Optimal. Leaf size=146 \[ \frac {e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac {5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac {5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6} \]
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Rubi [A] time = 0.25, antiderivative size = 146, 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 {e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac {5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac {5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6} \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)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^5 \, dx\\ &=\int \left (\frac {(b d-a e)^5 (a+b x)^5}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^6}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^7}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^8}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^9}{b^5}+\frac {e^5 (a+b x)^{10}}{b^5}\right ) \, dx\\ &=\frac {(b d-a e)^5 (a+b x)^6}{6 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^7}{7 b^6}+\frac {5 e^2 (b d-a e)^3 (a+b x)^8}{4 b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^9}{9 b^6}+\frac {e^4 (b d-a e) (a+b x)^{10}}{2 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 413, normalized size = 2.83 \begin {gather*} a^5 d^5 x+\frac {5}{2} a^4 d^4 x^2 (a e+b d)+\frac {5}{9} b^3 e^3 x^9 \left (2 a^2 e^2+5 a b d e+2 b^2 d^2\right )+\frac {5}{3} a^3 d^3 x^3 \left (2 a^2 e^2+5 a b d e+2 b^2 d^2\right )+\frac {5}{4} b^2 e^2 x^8 \left (a^3 e^3+5 a^2 b d e^2+5 a b^2 d^2 e+b^3 d^3\right )+\frac {5}{2} a^2 d^2 x^4 \left (a^3 e^3+5 a^2 b d e^2+5 a b^2 d^2 e+b^3 d^3\right )+\frac {5}{7} b e x^7 \left (a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right )+a d x^5 \left (a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right )+\frac {1}{6} x^6 \left (a^5 e^5+25 a^4 b d e^4+100 a^3 b^2 d^2 e^3+100 a^2 b^3 d^3 e^2+25 a b^4 d^4 e+b^5 d^5\right )+\frac {1}{2} b^4 e^4 x^{10} (a e+b d)+\frac {1}{11} b^5 e^5 x^{11} \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)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.37, size = 488, normalized size = 3.34 \begin {gather*} \frac {1}{11} x^{11} e^{5} b^{5} + \frac {1}{2} x^{10} e^{4} d b^{5} + \frac {1}{2} x^{10} e^{5} b^{4} a + \frac {10}{9} x^{9} e^{3} d^{2} b^{5} + \frac {25}{9} x^{9} e^{4} d b^{4} a + \frac {10}{9} x^{9} e^{5} b^{3} a^{2} + \frac {5}{4} x^{8} e^{2} d^{3} b^{5} + \frac {25}{4} x^{8} e^{3} d^{2} b^{4} a + \frac {25}{4} x^{8} e^{4} d b^{3} a^{2} + \frac {5}{4} x^{8} e^{5} b^{2} a^{3} + \frac {5}{7} x^{7} e d^{4} b^{5} + \frac {50}{7} x^{7} e^{2} d^{3} b^{4} a + \frac {100}{7} x^{7} e^{3} d^{2} b^{3} a^{2} + \frac {50}{7} x^{7} e^{4} d b^{2} a^{3} + \frac {5}{7} x^{7} e^{5} b a^{4} + \frac {1}{6} x^{6} d^{5} b^{5} + \frac {25}{6} x^{6} e d^{4} b^{4} a + \frac {50}{3} x^{6} e^{2} d^{3} b^{3} a^{2} + \frac {50}{3} x^{6} e^{3} d^{2} b^{2} a^{3} + \frac {25}{6} x^{6} e^{4} d b a^{4} + \frac {1}{6} x^{6} e^{5} a^{5} + x^{5} d^{5} b^{4} a + 10 x^{5} e d^{4} b^{3} a^{2} + 20 x^{5} e^{2} d^{3} b^{2} a^{3} + 10 x^{5} e^{3} d^{2} b a^{4} + x^{5} e^{4} d a^{5} + \frac {5}{2} x^{4} d^{5} b^{3} a^{2} + \frac {25}{2} x^{4} e d^{4} b^{2} a^{3} + \frac {25}{2} x^{4} e^{2} d^{3} b a^{4} + \frac {5}{2} x^{4} e^{3} d^{2} a^{5} + \frac {10}{3} x^{3} d^{5} b^{2} a^{3} + \frac {25}{3} x^{3} e d^{4} b a^{4} + \frac {10}{3} x^{3} e^{2} d^{3} a^{5} + \frac {5}{2} x^{2} d^{5} b a^{4} + \frac {5}{2} x^{2} e d^{4} a^{5} + x d^{5} a^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 470, normalized size = 3.22 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} e^{5} + \frac {1}{2} \, b^{5} d x^{10} e^{4} + \frac {10}{9} \, b^{5} d^{2} x^{9} e^{3} + \frac {5}{4} \, b^{5} d^{3} x^{8} e^{2} + \frac {5}{7} \, b^{5} d^{4} x^{7} e + \frac {1}{6} \, b^{5} d^{5} x^{6} + \frac {1}{2} \, a b^{4} x^{10} e^{5} + \frac {25}{9} \, a b^{4} d x^{9} e^{4} + \frac {25}{4} \, a b^{4} d^{2} x^{8} e^{3} + \frac {50}{7} \, a b^{4} d^{3} x^{7} e^{2} + \frac {25}{6} \, a b^{4} d^{4} x^{6} e + a b^{4} d^{5} x^{5} + \frac {10}{9} \, a^{2} b^{3} x^{9} e^{5} + \frac {25}{4} \, a^{2} b^{3} d x^{8} e^{4} + \frac {100}{7} \, a^{2} b^{3} d^{2} x^{7} e^{3} + \frac {50}{3} \, a^{2} b^{3} d^{3} x^{6} e^{2} + 10 \, a^{2} b^{3} d^{4} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{5} x^{4} + \frac {5}{4} \, a^{3} b^{2} x^{8} e^{5} + \frac {50}{7} \, a^{3} b^{2} d x^{7} e^{4} + \frac {50}{3} \, a^{3} b^{2} d^{2} x^{6} e^{3} + 20 \, a^{3} b^{2} d^{3} x^{5} e^{2} + \frac {25}{2} \, a^{3} b^{2} d^{4} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{5} x^{3} + \frac {5}{7} \, a^{4} b x^{7} e^{5} + \frac {25}{6} \, a^{4} b d x^{6} e^{4} + 10 \, a^{4} b d^{2} x^{5} e^{3} + \frac {25}{2} \, a^{4} b d^{3} x^{4} e^{2} + \frac {25}{3} \, a^{4} b d^{4} x^{3} e + \frac {5}{2} \, a^{4} b d^{5} x^{2} + \frac {1}{6} \, a^{5} x^{6} e^{5} + a^{5} d x^{5} e^{4} + \frac {5}{2} \, a^{5} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{5} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{5} d^{4} x^{2} e + a^{5} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 688, normalized size = 4.71 \begin {gather*} \frac {b^{5} e^{5} x^{11}}{11}+a^{5} d^{5} x +\frac {\left (4 a \,b^{4} e^{5}+\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{4}\right ) x^{10}}{10}+\frac {\left (6 a^{2} b^{3} e^{5}+4 \left (a \,e^{5}+5 b d \,e^{4}\right ) a \,b^{3}+\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (4 a^{3} b^{2} e^{5}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2} b^{2}+4 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a \,b^{3}+\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (a^{4} b \,e^{5}+4 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{3} b +6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2} b^{2}+4 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a \,b^{3}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{4}+4 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{3} b +6 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{2} b^{2}+4 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a \,b^{3}+\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (a \,b^{4} d^{5}+\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{4}+4 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{3} b +6 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{2} b^{2}+4 \left (5 a \,d^{4} e +b \,d^{5}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 a^{2} b^{3} d^{5}+\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{4}+4 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{3} b +6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 a^{3} b^{2} d^{5}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{4}+4 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{4}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 427, normalized size = 2.92 \begin {gather*} \frac {1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac {1}{2} \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac {5}{4} \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac {5}{7} \, {\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} + {\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 405, normalized size = 2.77 \begin {gather*} x^6\,\left (\frac {a^5\,e^5}{6}+\frac {25\,a^4\,b\,d\,e^4}{6}+\frac {50\,a^3\,b^2\,d^2\,e^3}{3}+\frac {50\,a^2\,b^3\,d^3\,e^2}{3}+\frac {25\,a\,b^4\,d^4\,e}{6}+\frac {b^5\,d^5}{6}\right )+x^5\,\left (a^5\,d\,e^4+10\,a^4\,b\,d^2\,e^3+20\,a^3\,b^2\,d^3\,e^2+10\,a^2\,b^3\,d^4\,e+a\,b^4\,d^5\right )+x^7\,\left (\frac {5\,a^4\,b\,e^5}{7}+\frac {50\,a^3\,b^2\,d\,e^4}{7}+\frac {100\,a^2\,b^3\,d^2\,e^3}{7}+\frac {50\,a\,b^4\,d^3\,e^2}{7}+\frac {5\,b^5\,d^4\,e}{7}\right )+a^5\,d^5\,x+\frac {b^5\,e^5\,x^{11}}{11}+\frac {5\,a^2\,d^2\,x^4\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{2}+\frac {5\,b^2\,e^2\,x^8\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{4}+\frac {5\,a^4\,d^4\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {b^4\,e^4\,x^{10}\,\left (a\,e+b\,d\right )}{2}+\frac {5\,a^3\,d^3\,x^3\,\left (2\,a^2\,e^2+5\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3}+\frac {5\,b^3\,e^3\,x^9\,\left (2\,a^2\,e^2+5\,a\,b\,d\,e+2\,b^2\,d^2\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 500, normalized size = 3.42 \begin {gather*} a^{5} d^{5} x + \frac {b^{5} e^{5} x^{11}}{11} + x^{10} \left (\frac {a b^{4} e^{5}}{2} + \frac {b^{5} d e^{4}}{2}\right ) + x^{9} \left (\frac {10 a^{2} b^{3} e^{5}}{9} + \frac {25 a b^{4} d e^{4}}{9} + \frac {10 b^{5} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac {5 a^{3} b^{2} e^{5}}{4} + \frac {25 a^{2} b^{3} d e^{4}}{4} + \frac {25 a b^{4} d^{2} e^{3}}{4} + \frac {5 b^{5} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac {5 a^{4} b e^{5}}{7} + \frac {50 a^{3} b^{2} d e^{4}}{7} + \frac {100 a^{2} b^{3} d^{2} e^{3}}{7} + \frac {50 a b^{4} d^{3} e^{2}}{7} + \frac {5 b^{5} d^{4} e}{7}\right ) + x^{6} \left (\frac {a^{5} e^{5}}{6} + \frac {25 a^{4} b d e^{4}}{6} + \frac {50 a^{3} b^{2} d^{2} e^{3}}{3} + \frac {50 a^{2} b^{3} d^{3} e^{2}}{3} + \frac {25 a b^{4} d^{4} e}{6} + \frac {b^{5} d^{5}}{6}\right ) + x^{5} \left (a^{5} d e^{4} + 10 a^{4} b d^{2} e^{3} + 20 a^{3} b^{2} d^{3} e^{2} + 10 a^{2} b^{3} d^{4} e + a b^{4} d^{5}\right ) + x^{4} \left (\frac {5 a^{5} d^{2} e^{3}}{2} + \frac {25 a^{4} b d^{3} e^{2}}{2} + \frac {25 a^{3} b^{2} d^{4} e}{2} + \frac {5 a^{2} b^{3} d^{5}}{2}\right ) + x^{3} \left (\frac {10 a^{5} d^{3} e^{2}}{3} + \frac {25 a^{4} b d^{4} e}{3} + \frac {10 a^{3} b^{2} d^{5}}{3}\right ) + x^{2} \left (\frac {5 a^{5} d^{4} e}{2} + \frac {5 a^{4} b d^{5}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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