Optimal. Leaf size=119 \[ \frac {4 e^3 (a+b x)^{11} (b d-a e)}{11 b^5}+\frac {3 e^2 (a+b x)^{10} (b d-a e)^2}{5 b^5}+\frac {4 e (a+b x)^9 (b d-a e)^3}{9 b^5}+\frac {(a+b x)^8 (b d-a e)^4}{8 b^5}+\frac {e^4 (a+b x)^{12}}{12 b^5} \]
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Rubi [A] time = 0.28, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} \frac {4 e^3 (a+b x)^{11} (b d-a e)}{11 b^5}+\frac {3 e^2 (a+b x)^{10} (b d-a e)^2}{5 b^5}+\frac {4 e (a+b x)^9 (b d-a e)^3}{9 b^5}+\frac {(a+b x)^8 (b d-a e)^4}{8 b^5}+\frac {e^4 (a+b x)^{12}}{12 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^4 \, dx\\ &=\int \left (\frac {(b d-a e)^4 (a+b x)^7}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^8}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^9}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^{10}}{b^4}+\frac {e^4 (a+b x)^{11}}{b^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (a+b x)^8}{8 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^9}{9 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^{10}}{5 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^{11}}{11 b^5}+\frac {e^4 (a+b x)^{12}}{12 b^5}\\ \end {align*}
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Mathematica [B] time = 0.12, size = 405, normalized size = 3.40 \begin {gather*} \frac {x \left (792 a^7 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+924 a^6 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+792 a^5 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+495 a^4 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+220 a^3 b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+66 a^2 b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )+12 a b^6 x^6 \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )+b^7 x^7 \left (495 d^4+1760 d^3 e x+2376 d^2 e^2 x^2+1440 d e^3 x^3+330 e^4 x^4\right )\right )}{3960} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.37, size = 546, normalized size = 4.59 \begin {gather*} \frac {1}{12} x^{12} e^{4} b^{7} + \frac {4}{11} x^{11} e^{3} d b^{7} + \frac {7}{11} x^{11} e^{4} b^{6} a + \frac {3}{5} x^{10} e^{2} d^{2} b^{7} + \frac {14}{5} x^{10} e^{3} d b^{6} a + \frac {21}{10} x^{10} e^{4} b^{5} a^{2} + \frac {4}{9} x^{9} e d^{3} b^{7} + \frac {14}{3} x^{9} e^{2} d^{2} b^{6} a + \frac {28}{3} x^{9} e^{3} d b^{5} a^{2} + \frac {35}{9} x^{9} e^{4} b^{4} a^{3} + \frac {1}{8} x^{8} d^{4} b^{7} + \frac {7}{2} x^{8} e d^{3} b^{6} a + \frac {63}{4} x^{8} e^{2} d^{2} b^{5} a^{2} + \frac {35}{2} x^{8} e^{3} d b^{4} a^{3} + \frac {35}{8} x^{8} e^{4} b^{3} a^{4} + x^{7} d^{4} b^{6} a + 12 x^{7} e d^{3} b^{5} a^{2} + 30 x^{7} e^{2} d^{2} b^{4} a^{3} + 20 x^{7} e^{3} d b^{3} a^{4} + 3 x^{7} e^{4} b^{2} a^{5} + \frac {7}{2} x^{6} d^{4} b^{5} a^{2} + \frac {70}{3} x^{6} e d^{3} b^{4} a^{3} + 35 x^{6} e^{2} d^{2} b^{3} a^{4} + 14 x^{6} e^{3} d b^{2} a^{5} + \frac {7}{6} x^{6} e^{4} b a^{6} + 7 x^{5} d^{4} b^{4} a^{3} + 28 x^{5} e d^{3} b^{3} a^{4} + \frac {126}{5} x^{5} e^{2} d^{2} b^{2} a^{5} + \frac {28}{5} x^{5} e^{3} d b a^{6} + \frac {1}{5} x^{5} e^{4} a^{7} + \frac {35}{4} x^{4} d^{4} b^{3} a^{4} + 21 x^{4} e d^{3} b^{2} a^{5} + \frac {21}{2} x^{4} e^{2} d^{2} b a^{6} + x^{4} e^{3} d a^{7} + 7 x^{3} d^{4} b^{2} a^{5} + \frac {28}{3} x^{3} e d^{3} b a^{6} + 2 x^{3} e^{2} d^{2} a^{7} + \frac {7}{2} x^{2} d^{4} b a^{6} + 2 x^{2} e d^{3} a^{7} + x d^{4} a^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 530, normalized size = 4.45 \begin {gather*} \frac {1}{12} \, b^{7} x^{12} e^{4} + \frac {4}{11} \, b^{7} d x^{11} e^{3} + \frac {3}{5} \, b^{7} d^{2} x^{10} e^{2} + \frac {4}{9} \, b^{7} d^{3} x^{9} e + \frac {1}{8} \, b^{7} d^{4} x^{8} + \frac {7}{11} \, a b^{6} x^{11} e^{4} + \frac {14}{5} \, a b^{6} d x^{10} e^{3} + \frac {14}{3} \, a b^{6} d^{2} x^{9} e^{2} + \frac {7}{2} \, a b^{6} d^{3} x^{8} e + a b^{6} d^{4} x^{7} + \frac {21}{10} \, a^{2} b^{5} x^{10} e^{4} + \frac {28}{3} \, a^{2} b^{5} d x^{9} e^{3} + \frac {63}{4} \, a^{2} b^{5} d^{2} x^{8} e^{2} + 12 \, a^{2} b^{5} d^{3} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{4} x^{6} + \frac {35}{9} \, a^{3} b^{4} x^{9} e^{4} + \frac {35}{2} \, a^{3} b^{4} d x^{8} e^{3} + 30 \, a^{3} b^{4} d^{2} x^{7} e^{2} + \frac {70}{3} \, a^{3} b^{4} d^{3} x^{6} e + 7 \, a^{3} b^{4} d^{4} x^{5} + \frac {35}{8} \, a^{4} b^{3} x^{8} e^{4} + 20 \, a^{4} b^{3} d x^{7} e^{3} + 35 \, a^{4} b^{3} d^{2} x^{6} e^{2} + 28 \, a^{4} b^{3} d^{3} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{4} x^{4} + 3 \, a^{5} b^{2} x^{7} e^{4} + 14 \, a^{5} b^{2} d x^{6} e^{3} + \frac {126}{5} \, a^{5} b^{2} d^{2} x^{5} e^{2} + 21 \, a^{5} b^{2} d^{3} x^{4} e + 7 \, a^{5} b^{2} d^{4} x^{3} + \frac {7}{6} \, a^{6} b x^{6} e^{4} + \frac {28}{5} \, a^{6} b d x^{5} e^{3} + \frac {21}{2} \, a^{6} b d^{2} x^{4} e^{2} + \frac {28}{3} \, a^{6} b d^{3} x^{3} e + \frac {7}{2} \, a^{6} b d^{4} x^{2} + \frac {1}{5} \, a^{7} x^{5} e^{4} + a^{7} d x^{4} e^{3} + 2 \, a^{7} d^{2} x^{3} e^{2} + 2 \, a^{7} d^{3} x^{2} e + a^{7} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 799, normalized size = 6.71 \begin {gather*} \frac {b^{7} e^{4} x^{12}}{12}+a^{7} d^{4} x +\frac {\left (6 a \,b^{6} e^{4}+\left (a \,e^{4}+4 b d \,e^{3}\right ) b^{6}\right ) x^{11}}{11}+\frac {\left (15 a^{2} b^{5} e^{4}+6 \left (a \,e^{4}+4 b d \,e^{3}\right ) a \,b^{5}+\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) b^{6}\right ) x^{10}}{10}+\frac {\left (20 a^{3} b^{4} e^{4}+15 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{2} b^{4}+6 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a \,b^{5}+\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) b^{6}\right ) x^{9}}{9}+\frac {\left (15 a^{4} b^{3} e^{4}+20 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{3} b^{3}+15 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{2} b^{4}+6 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a \,b^{5}+\left (4 a \,d^{3} e +b \,d^{4}\right ) b^{6}\right ) x^{8}}{8}+\frac {\left (6 a^{5} b^{2} e^{4}+a \,b^{6} d^{4}+15 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{4} b^{2}+20 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{3} b^{3}+15 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{2} b^{4}+6 \left (4 a \,d^{3} e +b \,d^{4}\right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (a^{6} b \,e^{4}+6 a^{2} b^{5} d^{4}+6 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{5} b +15 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{4} b^{2}+20 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{3} b^{3}+15 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (15 a^{3} b^{4} d^{4}+\left (a \,e^{4}+4 b d \,e^{3}\right ) a^{6}+6 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{5} b +15 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{4} b^{2}+20 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (20 a^{4} b^{3} d^{4}+\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{6}+6 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{5} b +15 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (15 a^{5} b^{2} d^{4}+\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{6}+6 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b \,d^{4}+\left (4 a \,d^{3} e +b \,d^{4}\right ) a^{6}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 489, normalized size = 4.11 \begin {gather*} \frac {1}{12} \, b^{7} e^{4} x^{12} + a^{7} d^{4} x + \frac {1}{11} \, {\left (4 \, b^{7} d e^{3} + 7 \, a b^{6} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, b^{7} d^{2} e^{2} + 28 \, a b^{6} d e^{3} + 21 \, a^{2} b^{5} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, b^{7} d^{3} e + 42 \, a b^{6} d^{2} e^{2} + 84 \, a^{2} b^{5} d e^{3} + 35 \, a^{3} b^{4} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{4} + 28 \, a b^{6} d^{3} e + 126 \, a^{2} b^{5} d^{2} e^{2} + 140 \, a^{3} b^{4} d e^{3} + 35 \, a^{4} b^{3} e^{4}\right )} x^{8} + {\left (a b^{6} d^{4} + 12 \, a^{2} b^{5} d^{3} e + 30 \, a^{3} b^{4} d^{2} e^{2} + 20 \, a^{4} b^{3} d e^{3} + 3 \, a^{5} b^{2} e^{4}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d^{4} + 20 \, a^{3} b^{4} d^{3} e + 30 \, a^{4} b^{3} d^{2} e^{2} + 12 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (35 \, a^{3} b^{4} d^{4} + 140 \, a^{4} b^{3} d^{3} e + 126 \, a^{5} b^{2} d^{2} e^{2} + 28 \, a^{6} b d e^{3} + a^{7} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, a^{4} b^{3} d^{4} + 84 \, a^{5} b^{2} d^{3} e + 42 \, a^{6} b d^{2} e^{2} + 4 \, a^{7} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{4} + 28 \, a^{6} b d^{3} e + 6 \, a^{7} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{4} + 4 \, a^{7} d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 470, normalized size = 3.95 \begin {gather*} x^5\,\left (\frac {a^7\,e^4}{5}+\frac {28\,a^6\,b\,d\,e^3}{5}+\frac {126\,a^5\,b^2\,d^2\,e^2}{5}+28\,a^4\,b^3\,d^3\,e+7\,a^3\,b^4\,d^4\right )+x^8\,\left (\frac {35\,a^4\,b^3\,e^4}{8}+\frac {35\,a^3\,b^4\,d\,e^3}{2}+\frac {63\,a^2\,b^5\,d^2\,e^2}{4}+\frac {7\,a\,b^6\,d^3\,e}{2}+\frac {b^7\,d^4}{8}\right )+x^4\,\left (a^7\,d\,e^3+\frac {21\,a^6\,b\,d^2\,e^2}{2}+21\,a^5\,b^2\,d^3\,e+\frac {35\,a^4\,b^3\,d^4}{4}\right )+x^9\,\left (\frac {35\,a^3\,b^4\,e^4}{9}+\frac {28\,a^2\,b^5\,d\,e^3}{3}+\frac {14\,a\,b^6\,d^2\,e^2}{3}+\frac {4\,b^7\,d^3\,e}{9}\right )+x^7\,\left (3\,a^5\,b^2\,e^4+20\,a^4\,b^3\,d\,e^3+30\,a^3\,b^4\,d^2\,e^2+12\,a^2\,b^5\,d^3\,e+a\,b^6\,d^4\right )+x^6\,\left (\frac {7\,a^6\,b\,e^4}{6}+14\,a^5\,b^2\,d\,e^3+35\,a^4\,b^3\,d^2\,e^2+\frac {70\,a^3\,b^4\,d^3\,e}{3}+\frac {7\,a^2\,b^5\,d^4}{2}\right )+a^7\,d^4\,x+\frac {b^7\,e^4\,x^{12}}{12}+\frac {a^6\,d^3\,x^2\,\left (4\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^3\,x^{11}\,\left (7\,a\,e+4\,b\,d\right )}{11}+\frac {a^5\,d^2\,x^3\,\left (6\,a^2\,e^2+28\,a\,b\,d\,e+21\,b^2\,d^2\right )}{3}+\frac {b^5\,e^2\,x^{10}\,\left (21\,a^2\,e^2+28\,a\,b\,d\,e+6\,b^2\,d^2\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 549, normalized size = 4.61 \begin {gather*} a^{7} d^{4} x + \frac {b^{7} e^{4} x^{12}}{12} + x^{11} \left (\frac {7 a b^{6} e^{4}}{11} + \frac {4 b^{7} d e^{3}}{11}\right ) + x^{10} \left (\frac {21 a^{2} b^{5} e^{4}}{10} + \frac {14 a b^{6} d e^{3}}{5} + \frac {3 b^{7} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac {35 a^{3} b^{4} e^{4}}{9} + \frac {28 a^{2} b^{5} d e^{3}}{3} + \frac {14 a b^{6} d^{2} e^{2}}{3} + \frac {4 b^{7} d^{3} e}{9}\right ) + x^{8} \left (\frac {35 a^{4} b^{3} e^{4}}{8} + \frac {35 a^{3} b^{4} d e^{3}}{2} + \frac {63 a^{2} b^{5} d^{2} e^{2}}{4} + \frac {7 a b^{6} d^{3} e}{2} + \frac {b^{7} d^{4}}{8}\right ) + x^{7} \left (3 a^{5} b^{2} e^{4} + 20 a^{4} b^{3} d e^{3} + 30 a^{3} b^{4} d^{2} e^{2} + 12 a^{2} b^{5} d^{3} e + a b^{6} d^{4}\right ) + x^{6} \left (\frac {7 a^{6} b e^{4}}{6} + 14 a^{5} b^{2} d e^{3} + 35 a^{4} b^{3} d^{2} e^{2} + \frac {70 a^{3} b^{4} d^{3} e}{3} + \frac {7 a^{2} b^{5} d^{4}}{2}\right ) + x^{5} \left (\frac {a^{7} e^{4}}{5} + \frac {28 a^{6} b d e^{3}}{5} + \frac {126 a^{5} b^{2} d^{2} e^{2}}{5} + 28 a^{4} b^{3} d^{3} e + 7 a^{3} b^{4} d^{4}\right ) + x^{4} \left (a^{7} d e^{3} + \frac {21 a^{6} b d^{2} e^{2}}{2} + 21 a^{5} b^{2} d^{3} e + \frac {35 a^{4} b^{3} d^{4}}{4}\right ) + x^{3} \left (2 a^{7} d^{2} e^{2} + \frac {28 a^{6} b d^{3} e}{3} + 7 a^{5} b^{2} d^{4}\right ) + x^{2} \left (2 a^{7} d^{3} e + \frac {7 a^{6} b d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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