Optimal. Leaf size=173 \[ \frac {6 e^5 (a+b x)^{13} (b d-a e)}{13 b^7}+\frac {5 e^4 (a+b x)^{12} (b d-a e)^2}{4 b^7}+\frac {20 e^3 (a+b x)^{11} (b d-a e)^3}{11 b^7}+\frac {3 e^2 (a+b x)^{10} (b d-a e)^4}{2 b^7}+\frac {2 e (a+b x)^9 (b d-a e)^5}{3 b^7}+\frac {(a+b x)^8 (b d-a e)^6}{8 b^7}+\frac {e^6 (a+b x)^{14}}{14 b^7} \]
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Rubi [A] time = 0.44, antiderivative size = 173, 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 {6 e^5 (a+b x)^{13} (b d-a e)}{13 b^7}+\frac {5 e^4 (a+b x)^{12} (b d-a e)^2}{4 b^7}+\frac {20 e^3 (a+b x)^{11} (b d-a e)^3}{11 b^7}+\frac {3 e^2 (a+b x)^{10} (b d-a e)^4}{2 b^7}+\frac {2 e (a+b x)^9 (b d-a e)^5}{3 b^7}+\frac {(a+b x)^8 (b d-a e)^6}{8 b^7}+\frac {e^6 (a+b x)^{14}}{14 b^7} \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)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^6 \, dx\\ &=\int \left (\frac {(b d-a e)^6 (a+b x)^7}{b^6}+\frac {6 e (b d-a e)^5 (a+b x)^8}{b^6}+\frac {15 e^2 (b d-a e)^4 (a+b x)^9}{b^6}+\frac {20 e^3 (b d-a e)^3 (a+b x)^{10}}{b^6}+\frac {15 e^4 (b d-a e)^2 (a+b x)^{11}}{b^6}+\frac {6 e^5 (b d-a e) (a+b x)^{12}}{b^6}+\frac {e^6 (a+b x)^{13}}{b^6}\right ) \, dx\\ &=\frac {(b d-a e)^6 (a+b x)^8}{8 b^7}+\frac {2 e (b d-a e)^5 (a+b x)^9}{3 b^7}+\frac {3 e^2 (b d-a e)^4 (a+b x)^{10}}{2 b^7}+\frac {20 e^3 (b d-a e)^3 (a+b x)^{11}}{11 b^7}+\frac {5 e^4 (b d-a e)^2 (a+b x)^{12}}{4 b^7}+\frac {6 e^5 (b d-a e) (a+b x)^{13}}{13 b^7}+\frac {e^6 (a+b x)^{14}}{14 b^7}\\ \end {align*}
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Mathematica [B] time = 0.17, size = 581, normalized size = 3.36 \begin {gather*} \frac {x \left (3432 a^7 \left (7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right )+3003 a^6 b x \left (28 d^6+112 d^5 e x+210 d^4 e^2 x^2+224 d^3 e^3 x^3+140 d^2 e^4 x^4+48 d e^5 x^5+7 e^6 x^6\right )+2002 a^5 b^2 x^2 \left (84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right )+1001 a^4 b^3 x^3 \left (210 d^6+1008 d^5 e x+2100 d^4 e^2 x^2+2400 d^3 e^3 x^3+1575 d^2 e^4 x^4+560 d e^5 x^5+84 e^6 x^6\right )+364 a^3 b^4 x^4 \left (462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right )+91 a^2 b^5 x^5 \left (924 d^6+4752 d^5 e x+10395 d^4 e^2 x^2+12320 d^3 e^3 x^3+8316 d^2 e^4 x^4+3024 d e^5 x^5+462 e^6 x^6\right )+14 a b^6 x^6 \left (1716 d^6+9009 d^5 e x+20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+6006 d e^5 x^5+924 e^6 x^6\right )+b^7 x^7 \left (3003 d^6+16016 d^5 e x+36036 d^4 e^2 x^2+43680 d^3 e^3 x^3+30030 d^2 e^4 x^4+11088 d e^5 x^5+1716 e^6 x^6\right )\right )}{24024} \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)^6 \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 = 798, normalized size = 4.61 \begin {gather*} \frac {1}{14} x^{14} e^{6} b^{7} + \frac {6}{13} x^{13} e^{5} d b^{7} + \frac {7}{13} x^{13} e^{6} b^{6} a + \frac {5}{4} x^{12} e^{4} d^{2} b^{7} + \frac {7}{2} x^{12} e^{5} d b^{6} a + \frac {7}{4} x^{12} e^{6} b^{5} a^{2} + \frac {20}{11} x^{11} e^{3} d^{3} b^{7} + \frac {105}{11} x^{11} e^{4} d^{2} b^{6} a + \frac {126}{11} x^{11} e^{5} d b^{5} a^{2} + \frac {35}{11} x^{11} e^{6} b^{4} a^{3} + \frac {3}{2} x^{10} e^{2} d^{4} b^{7} + 14 x^{10} e^{3} d^{3} b^{6} a + \frac {63}{2} x^{10} e^{4} d^{2} b^{5} a^{2} + 21 x^{10} e^{5} d b^{4} a^{3} + \frac {7}{2} x^{10} e^{6} b^{3} a^{4} + \frac {2}{3} x^{9} e d^{5} b^{7} + \frac {35}{3} x^{9} e^{2} d^{4} b^{6} a + \frac {140}{3} x^{9} e^{3} d^{3} b^{5} a^{2} + \frac {175}{3} x^{9} e^{4} d^{2} b^{4} a^{3} + \frac {70}{3} x^{9} e^{5} d b^{3} a^{4} + \frac {7}{3} x^{9} e^{6} b^{2} a^{5} + \frac {1}{8} x^{8} d^{6} b^{7} + \frac {21}{4} x^{8} e d^{5} b^{6} a + \frac {315}{8} x^{8} e^{2} d^{4} b^{5} a^{2} + \frac {175}{2} x^{8} e^{3} d^{3} b^{4} a^{3} + \frac {525}{8} x^{8} e^{4} d^{2} b^{3} a^{4} + \frac {63}{4} x^{8} e^{5} d b^{2} a^{5} + \frac {7}{8} x^{8} e^{6} b a^{6} + x^{7} d^{6} b^{6} a + 18 x^{7} e d^{5} b^{5} a^{2} + 75 x^{7} e^{2} d^{4} b^{4} a^{3} + 100 x^{7} e^{3} d^{3} b^{3} a^{4} + 45 x^{7} e^{4} d^{2} b^{2} a^{5} + 6 x^{7} e^{5} d b a^{6} + \frac {1}{7} x^{7} e^{6} a^{7} + \frac {7}{2} x^{6} d^{6} b^{5} a^{2} + 35 x^{6} e d^{5} b^{4} a^{3} + \frac {175}{2} x^{6} e^{2} d^{4} b^{3} a^{4} + 70 x^{6} e^{3} d^{3} b^{2} a^{5} + \frac {35}{2} x^{6} e^{4} d^{2} b a^{6} + x^{6} e^{5} d a^{7} + 7 x^{5} d^{6} b^{4} a^{3} + 42 x^{5} e d^{5} b^{3} a^{4} + 63 x^{5} e^{2} d^{4} b^{2} a^{5} + 28 x^{5} e^{3} d^{3} b a^{6} + 3 x^{5} e^{4} d^{2} a^{7} + \frac {35}{4} x^{4} d^{6} b^{3} a^{4} + \frac {63}{2} x^{4} e d^{5} b^{2} a^{5} + \frac {105}{4} x^{4} e^{2} d^{4} b a^{6} + 5 x^{4} e^{3} d^{3} a^{7} + 7 x^{3} d^{6} b^{2} a^{5} + 14 x^{3} e d^{5} b a^{6} + 5 x^{3} e^{2} d^{4} a^{7} + \frac {7}{2} x^{2} d^{6} b a^{6} + 3 x^{2} e d^{5} a^{7} + x d^{6} a^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 766, normalized size = 4.43 \begin {gather*} \frac {1}{14} \, b^{7} x^{14} e^{6} + \frac {6}{13} \, b^{7} d x^{13} e^{5} + \frac {5}{4} \, b^{7} d^{2} x^{12} e^{4} + \frac {20}{11} \, b^{7} d^{3} x^{11} e^{3} + \frac {3}{2} \, b^{7} d^{4} x^{10} e^{2} + \frac {2}{3} \, b^{7} d^{5} x^{9} e + \frac {1}{8} \, b^{7} d^{6} x^{8} + \frac {7}{13} \, a b^{6} x^{13} e^{6} + \frac {7}{2} \, a b^{6} d x^{12} e^{5} + \frac {105}{11} \, a b^{6} d^{2} x^{11} e^{4} + 14 \, a b^{6} d^{3} x^{10} e^{3} + \frac {35}{3} \, a b^{6} d^{4} x^{9} e^{2} + \frac {21}{4} \, a b^{6} d^{5} x^{8} e + a b^{6} d^{6} x^{7} + \frac {7}{4} \, a^{2} b^{5} x^{12} e^{6} + \frac {126}{11} \, a^{2} b^{5} d x^{11} e^{5} + \frac {63}{2} \, a^{2} b^{5} d^{2} x^{10} e^{4} + \frac {140}{3} \, a^{2} b^{5} d^{3} x^{9} e^{3} + \frac {315}{8} \, a^{2} b^{5} d^{4} x^{8} e^{2} + 18 \, a^{2} b^{5} d^{5} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{6} x^{6} + \frac {35}{11} \, a^{3} b^{4} x^{11} e^{6} + 21 \, a^{3} b^{4} d x^{10} e^{5} + \frac {175}{3} \, a^{3} b^{4} d^{2} x^{9} e^{4} + \frac {175}{2} \, a^{3} b^{4} d^{3} x^{8} e^{3} + 75 \, a^{3} b^{4} d^{4} x^{7} e^{2} + 35 \, a^{3} b^{4} d^{5} x^{6} e + 7 \, a^{3} b^{4} d^{6} x^{5} + \frac {7}{2} \, a^{4} b^{3} x^{10} e^{6} + \frac {70}{3} \, a^{4} b^{3} d x^{9} e^{5} + \frac {525}{8} \, a^{4} b^{3} d^{2} x^{8} e^{4} + 100 \, a^{4} b^{3} d^{3} x^{7} e^{3} + \frac {175}{2} \, a^{4} b^{3} d^{4} x^{6} e^{2} + 42 \, a^{4} b^{3} d^{5} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{6} x^{4} + \frac {7}{3} \, a^{5} b^{2} x^{9} e^{6} + \frac {63}{4} \, a^{5} b^{2} d x^{8} e^{5} + 45 \, a^{5} b^{2} d^{2} x^{7} e^{4} + 70 \, a^{5} b^{2} d^{3} x^{6} e^{3} + 63 \, a^{5} b^{2} d^{4} x^{5} e^{2} + \frac {63}{2} \, a^{5} b^{2} d^{5} x^{4} e + 7 \, a^{5} b^{2} d^{6} x^{3} + \frac {7}{8} \, a^{6} b x^{8} e^{6} + 6 \, a^{6} b d x^{7} e^{5} + \frac {35}{2} \, a^{6} b d^{2} x^{6} e^{4} + 28 \, a^{6} b d^{3} x^{5} e^{3} + \frac {105}{4} \, a^{6} b d^{4} x^{4} e^{2} + 14 \, a^{6} b d^{5} x^{3} e + \frac {7}{2} \, a^{6} b d^{6} x^{2} + \frac {1}{7} \, a^{7} x^{7} e^{6} + a^{7} d x^{6} e^{5} + 3 \, a^{7} d^{2} x^{5} e^{4} + 5 \, a^{7} d^{3} x^{4} e^{3} + 5 \, a^{7} d^{4} x^{3} e^{2} + 3 \, a^{7} d^{5} x^{2} e + a^{7} d^{6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1165, normalized size = 6.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 706, normalized size = 4.08 \begin {gather*} \frac {1}{14} \, b^{7} e^{6} x^{14} + a^{7} d^{6} x + \frac {1}{13} \, {\left (6 \, b^{7} d e^{5} + 7 \, a b^{6} e^{6}\right )} x^{13} + \frac {1}{4} \, {\left (5 \, b^{7} d^{2} e^{4} + 14 \, a b^{6} d e^{5} + 7 \, a^{2} b^{5} e^{6}\right )} x^{12} + \frac {1}{11} \, {\left (20 \, b^{7} d^{3} e^{3} + 105 \, a b^{6} d^{2} e^{4} + 126 \, a^{2} b^{5} d e^{5} + 35 \, a^{3} b^{4} e^{6}\right )} x^{11} + \frac {1}{2} \, {\left (3 \, b^{7} d^{4} e^{2} + 28 \, a b^{6} d^{3} e^{3} + 63 \, a^{2} b^{5} d^{2} e^{4} + 42 \, a^{3} b^{4} d e^{5} + 7 \, a^{4} b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, b^{7} d^{5} e + 35 \, a b^{6} d^{4} e^{2} + 140 \, a^{2} b^{5} d^{3} e^{3} + 175 \, a^{3} b^{4} d^{2} e^{4} + 70 \, a^{4} b^{3} d e^{5} + 7 \, a^{5} b^{2} e^{6}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{6} + 42 \, a b^{6} d^{5} e + 315 \, a^{2} b^{5} d^{4} e^{2} + 700 \, a^{3} b^{4} d^{3} e^{3} + 525 \, a^{4} b^{3} d^{2} e^{4} + 126 \, a^{5} b^{2} d e^{5} + 7 \, a^{6} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (7 \, a b^{6} d^{6} + 126 \, a^{2} b^{5} d^{5} e + 525 \, a^{3} b^{4} d^{4} e^{2} + 700 \, a^{4} b^{3} d^{3} e^{3} + 315 \, a^{5} b^{2} d^{2} e^{4} + 42 \, a^{6} b d e^{5} + a^{7} e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (7 \, a^{2} b^{5} d^{6} + 70 \, a^{3} b^{4} d^{5} e + 175 \, a^{4} b^{3} d^{4} e^{2} + 140 \, a^{5} b^{2} d^{3} e^{3} + 35 \, a^{6} b d^{2} e^{4} + 2 \, a^{7} d e^{5}\right )} x^{6} + {\left (7 \, a^{3} b^{4} d^{6} + 42 \, a^{4} b^{3} d^{5} e + 63 \, a^{5} b^{2} d^{4} e^{2} + 28 \, a^{6} b d^{3} e^{3} + 3 \, a^{7} d^{2} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, a^{4} b^{3} d^{6} + 126 \, a^{5} b^{2} d^{5} e + 105 \, a^{6} b d^{4} e^{2} + 20 \, a^{7} d^{3} e^{3}\right )} x^{4} + {\left (7 \, a^{5} b^{2} d^{6} + 14 \, a^{6} b d^{5} e + 5 \, a^{7} d^{4} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{6} + 6 \, a^{7} d^{5} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 683, normalized size = 3.95 \begin {gather*} x^5\,\left (3\,a^7\,d^2\,e^4+28\,a^6\,b\,d^3\,e^3+63\,a^5\,b^2\,d^4\,e^2+42\,a^4\,b^3\,d^5\,e+7\,a^3\,b^4\,d^6\right )+x^{10}\,\left (\frac {7\,a^4\,b^3\,e^6}{2}+21\,a^3\,b^4\,d\,e^5+\frac {63\,a^2\,b^5\,d^2\,e^4}{2}+14\,a\,b^6\,d^3\,e^3+\frac {3\,b^7\,d^4\,e^2}{2}\right )+x^6\,\left (a^7\,d\,e^5+\frac {35\,a^6\,b\,d^2\,e^4}{2}+70\,a^5\,b^2\,d^3\,e^3+\frac {175\,a^4\,b^3\,d^4\,e^2}{2}+35\,a^3\,b^4\,d^5\,e+\frac {7\,a^2\,b^5\,d^6}{2}\right )+x^9\,\left (\frac {7\,a^5\,b^2\,e^6}{3}+\frac {70\,a^4\,b^3\,d\,e^5}{3}+\frac {175\,a^3\,b^4\,d^2\,e^4}{3}+\frac {140\,a^2\,b^5\,d^3\,e^3}{3}+\frac {35\,a\,b^6\,d^4\,e^2}{3}+\frac {2\,b^7\,d^5\,e}{3}\right )+x^7\,\left (\frac {a^7\,e^6}{7}+6\,a^6\,b\,d\,e^5+45\,a^5\,b^2\,d^2\,e^4+100\,a^4\,b^3\,d^3\,e^3+75\,a^3\,b^4\,d^4\,e^2+18\,a^2\,b^5\,d^5\,e+a\,b^6\,d^6\right )+x^8\,\left (\frac {7\,a^6\,b\,e^6}{8}+\frac {63\,a^5\,b^2\,d\,e^5}{4}+\frac {525\,a^4\,b^3\,d^2\,e^4}{8}+\frac {175\,a^3\,b^4\,d^3\,e^3}{2}+\frac {315\,a^2\,b^5\,d^4\,e^2}{8}+\frac {21\,a\,b^6\,d^5\,e}{4}+\frac {b^7\,d^6}{8}\right )+x^4\,\left (5\,a^7\,d^3\,e^3+\frac {105\,a^6\,b\,d^4\,e^2}{4}+\frac {63\,a^5\,b^2\,d^5\,e}{2}+\frac {35\,a^4\,b^3\,d^6}{4}\right )+x^{11}\,\left (\frac {35\,a^3\,b^4\,e^6}{11}+\frac {126\,a^2\,b^5\,d\,e^5}{11}+\frac {105\,a\,b^6\,d^2\,e^4}{11}+\frac {20\,b^7\,d^3\,e^3}{11}\right )+a^7\,d^6\,x+\frac {b^7\,e^6\,x^{14}}{14}+\frac {a^6\,d^5\,x^2\,\left (6\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^5\,x^{13}\,\left (7\,a\,e+6\,b\,d\right )}{13}+a^5\,d^4\,x^3\,\left (5\,a^2\,e^2+14\,a\,b\,d\,e+7\,b^2\,d^2\right )+\frac {b^5\,e^4\,x^{12}\,\left (7\,a^2\,e^2+14\,a\,b\,d\,e+5\,b^2\,d^2\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 796, normalized size = 4.60 \begin {gather*} a^{7} d^{6} x + \frac {b^{7} e^{6} x^{14}}{14} + x^{13} \left (\frac {7 a b^{6} e^{6}}{13} + \frac {6 b^{7} d e^{5}}{13}\right ) + x^{12} \left (\frac {7 a^{2} b^{5} e^{6}}{4} + \frac {7 a b^{6} d e^{5}}{2} + \frac {5 b^{7} d^{2} e^{4}}{4}\right ) + x^{11} \left (\frac {35 a^{3} b^{4} e^{6}}{11} + \frac {126 a^{2} b^{5} d e^{5}}{11} + \frac {105 a b^{6} d^{2} e^{4}}{11} + \frac {20 b^{7} d^{3} e^{3}}{11}\right ) + x^{10} \left (\frac {7 a^{4} b^{3} e^{6}}{2} + 21 a^{3} b^{4} d e^{5} + \frac {63 a^{2} b^{5} d^{2} e^{4}}{2} + 14 a b^{6} d^{3} e^{3} + \frac {3 b^{7} d^{4} e^{2}}{2}\right ) + x^{9} \left (\frac {7 a^{5} b^{2} e^{6}}{3} + \frac {70 a^{4} b^{3} d e^{5}}{3} + \frac {175 a^{3} b^{4} d^{2} e^{4}}{3} + \frac {140 a^{2} b^{5} d^{3} e^{3}}{3} + \frac {35 a b^{6} d^{4} e^{2}}{3} + \frac {2 b^{7} d^{5} e}{3}\right ) + x^{8} \left (\frac {7 a^{6} b e^{6}}{8} + \frac {63 a^{5} b^{2} d e^{5}}{4} + \frac {525 a^{4} b^{3} d^{2} e^{4}}{8} + \frac {175 a^{3} b^{4} d^{3} e^{3}}{2} + \frac {315 a^{2} b^{5} d^{4} e^{2}}{8} + \frac {21 a b^{6} d^{5} e}{4} + \frac {b^{7} d^{6}}{8}\right ) + x^{7} \left (\frac {a^{7} e^{6}}{7} + 6 a^{6} b d e^{5} + 45 a^{5} b^{2} d^{2} e^{4} + 100 a^{4} b^{3} d^{3} e^{3} + 75 a^{3} b^{4} d^{4} e^{2} + 18 a^{2} b^{5} d^{5} e + a b^{6} d^{6}\right ) + x^{6} \left (a^{7} d e^{5} + \frac {35 a^{6} b d^{2} e^{4}}{2} + 70 a^{5} b^{2} d^{3} e^{3} + \frac {175 a^{4} b^{3} d^{4} e^{2}}{2} + 35 a^{3} b^{4} d^{5} e + \frac {7 a^{2} b^{5} d^{6}}{2}\right ) + x^{5} \left (3 a^{7} d^{2} e^{4} + 28 a^{6} b d^{3} e^{3} + 63 a^{5} b^{2} d^{4} e^{2} + 42 a^{4} b^{3} d^{5} e + 7 a^{3} b^{4} d^{6}\right ) + x^{4} \left (5 a^{7} d^{3} e^{3} + \frac {105 a^{6} b d^{4} e^{2}}{4} + \frac {63 a^{5} b^{2} d^{5} e}{2} + \frac {35 a^{4} b^{3} d^{6}}{4}\right ) + x^{3} \left (5 a^{7} d^{4} e^{2} + 14 a^{6} b d^{5} e + 7 a^{5} b^{2} d^{6}\right ) + x^{2} \left (3 a^{7} d^{5} e + \frac {7 a^{6} b d^{6}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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