Optimal. Leaf size=132 \[ a^4 d^2 x+\frac {1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+\frac {2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac {1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac {2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^5}{5 c}+\frac {1}{11} c^4 e^2 x^{11} \]
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Rubi [A] time = 0.10, antiderivative size = 132, 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 = {696, 1810} \begin {gather*} \frac {2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac {1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+a^4 d^2 x+\frac {1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac {2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^5}{5 c}+\frac {1}{11} c^4 e^2 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 696
Rule 1810
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx &=\frac {d e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac {d e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^2+a^3 \left (4 c d^2+a e^2\right ) x^2+2 a^2 c \left (3 c d^2+2 a e^2\right ) x^4+2 a c^2 \left (2 c d^2+3 a e^2\right ) x^6+c^3 \left (c d^2+4 a e^2\right ) x^8+c^4 e^2 x^{10}\right ) \, dx\\ &=a^4 d^2 x+\frac {1}{3} a^3 \left (4 c d^2+a e^2\right ) x^3+\frac {2}{5} a^2 c \left (3 c d^2+2 a e^2\right ) x^5+\frac {2}{7} a c^2 \left (2 c d^2+3 a e^2\right ) x^7+\frac {1}{9} c^3 \left (c d^2+4 a e^2\right ) x^9+\frac {1}{11} c^4 e^2 x^{11}+\frac {d e \left (a+c x^2\right )^5}{5 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 148, normalized size = 1.12 \begin {gather*} a^4 \left (d^2 x+d e x^2+\frac {e^2 x^3}{3}\right )+\frac {2}{15} a^3 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+\frac {2}{35} a^2 c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac {1}{63} a c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right )+\frac {1}{495} c^4 x^9 \left (55 d^2+99 d e x+45 e^2 x^2\right ) \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)^2 \left (a+c x^2\right )^4 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.34, size = 171, normalized size = 1.30 \begin {gather*} \frac {1}{11} x^{11} e^{2} c^{4} + \frac {1}{5} x^{10} e d c^{4} + \frac {1}{9} x^{9} d^{2} c^{4} + \frac {4}{9} x^{9} e^{2} c^{3} a + x^{8} e d c^{3} a + \frac {4}{7} x^{7} d^{2} c^{3} a + \frac {6}{7} x^{7} e^{2} c^{2} a^{2} + 2 x^{6} e d c^{2} a^{2} + \frac {6}{5} x^{5} d^{2} c^{2} a^{2} + \frac {4}{5} x^{5} e^{2} c a^{3} + 2 x^{4} e d c a^{3} + \frac {4}{3} x^{3} d^{2} c a^{3} + \frac {1}{3} x^{3} e^{2} a^{4} + x^{2} e d a^{4} + x d^{2} a^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 171, normalized size = 1.30 \begin {gather*} \frac {1}{11} \, c^{4} x^{11} e^{2} + \frac {1}{5} \, c^{4} d x^{10} e + \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {4}{9} \, a c^{3} x^{9} e^{2} + a c^{3} d x^{8} e + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {6}{7} \, a^{2} c^{2} x^{7} e^{2} + 2 \, a^{2} c^{2} d x^{6} e + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {4}{5} \, a^{3} c x^{5} e^{2} + 2 \, a^{3} c d x^{4} e + \frac {4}{3} \, a^{3} c d^{2} x^{3} + \frac {1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 170, normalized size = 1.29 \begin {gather*} \frac {c^{4} e^{2} x^{11}}{11}+\frac {c^{4} d e \,x^{10}}{5}+a \,c^{3} d e \,x^{8}+2 a^{2} c^{2} d e \,x^{6}+2 a^{3} c d e \,x^{4}+\frac {\left (4 e^{2} a \,c^{3}+c^{4} d^{2}\right ) x^{9}}{9}+a^{4} d e \,x^{2}+a^{4} d^{2} x +\frac {\left (6 e^{2} a^{2} c^{2}+4 d^{2} a \,c^{3}\right ) x^{7}}{7}+\frac {\left (4 e^{2} a^{3} c +6 d^{2} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (e^{2} a^{4}+4 d^{2} 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 = 1.34, size = 169, normalized size = 1.28 \begin {gather*} \frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, c^{4} d e x^{10} + a c^{3} d e x^{8} + 2 \, a^{2} c^{2} d e x^{6} + 2 \, a^{3} c d e x^{4} + \frac {1}{9} \, {\left (c^{4} d^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + a^{4} d e x^{2} + \frac {2}{7} \, {\left (2 \, a c^{3} d^{2} + 3 \, a^{2} c^{2} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac {2}{5} \, {\left (3 \, a^{2} c^{2} d^{2} + 2 \, a^{3} c e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{2} + a^{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 = 0.07, size = 161, normalized size = 1.22 \begin {gather*} x^3\,\left (\frac {a^4\,e^2}{3}+\frac {4\,c\,a^3\,d^2}{3}\right )+x^9\,\left (\frac {c^4\,d^2}{9}+\frac {4\,a\,c^3\,e^2}{9}\right )+a^4\,d^2\,x+\frac {c^4\,e^2\,x^{11}}{11}+a^4\,d\,e\,x^2+\frac {c^4\,d\,e\,x^{10}}{5}+\frac {2\,a^2\,c\,x^5\,\left (3\,c\,d^2+2\,a\,e^2\right )}{5}+\frac {2\,a\,c^2\,x^7\,\left (2\,c\,d^2+3\,a\,e^2\right )}{7}+2\,a^3\,c\,d\,e\,x^4+a\,c^3\,d\,e\,x^8+2\,a^2\,c^2\,d\,e\,x^6 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 187, normalized size = 1.42 \begin {gather*} a^{4} d^{2} x + a^{4} d e x^{2} + 2 a^{3} c d e x^{4} + 2 a^{2} c^{2} d e x^{6} + a c^{3} d e x^{8} + \frac {c^{4} d e x^{10}}{5} + \frac {c^{4} e^{2} x^{11}}{11} + x^{9} \left (\frac {4 a c^{3} e^{2}}{9} + \frac {c^{4} d^{2}}{9}\right ) + x^{7} \left (\frac {6 a^{2} c^{2} e^{2}}{7} + \frac {4 a c^{3} d^{2}}{7}\right ) + x^{5} \left (\frac {4 a^{3} c e^{2}}{5} + \frac {6 a^{2} c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {4 a^{3} c d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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