Optimal. Leaf size=92 \[ 16 a^2 c^2 x+\frac {8}{5} c x^5 \left (a d^2+2 c^3\right )+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9} \]
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Rubi [A] time = 0.04, antiderivative size = 92, 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 = {2061} \[ 16 a^2 c^2 x+\frac {8}{5} c x^5 \left (a d^2+2 c^3\right )+8 a c^2 d x^4+\frac {32}{3} a c^3 x^3+\frac {24}{7} c^2 d^2 x^7+\frac {16}{3} c^3 d x^6+c d^3 x^8+\frac {d^4 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 2061
Rubi steps
\begin {align*} \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx &=\int \left (16 a^2 c^2+32 a c^3 x^2+32 a c^2 d x^3+8 c \left (2 c^3+a d^2\right ) x^4+32 c^3 d x^5+24 c^2 d^2 x^6+8 c d^3 x^7+d^4 x^8\right ) \, dx\\ &=16 a^2 c^2 x+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {8}{5} c \left (2 c^3+a d^2\right ) x^5+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 92, normalized size = 1.00 \[ 16 a^2 c^2 x+\frac {8}{5} c x^5 \left (a d^2+2 c^3\right )+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 83, normalized size = 0.90 \[ \frac {1}{9} x^{9} d^{4} + x^{8} d^{3} c + \frac {24}{7} x^{7} d^{2} c^{2} + \frac {16}{3} x^{6} d c^{3} + \frac {16}{5} x^{5} c^{4} + \frac {8}{5} x^{5} d^{2} c a + 8 x^{4} d c^{2} a + \frac {32}{3} x^{3} c^{3} a + 16 x c^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 83, normalized size = 0.90 \[ \frac {1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac {24}{7} \, c^{2} d^{2} x^{7} + \frac {16}{3} \, c^{3} d x^{6} + \frac {16}{5} \, c^{4} x^{5} + \frac {8}{5} \, a c d^{2} x^{5} + 8 \, a c^{2} d x^{4} + \frac {32}{3} \, a c^{3} x^{3} + 16 \, a^{2} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 84, normalized size = 0.91 \[ \frac {d^{4} x^{9}}{9}+c \,d^{3} x^{8}+\frac {24 c^{2} d^{2} x^{7}}{7}+\frac {16 c^{3} d \,x^{6}}{3}+8 a \,c^{2} d \,x^{4}+\frac {32 a \,c^{3} x^{3}}{3}+16 a^{2} c^{2} x +\frac {\left (8 a c \,d^{2}+16 c^{4}\right ) x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 94, normalized size = 1.02 \[ \frac {1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac {16}{7} \, c^{2} d^{2} x^{7} + \frac {16}{5} \, c^{4} x^{5} + 16 \, a^{2} c^{2} x + \frac {8}{15} \, {\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a c + \frac {8}{21} \, {\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 82, normalized size = 0.89 \[ x^5\,\left (\frac {16\,c^4}{5}+\frac {8\,a\,c\,d^2}{5}\right )+\frac {d^4\,x^9}{9}+16\,a^2\,c^2\,x+\frac {32\,a\,c^3\,x^3}{3}+\frac {16\,c^3\,d\,x^6}{3}+c\,d^3\,x^8+\frac {24\,c^2\,d^2\,x^7}{7}+8\,a\,c^2\,d\,x^4 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 95, normalized size = 1.03 \[ 16 a^{2} c^{2} x + \frac {32 a c^{3} x^{3}}{3} + 8 a c^{2} d x^{4} + \frac {16 c^{3} d x^{6}}{3} + \frac {24 c^{2} d^{2} x^{7}}{7} + c d^{3} x^{8} + \frac {d^{4} x^{9}}{9} + x^{5} \left (\frac {8 a c d^{2}}{5} + \frac {16 c^{4}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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