Optimal. Leaf size=89 \[ \frac {a^2 (d+e x)^4}{4 e}+\frac {\left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac {a b (d+e x)^6}{3 e}+\frac {b c (d+e x)^{10}}{5 e}+\frac {c^2 (d+e x)^{12}}{12 e} \]
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Rubi [A] time = 0.19, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1142, 1114, 631} \[ \frac {a^2 (d+e x)^4}{4 e}+\frac {\left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac {a b (d+e x)^6}{3 e}+\frac {b c (d+e x)^{10}}{5 e}+\frac {c^2 (d+e x)^{12}}{12 e} \]
Antiderivative was successfully verified.
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Rule 631
Rule 1114
Rule 1142
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right )^2 \, dx,x,d+e x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int x \left (a+b x+c x^2\right )^2 \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 x+2 a b x^2+\left (b^2+2 a c\right ) x^3+2 b c x^4+c^2 x^5\right ) \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {a^2 (d+e x)^4}{4 e}+\frac {a b (d+e x)^6}{3 e}+\frac {\left (b^2+2 a c\right ) (d+e x)^8}{8 e}+\frac {b c (d+e x)^{10}}{5 e}+\frac {c^2 (d+e x)^{12}}{12 e}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 401, normalized size = 4.51 \[ \frac {1}{4} e^3 x^4 \left (a^2+20 a b d^2+70 a c d^4+35 b^2 d^4+168 b c d^6+165 c^2 d^8\right )+\frac {1}{3} d e^2 x^3 \left (3 a^2+20 a b d^2+42 a c d^4+21 b^2 d^4+72 b c d^6+55 c^2 d^8\right )+\frac {1}{2} d^2 e x^2 \left (3 a^2+10 a b d^2+14 a c d^4+7 b^2 d^4+18 b c d^6+11 c^2 d^8\right )+\frac {1}{8} e^7 x^8 \left (2 a c+b^2+72 b c d^2+330 c^2 d^4\right )+d e^6 x^7 \left (2 a c+b^2+24 b c d^2+66 c^2 d^4\right )+\frac {1}{6} e^5 x^6 \left (2 a b+42 a c d^2+21 b^2 d^2+252 b c d^4+462 c^2 d^6\right )+\frac {1}{5} d e^4 x^5 \left (10 a b+70 a c d^2+35 b^2 d^2+252 b c d^4+330 c^2 d^6\right )+d^3 x \left (a+b d^2+c d^4\right )^2+\frac {1}{10} c e^9 x^{10} \left (2 b+55 c d^2\right )+\frac {1}{3} c d e^8 x^9 \left (6 b+55 c d^2\right )+c^2 d e^{10} x^{11}+\frac {1}{12} c^2 e^{11} x^{12} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 571, normalized size = 6.42 \[ \frac {1}{12} x^{12} e^{11} c^{2} + x^{11} e^{10} d c^{2} + \frac {11}{2} x^{10} e^{9} d^{2} c^{2} + \frac {55}{3} x^{9} e^{8} d^{3} c^{2} + \frac {165}{4} x^{8} e^{7} d^{4} c^{2} + \frac {1}{5} x^{10} e^{9} c b + 66 x^{7} e^{6} d^{5} c^{2} + 2 x^{9} e^{8} d c b + 77 x^{6} e^{5} d^{6} c^{2} + 9 x^{8} e^{7} d^{2} c b + 66 x^{5} e^{4} d^{7} c^{2} + 24 x^{7} e^{6} d^{3} c b + \frac {165}{4} x^{4} e^{3} d^{8} c^{2} + 42 x^{6} e^{5} d^{4} c b + \frac {1}{8} x^{8} e^{7} b^{2} + \frac {1}{4} x^{8} e^{7} c a + \frac {55}{3} x^{3} e^{2} d^{9} c^{2} + \frac {252}{5} x^{5} e^{4} d^{5} c b + x^{7} e^{6} d b^{2} + 2 x^{7} e^{6} d c a + \frac {11}{2} x^{2} e d^{10} c^{2} + 42 x^{4} e^{3} d^{6} c b + \frac {7}{2} x^{6} e^{5} d^{2} b^{2} + 7 x^{6} e^{5} d^{2} c a + x d^{11} c^{2} + 24 x^{3} e^{2} d^{7} c b + 7 x^{5} e^{4} d^{3} b^{2} + 14 x^{5} e^{4} d^{3} c a + 9 x^{2} e d^{8} c b + \frac {35}{4} x^{4} e^{3} d^{4} b^{2} + \frac {35}{2} x^{4} e^{3} d^{4} c a + \frac {1}{3} x^{6} e^{5} b a + 2 x d^{9} c b + 7 x^{3} e^{2} d^{5} b^{2} + 14 x^{3} e^{2} d^{5} c a + 2 x^{5} e^{4} d b a + \frac {7}{2} x^{2} e d^{6} b^{2} + 7 x^{2} e d^{6} c a + 5 x^{4} e^{3} d^{2} b a + x d^{7} b^{2} + 2 x d^{7} c a + \frac {20}{3} x^{3} e^{2} d^{3} b a + 5 x^{2} e d^{4} b a + \frac {1}{4} x^{4} e^{3} a^{2} + 2 x d^{5} b a + x^{3} e^{2} d a^{2} + \frac {3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 493, normalized size = 5.54 \[ \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} c^{2} d^{10} + \frac {5}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} c^{2} d^{8} e + \frac {5}{3} \, {\left (x^{2} e + 2 \, d x\right )}^{3} c^{2} d^{6} e^{2} + {\left (x^{2} e + 2 \, d x\right )} b c d^{8} + \frac {5}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{4} c^{2} d^{4} e^{3} + 2 \, {\left (x^{2} e + 2 \, d x\right )}^{2} b c d^{6} e + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{5} c^{2} d^{2} e^{4} + 2 \, {\left (x^{2} e + 2 \, d x\right )}^{3} b c d^{4} e^{2} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} b^{2} d^{6} + {\left (x^{2} e + 2 \, d x\right )} a c d^{6} + \frac {1}{12} \, {\left (x^{2} e + 2 \, d x\right )}^{6} c^{2} e^{5} + {\left (x^{2} e + 2 \, d x\right )}^{4} b c d^{2} e^{3} + \frac {3}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} b^{2} d^{4} e + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a c d^{4} e + \frac {1}{5} \, {\left (x^{2} e + 2 \, d x\right )}^{5} b c e^{4} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} b^{2} d^{2} e^{2} + {\left (x^{2} e + 2 \, d x\right )}^{3} a c d^{2} e^{2} + {\left (x^{2} e + 2 \, d x\right )} a b d^{4} + \frac {1}{8} \, {\left (x^{2} e + 2 \, d x\right )}^{4} b^{2} e^{3} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{4} a c e^{3} + {\left (x^{2} e + 2 \, d x\right )}^{2} a b d^{2} e + \frac {1}{3} \, {\left (x^{2} e + 2 \, d x\right )}^{3} a b e^{2} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} a^{2} d^{2} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a^{2} e \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 1314, normalized size = 14.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 403, normalized size = 4.53 \[ \frac {1}{12} \, c^{2} e^{11} x^{12} + c^{2} d e^{10} x^{11} + \frac {1}{10} \, {\left (55 \, c^{2} d^{2} + 2 \, b c\right )} e^{9} x^{10} + \frac {1}{3} \, {\left (55 \, c^{2} d^{3} + 6 \, b c d\right )} e^{8} x^{9} + \frac {1}{8} \, {\left (330 \, c^{2} d^{4} + 72 \, b c d^{2} + b^{2} + 2 \, a c\right )} e^{7} x^{8} + {\left (66 \, c^{2} d^{5} + 24 \, b c d^{3} + {\left (b^{2} + 2 \, a c\right )} d\right )} e^{6} x^{7} + \frac {1}{6} \, {\left (462 \, c^{2} d^{6} + 252 \, b c d^{4} + 21 \, {\left (b^{2} + 2 \, a c\right )} d^{2} + 2 \, a b\right )} e^{5} x^{6} + \frac {1}{5} \, {\left (330 \, c^{2} d^{7} + 252 \, b c d^{5} + 35 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 10 \, a b d\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (165 \, c^{2} d^{8} + 168 \, b c d^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} d^{4} + 20 \, a b d^{2} + a^{2}\right )} e^{3} x^{4} + \frac {1}{3} \, {\left (55 \, c^{2} d^{9} + 72 \, b c d^{7} + 21 \, {\left (b^{2} + 2 \, a c\right )} d^{5} + 20 \, a b d^{3} + 3 \, a^{2} d\right )} e^{2} x^{3} + \frac {1}{2} \, {\left (11 \, c^{2} d^{10} + 18 \, b c d^{8} + 7 \, {\left (b^{2} + 2 \, a c\right )} d^{6} + 10 \, a b d^{4} + 3 \, a^{2} d^{2}\right )} e x^{2} + {\left (c^{2} d^{11} + 2 \, b c d^{9} + {\left (b^{2} + 2 \, a c\right )} d^{7} + 2 \, a b d^{5} + a^{2} d^{3}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 383, normalized size = 4.30 \[ \frac {e^7\,x^8\,\left (b^2+72\,b\,c\,d^2+330\,c^2\,d^4+2\,a\,c\right )}{8}+\frac {e^5\,x^6\,\left (21\,b^2\,d^2+252\,b\,c\,d^4+2\,a\,b+462\,c^2\,d^6+42\,a\,c\,d^2\right )}{6}+\frac {e^3\,x^4\,\left (a^2+20\,a\,b\,d^2+70\,a\,c\,d^4+35\,b^2\,d^4+168\,b\,c\,d^6+165\,c^2\,d^8\right )}{4}+\frac {c^2\,e^{11}\,x^{12}}{12}+d^3\,x\,{\left (c\,d^4+b\,d^2+a\right )}^2+\frac {c\,e^9\,x^{10}\,\left (55\,c\,d^2+2\,b\right )}{10}+c^2\,d\,e^{10}\,x^{11}+\frac {d^2\,e\,x^2\,\left (3\,a^2+10\,a\,b\,d^2+14\,a\,c\,d^4+7\,b^2\,d^4+18\,b\,c\,d^6+11\,c^2\,d^8\right )}{2}+\frac {d\,e^2\,x^3\,\left (3\,a^2+20\,a\,b\,d^2+42\,a\,c\,d^4+21\,b^2\,d^4+72\,b\,c\,d^6+55\,c^2\,d^8\right )}{3}+d\,e^6\,x^7\,\left (b^2+24\,b\,c\,d^2+66\,c^2\,d^4+2\,a\,c\right )+\frac {d\,e^4\,x^5\,\left (35\,b^2\,d^2+252\,b\,c\,d^4+10\,a\,b+330\,c^2\,d^6+70\,a\,c\,d^2\right )}{5}+\frac {c\,d\,e^8\,x^9\,\left (55\,c\,d^2+6\,b\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 559, normalized size = 6.28 \[ c^{2} d e^{10} x^{11} + \frac {c^{2} e^{11} x^{12}}{12} + x^{10} \left (\frac {b c e^{9}}{5} + \frac {11 c^{2} d^{2} e^{9}}{2}\right ) + x^{9} \left (2 b c d e^{8} + \frac {55 c^{2} d^{3} e^{8}}{3}\right ) + x^{8} \left (\frac {a c e^{7}}{4} + \frac {b^{2} e^{7}}{8} + 9 b c d^{2} e^{7} + \frac {165 c^{2} d^{4} e^{7}}{4}\right ) + x^{7} \left (2 a c d e^{6} + b^{2} d e^{6} + 24 b c d^{3} e^{6} + 66 c^{2} d^{5} e^{6}\right ) + x^{6} \left (\frac {a b e^{5}}{3} + 7 a c d^{2} e^{5} + \frac {7 b^{2} d^{2} e^{5}}{2} + 42 b c d^{4} e^{5} + 77 c^{2} d^{6} e^{5}\right ) + x^{5} \left (2 a b d e^{4} + 14 a c d^{3} e^{4} + 7 b^{2} d^{3} e^{4} + \frac {252 b c d^{5} e^{4}}{5} + 66 c^{2} d^{7} e^{4}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + 5 a b d^{2} e^{3} + \frac {35 a c d^{4} e^{3}}{2} + \frac {35 b^{2} d^{4} e^{3}}{4} + 42 b c d^{6} e^{3} + \frac {165 c^{2} d^{8} e^{3}}{4}\right ) + x^{3} \left (a^{2} d e^{2} + \frac {20 a b d^{3} e^{2}}{3} + 14 a c d^{5} e^{2} + 7 b^{2} d^{5} e^{2} + 24 b c d^{7} e^{2} + \frac {55 c^{2} d^{9} e^{2}}{3}\right ) + x^{2} \left (\frac {3 a^{2} d^{2} e}{2} + 5 a b d^{4} e + 7 a c d^{6} e + \frac {7 b^{2} d^{6} e}{2} + 9 b c d^{8} e + \frac {11 c^{2} d^{10} e}{2}\right ) + x \left (a^{2} d^{3} + 2 a b d^{5} + 2 a c d^{7} + b^{2} d^{7} + 2 b c d^{9} + c^{2} d^{11}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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