^3\,e\,x^2+\frac {c^4\,d\,e^3\,x^{12}}{3}+\frac {4\,a\,c\,x^7\,\left (a^2\,e^4+9\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{7}+a^3\,d\,e\,x^4\,\left (4\,c\,d^2+a\,e^2\right )+\frac {2\,c^3\,d\,e\,x^{10}\,\left (c\,d^2+4\,a\,e^2\right )}{5}+\frac {4\,a^2\,c\,d\,e\,x^6\,\left (3\,c\,d^2+2\,a\,e^2\right )}{3}+a\,c^2\,d\,e\,x^8\,\left (2\,c\,d^2+3\,a\,e^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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