Optimal. Leaf size=119 \[ \frac {4 e^3 (a+b x)^9 (b d-a e)}{9 b^5}+\frac {3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac {4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5} \]
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Rubi [A] time = 0.22, 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)^9 (b d-a e)}{9 b^5}+\frac {3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac {4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac {e^4 (a+b x)^{10}}{10 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 )^2 \, dx &=\int (a+b x)^5 (d+e x)^4 \, dx\\ &=\int \left (\frac {(b d-a e)^4 (a+b x)^5}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^6}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^7}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^8}{b^4}+\frac {e^4 (a+b x)^9}{b^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (a+b x)^6}{6 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^7}{7 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^8}{4 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^9}{9 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 301, normalized size = 2.53 \begin {gather*} \frac {x \left (252 a^5 \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 )+210 a^4 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 )+120 a^3 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 )+45 a^2 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 )+10 a 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 )+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 )\right )}{1260} \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 )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.49, size = 396, normalized size = 3.33 \begin {gather*} \frac {1}{10} x^{10} e^{4} b^{5} + \frac {4}{9} x^{9} e^{3} d b^{5} + \frac {5}{9} x^{9} e^{4} b^{4} a + \frac {3}{4} x^{8} e^{2} d^{2} b^{5} + \frac {5}{2} x^{8} e^{3} d b^{4} a + \frac {5}{4} x^{8} e^{4} b^{3} a^{2} + \frac {4}{7} x^{7} e d^{3} b^{5} + \frac {30}{7} x^{7} e^{2} d^{2} b^{4} a + \frac {40}{7} x^{7} e^{3} d b^{3} a^{2} + \frac {10}{7} x^{7} e^{4} b^{2} a^{3} + \frac {1}{6} x^{6} d^{4} b^{5} + \frac {10}{3} x^{6} e d^{3} b^{4} a + 10 x^{6} e^{2} d^{2} b^{3} a^{2} + \frac {20}{3} x^{6} e^{3} d b^{2} a^{3} + \frac {5}{6} x^{6} e^{4} b a^{4} + x^{5} d^{4} b^{4} a + 8 x^{5} e d^{3} b^{3} a^{2} + 12 x^{5} e^{2} d^{2} b^{2} a^{3} + 4 x^{5} e^{3} d b a^{4} + \frac {1}{5} x^{5} e^{4} a^{5} + \frac {5}{2} x^{4} d^{4} b^{3} a^{2} + 10 x^{4} e d^{3} b^{2} a^{3} + \frac {15}{2} x^{4} e^{2} d^{2} b a^{4} + x^{4} e^{3} d a^{5} + \frac {10}{3} x^{3} d^{4} b^{2} a^{3} + \frac {20}{3} x^{3} e d^{3} b a^{4} + 2 x^{3} e^{2} d^{2} a^{5} + \frac {5}{2} x^{2} d^{4} b a^{4} + 2 x^{2} e d^{3} a^{5} + x d^{4} a^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 384, normalized size = 3.23 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} e^{4} + \frac {4}{9} \, b^{5} d x^{9} e^{3} + \frac {3}{4} \, b^{5} d^{2} x^{8} e^{2} + \frac {4}{7} \, b^{5} d^{3} x^{7} e + \frac {1}{6} \, b^{5} d^{4} x^{6} + \frac {5}{9} \, a b^{4} x^{9} e^{4} + \frac {5}{2} \, a b^{4} d x^{8} e^{3} + \frac {30}{7} \, a b^{4} d^{2} x^{7} e^{2} + \frac {10}{3} \, a b^{4} d^{3} x^{6} e + a b^{4} d^{4} x^{5} + \frac {5}{4} \, a^{2} b^{3} x^{8} e^{4} + \frac {40}{7} \, a^{2} b^{3} d x^{7} e^{3} + 10 \, a^{2} b^{3} d^{2} x^{6} e^{2} + 8 \, a^{2} b^{3} d^{3} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} + \frac {10}{7} \, a^{3} b^{2} x^{7} e^{4} + \frac {20}{3} \, a^{3} b^{2} d x^{6} e^{3} + 12 \, a^{3} b^{2} d^{2} x^{5} e^{2} + 10 \, a^{3} b^{2} d^{3} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} + \frac {5}{6} \, a^{4} b x^{6} e^{4} + 4 \, a^{4} b d x^{5} e^{3} + \frac {15}{2} \, a^{4} b d^{2} x^{4} e^{2} + \frac {20}{3} \, a^{4} b d^{3} x^{3} e + \frac {5}{2} \, a^{4} b d^{4} x^{2} + \frac {1}{5} \, a^{5} x^{5} e^{4} + a^{5} d x^{4} e^{3} + 2 \, a^{5} d^{2} x^{3} e^{2} + 2 \, a^{5} d^{3} x^{2} e + a^{5} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 559, normalized size = 4.70 \begin {gather*} \frac {b^{5} e^{4} x^{10}}{10}+a^{5} d^{4} x +\frac {\left (4 a \,b^{4} e^{4}+\left (a \,e^{4}+4 b d \,e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (6 a^{2} b^{3} e^{4}+4 \left (a \,e^{4}+4 b d \,e^{3}\right ) a \,b^{3}+\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (4 a^{3} b^{2} e^{4}+6 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{2} b^{2}+4 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a \,b^{3}+\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (a^{4} b \,e^{4}+4 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{3} b +6 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{2} b^{2}+4 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a \,b^{3}+\left (4 a \,d^{3} e +b \,d^{4}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (a \,b^{4} d^{4}+\left (a \,e^{4}+4 b d \,e^{3}\right ) a^{4}+4 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{3} b +6 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{2} b^{2}+4 \left (4 a \,d^{3} e +b \,d^{4}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 a^{2} b^{3} d^{4}+\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{4}+4 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{3} b +6 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 a^{3} b^{2} d^{4}+\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{4}+4 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{4}+\left (4 a \,d^{3} e +b \,d^{4}\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 = 0.64, size = 360, normalized size = 3.03 \begin {gather*} \frac {1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac {1}{9} \, {\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{4} + 4 \, a^{5} 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 = 2.07, size = 340, normalized size = 2.86 \begin {gather*} x^4\,\left (a^5\,d\,e^3+\frac {15\,a^4\,b\,d^2\,e^2}{2}+10\,a^3\,b^2\,d^3\,e+\frac {5\,a^2\,b^3\,d^4}{2}\right )+x^7\,\left (\frac {10\,a^3\,b^2\,e^4}{7}+\frac {40\,a^2\,b^3\,d\,e^3}{7}+\frac {30\,a\,b^4\,d^2\,e^2}{7}+\frac {4\,b^5\,d^3\,e}{7}\right )+x^5\,\left (\frac {a^5\,e^4}{5}+4\,a^4\,b\,d\,e^3+12\,a^3\,b^2\,d^2\,e^2+8\,a^2\,b^3\,d^3\,e+a\,b^4\,d^4\right )+x^6\,\left (\frac {5\,a^4\,b\,e^4}{6}+\frac {20\,a^3\,b^2\,d\,e^3}{3}+10\,a^2\,b^3\,d^2\,e^2+\frac {10\,a\,b^4\,d^3\,e}{3}+\frac {b^5\,d^4}{6}\right )+a^5\,d^4\,x+\frac {b^5\,e^4\,x^{10}}{10}+\frac {a^4\,d^3\,x^2\,\left (4\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e^3\,x^9\,\left (5\,a\,e+4\,b\,d\right )}{9}+\frac {2\,a^3\,d^2\,x^3\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{3}+\frac {b^3\,e^2\,x^8\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,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 = 0.14, size = 401, normalized size = 3.37 \begin {gather*} a^{5} d^{4} x + \frac {b^{5} e^{4} x^{10}}{10} + x^{9} \left (\frac {5 a b^{4} e^{4}}{9} + \frac {4 b^{5} d e^{3}}{9}\right ) + x^{8} \left (\frac {5 a^{2} b^{3} e^{4}}{4} + \frac {5 a b^{4} d e^{3}}{2} + \frac {3 b^{5} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac {10 a^{3} b^{2} e^{4}}{7} + \frac {40 a^{2} b^{3} d e^{3}}{7} + \frac {30 a b^{4} d^{2} e^{2}}{7} + \frac {4 b^{5} d^{3} e}{7}\right ) + x^{6} \left (\frac {5 a^{4} b e^{4}}{6} + \frac {20 a^{3} b^{2} d e^{3}}{3} + 10 a^{2} b^{3} d^{2} e^{2} + \frac {10 a b^{4} d^{3} e}{3} + \frac {b^{5} d^{4}}{6}\right ) + x^{5} \left (\frac {a^{5} e^{4}}{5} + 4 a^{4} b d e^{3} + 12 a^{3} b^{2} d^{2} e^{2} + 8 a^{2} b^{3} d^{3} e + a b^{4} d^{4}\right ) + x^{4} \left (a^{5} d e^{3} + \frac {15 a^{4} b d^{2} e^{2}}{2} + 10 a^{3} b^{2} d^{3} e + \frac {5 a^{2} b^{3} d^{4}}{2}\right ) + x^{3} \left (2 a^{5} d^{2} e^{2} + \frac {20 a^{4} b d^{3} e}{3} + \frac {10 a^{3} b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{5} d^{3} e + \frac {5 a^{4} b d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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