Optimal. Leaf size=171 \[ 64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac {48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac {32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac {4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \]
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Rubi [A]
time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2086}
\begin {gather*} 64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+64 a c^4 d x^6+\frac {4}{3} c d^2 x^9 \left (a d^2+20 c^3\right )+\frac {32}{7} c^3 x^7 \left (9 a d^2+2 c^3\right )+12 c^2 d x^8 \left (a d^2+2 c^3\right )+\frac {48}{5} a c^2 x^5 \left (a d^2+4 c^3\right )+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 2086
Rubi steps
\begin {align*} \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx &=\int \left (64 a^3 c^3+192 a^2 c^4 x^2+192 a^2 c^3 d x^3+48 a c^2 \left (4 c^3+a d^2\right ) x^4+384 a c^4 d x^5+32 c^3 \left (2 c^3+9 a d^2\right ) x^6+96 c^2 d \left (2 c^3+a d^2\right ) x^7+12 c d^2 \left (20 c^3+a d^2\right ) x^8+160 c^3 d^3 x^9+60 c^2 d^4 x^{10}+12 c d^5 x^{11}+d^6 x^{12}\right ) \, dx\\ &=64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac {48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac {32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac {4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 171, normalized size = 1.00 \begin {gather*} 64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac {48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac {32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac {4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac {60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac {d^6 x^{13}}{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 231, normalized size = 1.35
method | result | size |
norman | \(\frac {d^{6} x^{13}}{13}+c \,d^{5} x^{12}+\frac {60 c^{2} d^{4} x^{11}}{11}+16 c^{3} d^{3} x^{10}+\left (\frac {4}{3} a c \,d^{4}+\frac {80}{3} c^{4} d^{2}\right ) x^{9}+\left (12 a \,c^{2} d^{3}+24 c^{5} d \right ) x^{8}+\left (\frac {288}{7} a \,c^{3} d^{2}+\frac {64}{7} c^{6}\right ) x^{7}+64 a \,c^{4} d \,x^{6}+\left (\frac {48}{5} a^{2} c^{2} d^{2}+\frac {192}{5} a \,c^{5}\right ) x^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\) | \(163\) |
gosper | \(\frac {1}{13} d^{6} x^{13}+c \,d^{5} x^{12}+\frac {60}{11} c^{2} d^{4} x^{11}+16 c^{3} d^{3} x^{10}+\frac {4}{3} x^{9} a c \,d^{4}+\frac {80}{3} x^{9} c^{4} d^{2}+12 a \,c^{2} d^{3} x^{8}+24 c^{5} d \,x^{8}+\frac {288}{7} x^{7} a \,c^{3} d^{2}+\frac {64}{7} x^{7} c^{6}+64 a \,c^{4} d \,x^{6}+\frac {48}{5} x^{5} a^{2} c^{2} d^{2}+\frac {192}{5} x^{5} a \,c^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\) | \(167\) |
risch | \(\frac {1}{13} d^{6} x^{13}+c \,d^{5} x^{12}+\frac {60}{11} c^{2} d^{4} x^{11}+16 c^{3} d^{3} x^{10}+\frac {4}{3} x^{9} a c \,d^{4}+\frac {80}{3} x^{9} c^{4} d^{2}+12 a \,c^{2} d^{3} x^{8}+24 c^{5} d \,x^{8}+\frac {288}{7} x^{7} a \,c^{3} d^{2}+\frac {64}{7} x^{7} c^{6}+64 a \,c^{4} d \,x^{6}+\frac {48}{5} x^{5} a^{2} c^{2} d^{2}+\frac {192}{5} x^{5} a \,c^{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\) | \(167\) |
default | \(\frac {d^{6} x^{13}}{13}+c \,d^{5} x^{12}+\frac {60 c^{2} d^{4} x^{11}}{11}+16 c^{3} d^{3} x^{10}+\frac {\left (4 a c \,d^{4}+224 c^{4} d^{2}+d^{2} \left (8 a c \,d^{2}+16 c^{4}\right )\right ) x^{9}}{9}+\frac {\left (64 a \,c^{2} d^{3}+128 c^{5} d +4 c d \left (8 a c \,d^{2}+16 c^{4}\right )\right ) x^{8}}{8}+\frac {\left (256 a \,c^{3} d^{2}+4 c^{2} \left (8 a c \,d^{2}+16 c^{4}\right )\right ) x^{7}}{7}+64 a \,c^{4} d \,x^{6}+\frac {\left (4 a c \left (8 a c \,d^{2}+16 c^{4}\right )+128 a \,c^{5}+16 a^{2} c^{2} d^{2}\right ) x^{5}}{5}+48 a^{2} c^{3} d \,x^{4}+64 a^{2} c^{4} x^{3}+64 a^{3} c^{3} x\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 205, normalized size = 1.20 \begin {gather*} \frac {1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac {48}{11} \, c^{2} d^{4} x^{11} + \frac {32}{5} \, c^{3} d^{3} x^{10} + \frac {64}{7} \, c^{6} x^{7} + 64 \, a^{3} c^{3} x + \frac {16}{5} \, {\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a^{2} c^{2} + \frac {8}{3} \, {\left (2 \, d^{2} x^{9} + 9 \, c d x^{8}\right )} c^{4} + \frac {4}{105} \, {\left (35 \, d^{4} x^{9} + 315 \, c d^{3} x^{8} + 720 \, c^{2} d^{2} x^{7} + 1008 \, c^{4} x^{5} + 120 \, {\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2}\right )} a c + \frac {4}{165} \, {\left (45 \, d^{4} x^{11} + 396 \, c d^{3} x^{10} + 880 \, c^{2} d^{2} x^{9}\right )} c^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 163, normalized size = 0.95 \begin {gather*} \frac {1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac {60}{11} \, c^{2} d^{4} x^{11} + 16 \, c^{3} d^{3} x^{10} + 64 \, a c^{4} d x^{6} + 48 \, a^{2} c^{3} d x^{4} + \frac {4}{3} \, {\left (20 \, c^{4} d^{2} + a c d^{4}\right )} x^{9} + 64 \, a^{2} c^{4} x^{3} + 12 \, {\left (2 \, c^{5} d + a c^{2} d^{3}\right )} x^{8} + \frac {32}{7} \, {\left (2 \, c^{6} + 9 \, a c^{3} d^{2}\right )} x^{7} + 64 \, a^{3} c^{3} x + \frac {48}{5} \, {\left (4 \, a c^{5} + a^{2} c^{2} d^{2}\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 180, normalized size = 1.05 \begin {gather*} 64 a^{3} c^{3} x + 64 a^{2} c^{4} x^{3} + 48 a^{2} c^{3} d x^{4} + 64 a c^{4} d x^{6} + 16 c^{3} d^{3} x^{10} + \frac {60 c^{2} d^{4} x^{11}}{11} + c d^{5} x^{12} + \frac {d^{6} x^{13}}{13} + x^{9} \cdot \left (\frac {4 a c d^{4}}{3} + \frac {80 c^{4} d^{2}}{3}\right ) + x^{8} \cdot \left (12 a c^{2} d^{3} + 24 c^{5} d\right ) + x^{7} \cdot \left (\frac {288 a c^{3} d^{2}}{7} + \frac {64 c^{6}}{7}\right ) + x^{5} \cdot \left (\frac {48 a^{2} c^{2} d^{2}}{5} + \frac {192 a c^{5}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.39, size = 166, normalized size = 0.97 \begin {gather*} \frac {1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac {60}{11} \, c^{2} d^{4} x^{11} + 16 \, c^{3} d^{3} x^{10} + \frac {80}{3} \, c^{4} d^{2} x^{9} + \frac {4}{3} \, a c d^{4} x^{9} + 24 \, c^{5} d x^{8} + 12 \, a c^{2} d^{3} x^{8} + \frac {64}{7} \, c^{6} x^{7} + \frac {288}{7} \, a c^{3} d^{2} x^{7} + 64 \, a c^{4} d x^{6} + \frac {192}{5} \, a c^{5} x^{5} + \frac {48}{5} \, a^{2} c^{2} d^{2} x^{5} + 48 \, a^{2} c^{3} d x^{4} + 64 \, a^{2} c^{4} x^{3} + 64 \, a^{3} c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.16, size = 160, normalized size = 0.94 \begin {gather*} x^8\,\left (24\,c^5\,d+12\,a\,c^2\,d^3\right )+x^9\,\left (\frac {80\,c^4\,d^2}{3}+\frac {4\,a\,c\,d^4}{3}\right )+\frac {d^6\,x^{13}}{13}+x^7\,\left (\frac {64\,c^6}{7}+\frac {288\,a\,c^3\,d^2}{7}\right )+64\,a^3\,c^3\,x+c\,d^5\,x^{12}+64\,a^2\,c^4\,x^3+16\,c^3\,d^3\,x^{10}+\frac {60\,c^2\,d^4\,x^{11}}{11}+48\,a^2\,c^3\,d\,x^4+\frac {48\,a\,c^2\,x^5\,\left (4\,c^3+a\,d^2\right )}{5}+64\,a\,c^4\,d\,x^6 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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