Optimal. Leaf size=119 \[ -\frac {4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac {3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac {4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac {b^4 (d+e x)^{10}}{10 e^5} \]
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Rubi [A] time = 0.21, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac {3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac {4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac {b^4 (d+e x)^{10}}{10 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^5 \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^5}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^6}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^7}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^8}{e^4}+\frac {b^4 (d+e x)^9}{e^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (d+e x)^6}{6 e^5}-\frac {4 b (b d-a e)^3 (d+e x)^7}{7 e^5}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8}{4 e^5}-\frac {4 b^3 (b d-a e) (d+e x)^9}{9 e^5}+\frac {b^4 (d+e x)^{10}}{10 e^5}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 350, normalized size = 2.94 \begin {gather*} a^4 d^5 x+\frac {1}{2} a^3 d^4 x^2 (5 a e+4 b d)+\frac {1}{4} b^2 e^3 x^8 \left (3 a^2 e^2+10 a b d e+5 b^2 d^2\right )+\frac {2}{3} a^2 d^3 x^3 \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )+\frac {2}{7} b e^2 x^7 \left (2 a^3 e^3+15 a^2 b d e^2+20 a b^2 d^2 e+5 b^3 d^3\right )+\frac {1}{2} a d^2 x^4 \left (5 a^3 e^3+20 a^2 b d e^2+15 a b^2 d^2 e+2 b^3 d^3\right )+\frac {1}{6} e x^6 \left (a^4 e^4+20 a^3 b d e^3+60 a^2 b^2 d^2 e^2+40 a b^3 d^3 e+5 b^4 d^4\right )+\frac {1}{5} d x^5 \left (5 a^4 e^4+40 a^3 b d e^3+60 a^2 b^2 d^2 e^2+20 a b^3 d^3 e+b^4 d^4\right )+\frac {1}{9} b^3 e^4 x^9 (4 a e+5 b d)+\frac {1}{10} b^4 e^5 x^{10} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (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 = 0.34, size = 396, normalized size = 3.33 \begin {gather*} \frac {1}{10} x^{10} e^{5} b^{4} + \frac {5}{9} x^{9} e^{4} d b^{4} + \frac {4}{9} x^{9} e^{5} b^{3} a + \frac {5}{4} x^{8} e^{3} d^{2} b^{4} + \frac {5}{2} x^{8} e^{4} d b^{3} a + \frac {3}{4} x^{8} e^{5} b^{2} a^{2} + \frac {10}{7} x^{7} e^{2} d^{3} b^{4} + \frac {40}{7} x^{7} e^{3} d^{2} b^{3} a + \frac {30}{7} x^{7} e^{4} d b^{2} a^{2} + \frac {4}{7} x^{7} e^{5} b a^{3} + \frac {5}{6} x^{6} e d^{4} b^{4} + \frac {20}{3} x^{6} e^{2} d^{3} b^{3} a + 10 x^{6} e^{3} d^{2} b^{2} a^{2} + \frac {10}{3} x^{6} e^{4} d b a^{3} + \frac {1}{6} x^{6} e^{5} a^{4} + \frac {1}{5} x^{5} d^{5} b^{4} + 4 x^{5} e d^{4} b^{3} a + 12 x^{5} e^{2} d^{3} b^{2} a^{2} + 8 x^{5} e^{3} d^{2} b a^{3} + x^{5} e^{4} d a^{4} + x^{4} d^{5} b^{3} a + \frac {15}{2} x^{4} e d^{4} b^{2} a^{2} + 10 x^{4} e^{2} d^{3} b a^{3} + \frac {5}{2} x^{4} e^{3} d^{2} a^{4} + 2 x^{3} d^{5} b^{2} a^{2} + \frac {20}{3} x^{3} e d^{4} b a^{3} + \frac {10}{3} x^{3} e^{2} d^{3} a^{4} + 2 x^{2} d^{5} b a^{3} + \frac {5}{2} x^{2} e d^{4} a^{4} + x d^{5} a^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 381, normalized size = 3.20 \begin {gather*} \frac {1}{10} \, b^{4} x^{10} e^{5} + \frac {5}{9} \, b^{4} d x^{9} e^{4} + \frac {5}{4} \, b^{4} d^{2} x^{8} e^{3} + \frac {10}{7} \, b^{4} d^{3} x^{7} e^{2} + \frac {5}{6} \, b^{4} d^{4} x^{6} e + \frac {1}{5} \, b^{4} d^{5} x^{5} + \frac {4}{9} \, a b^{3} x^{9} e^{5} + \frac {5}{2} \, a b^{3} d x^{8} e^{4} + \frac {40}{7} \, a b^{3} d^{2} x^{7} e^{3} + \frac {20}{3} \, a b^{3} d^{3} x^{6} e^{2} + 4 \, a b^{3} d^{4} x^{5} e + a b^{3} d^{5} x^{4} + \frac {3}{4} \, a^{2} b^{2} x^{8} e^{5} + \frac {30}{7} \, a^{2} b^{2} d x^{7} e^{4} + 10 \, a^{2} b^{2} d^{2} x^{6} e^{3} + 12 \, a^{2} b^{2} d^{3} x^{5} e^{2} + \frac {15}{2} \, a^{2} b^{2} d^{4} x^{4} e + 2 \, a^{2} b^{2} d^{5} x^{3} + \frac {4}{7} \, a^{3} b x^{7} e^{5} + \frac {10}{3} \, a^{3} b d x^{6} e^{4} + 8 \, a^{3} b d^{2} x^{5} e^{3} + 10 \, a^{3} b d^{3} x^{4} e^{2} + \frac {20}{3} \, a^{3} b d^{4} x^{3} e + 2 \, a^{3} b d^{5} x^{2} + \frac {1}{6} \, a^{4} x^{6} e^{5} + a^{4} d x^{5} e^{4} + \frac {5}{2} \, a^{4} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{4} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{4} d^{4} x^{2} e + a^{4} 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 = 361, normalized size = 3.03 \begin {gather*} \frac {b^{4} e^{5} x^{10}}{10}+a^{4} d^{5} x +\frac {\left (4 a \,b^{3} e^{5}+5 d \,e^{4} b^{4}\right ) x^{9}}{9}+\frac {\left (6 e^{5} b^{2} a^{2}+20 d \,e^{4} a \,b^{3}+10 d^{2} e^{3} b^{4}\right ) x^{8}}{8}+\frac {\left (4 e^{5} a^{3} b +30 d \,e^{4} b^{2} a^{2}+40 d^{2} e^{3} a \,b^{3}+10 d^{3} e^{2} b^{4}\right ) x^{7}}{7}+\frac {\left (e^{5} a^{4}+20 d \,e^{4} a^{3} b +60 d^{2} e^{3} b^{2} a^{2}+40 d^{3} e^{2} a \,b^{3}+5 d^{4} e \,b^{4}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{4}+40 d^{2} e^{3} a^{3} b +60 d^{3} e^{2} b^{2} a^{2}+20 d^{4} e a \,b^{3}+d^{5} b^{4}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{4}+40 d^{3} e^{2} a^{3} b +30 d^{4} e \,b^{2} a^{2}+4 d^{5} a \,b^{3}\right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{4}+20 d^{4} e \,a^{3} b +6 d^{5} b^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (5 d^{4} e \,a^{4}+4 d^{5} a^{3} b \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 360, normalized size = 3.03 \begin {gather*} \frac {1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac {1}{4} \, {\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac {2}{7} \, {\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{5} + 5 \, a^{4} 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 = 0.13, size = 340, normalized size = 2.86 \begin {gather*} x^4\,\left (\frac {5\,a^4\,d^2\,e^3}{2}+10\,a^3\,b\,d^3\,e^2+\frac {15\,a^2\,b^2\,d^4\,e}{2}+a\,b^3\,d^5\right )+x^7\,\left (\frac {4\,a^3\,b\,e^5}{7}+\frac {30\,a^2\,b^2\,d\,e^4}{7}+\frac {40\,a\,b^3\,d^2\,e^3}{7}+\frac {10\,b^4\,d^3\,e^2}{7}\right )+x^5\,\left (a^4\,d\,e^4+8\,a^3\,b\,d^2\,e^3+12\,a^2\,b^2\,d^3\,e^2+4\,a\,b^3\,d^4\,e+\frac {b^4\,d^5}{5}\right )+x^6\,\left (\frac {a^4\,e^5}{6}+\frac {10\,a^3\,b\,d\,e^4}{3}+10\,a^2\,b^2\,d^2\,e^3+\frac {20\,a\,b^3\,d^3\,e^2}{3}+\frac {5\,b^4\,d^4\,e}{6}\right )+a^4\,d^5\,x+\frac {b^4\,e^5\,x^{10}}{10}+\frac {a^3\,d^4\,x^2\,\left (5\,a\,e+4\,b\,d\right )}{2}+\frac {b^3\,e^4\,x^9\,\left (4\,a\,e+5\,b\,d\right )}{9}+\frac {2\,a^2\,d^3\,x^3\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b^2\,e^3\,x^8\,\left (3\,a^2\,e^2+10\,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 = 0.13, size = 401, normalized size = 3.37 \begin {gather*} a^{4} d^{5} x + \frac {b^{4} e^{5} x^{10}}{10} + x^{9} \left (\frac {4 a b^{3} e^{5}}{9} + \frac {5 b^{4} d e^{4}}{9}\right ) + x^{8} \left (\frac {3 a^{2} b^{2} e^{5}}{4} + \frac {5 a b^{3} d e^{4}}{2} + \frac {5 b^{4} d^{2} e^{3}}{4}\right ) + x^{7} \left (\frac {4 a^{3} b e^{5}}{7} + \frac {30 a^{2} b^{2} d e^{4}}{7} + \frac {40 a b^{3} d^{2} e^{3}}{7} + \frac {10 b^{4} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac {a^{4} e^{5}}{6} + \frac {10 a^{3} b d e^{4}}{3} + 10 a^{2} b^{2} d^{2} e^{3} + \frac {20 a b^{3} d^{3} e^{2}}{3} + \frac {5 b^{4} d^{4} e}{6}\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} b d^{2} e^{3} + 12 a^{2} b^{2} d^{3} e^{2} + 4 a b^{3} d^{4} e + \frac {b^{4} d^{5}}{5}\right ) + x^{4} \left (\frac {5 a^{4} d^{2} e^{3}}{2} + 10 a^{3} b d^{3} e^{2} + \frac {15 a^{2} b^{2} d^{4} e}{2} + a b^{3} d^{5}\right ) + x^{3} \left (\frac {10 a^{4} d^{3} e^{2}}{3} + \frac {20 a^{3} b d^{4} e}{3} + 2 a^{2} b^{2} d^{5}\right ) + x^{2} \left (\frac {5 a^{4} d^{4} e}{2} + 2 a^{3} b d^{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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