Optimal. Leaf size=65 \[ \frac {2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac {(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3} \]
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Rubi [A] time = 0.15, antiderivative size = 65, 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 {2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac {(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3} \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)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^2 \, dx\\ &=\int \left (\frac {(b d-a e)^2 (a+b x)^7}{b^2}+\frac {2 e (b d-a e) (a+b x)^8}{b^2}+\frac {e^2 (a+b x)^9}{b^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (a+b x)^8}{8 b^3}+\frac {2 e (b d-a e) (a+b x)^9}{9 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 229, normalized size = 3.52 \begin {gather*} \frac {1}{360} x \left (120 a^7 \left (3 d^2+3 d e x+e^2 x^2\right )+210 a^6 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+252 a^5 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+210 a^4 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+120 a^3 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+45 a^2 b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+10 a b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )+b^7 x^7 \left (45 d^2+80 d e x+36 e^2 x^2\right )\right ) \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)^2 \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 = 0.40, size = 294, normalized size = 4.52 \begin {gather*} \frac {1}{10} x^{10} e^{2} b^{7} + \frac {2}{9} x^{9} e d b^{7} + \frac {7}{9} x^{9} e^{2} b^{6} a + \frac {1}{8} x^{8} d^{2} b^{7} + \frac {7}{4} x^{8} e d b^{6} a + \frac {21}{8} x^{8} e^{2} b^{5} a^{2} + x^{7} d^{2} b^{6} a + 6 x^{7} e d b^{5} a^{2} + 5 x^{7} e^{2} b^{4} a^{3} + \frac {7}{2} x^{6} d^{2} b^{5} a^{2} + \frac {35}{3} x^{6} e d b^{4} a^{3} + \frac {35}{6} x^{6} e^{2} b^{3} a^{4} + 7 x^{5} d^{2} b^{4} a^{3} + 14 x^{5} e d b^{3} a^{4} + \frac {21}{5} x^{5} e^{2} b^{2} a^{5} + \frac {35}{4} x^{4} d^{2} b^{3} a^{4} + \frac {21}{2} x^{4} e d b^{2} a^{5} + \frac {7}{4} x^{4} e^{2} b a^{6} + 7 x^{3} d^{2} b^{2} a^{5} + \frac {14}{3} x^{3} e d b a^{6} + \frac {1}{3} x^{3} e^{2} a^{7} + \frac {7}{2} x^{2} d^{2} b a^{6} + x^{2} e d a^{7} + x d^{2} a^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 294, normalized size = 4.52 \begin {gather*} \frac {1}{10} \, b^{7} x^{10} e^{2} + \frac {2}{9} \, b^{7} d x^{9} e + \frac {1}{8} \, b^{7} d^{2} x^{8} + \frac {7}{9} \, a b^{6} x^{9} e^{2} + \frac {7}{4} \, a b^{6} d x^{8} e + a b^{6} d^{2} x^{7} + \frac {21}{8} \, a^{2} b^{5} x^{8} e^{2} + 6 \, a^{2} b^{5} d x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{2} x^{6} + 5 \, a^{3} b^{4} x^{7} e^{2} + \frac {35}{3} \, a^{3} b^{4} d x^{6} e + 7 \, a^{3} b^{4} d^{2} x^{5} + \frac {35}{6} \, a^{4} b^{3} x^{6} e^{2} + 14 \, a^{4} b^{3} d x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{2} x^{4} + \frac {21}{5} \, a^{5} b^{2} x^{5} e^{2} + \frac {21}{2} \, a^{5} b^{2} d x^{4} e + 7 \, a^{5} b^{2} d^{2} x^{3} + \frac {7}{4} \, a^{6} b x^{4} e^{2} + \frac {14}{3} \, a^{6} b d x^{3} e + \frac {7}{2} \, a^{6} b d^{2} x^{2} + \frac {1}{3} \, a^{7} x^{3} e^{2} + a^{7} d x^{2} e + a^{7} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 433, normalized size = 6.66 \begin {gather*} \frac {b^{7} e^{2} x^{10}}{10}+a^{7} d^{2} x +\frac {\left (6 a \,b^{6} e^{2}+\left (a \,e^{2}+2 b d e \right ) b^{6}\right ) x^{9}}{9}+\frac {\left (15 a^{2} b^{5} e^{2}+6 \left (a \,e^{2}+2 b d e \right ) a \,b^{5}+\left (2 a d e +b \,d^{2}\right ) b^{6}\right ) x^{8}}{8}+\frac {\left (20 a^{3} b^{4} e^{2}+a \,b^{6} d^{2}+15 \left (a \,e^{2}+2 b d e \right ) a^{2} b^{4}+6 \left (2 a d e +b \,d^{2}\right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (15 a^{4} b^{3} e^{2}+6 a^{2} b^{5} d^{2}+20 \left (a \,e^{2}+2 b d e \right ) a^{3} b^{3}+15 \left (2 a d e +b \,d^{2}\right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (6 a^{5} b^{2} e^{2}+15 a^{3} b^{4} d^{2}+15 \left (a \,e^{2}+2 b d e \right ) a^{4} b^{2}+20 \left (2 a d e +b \,d^{2}\right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (a^{6} b \,e^{2}+20 a^{4} b^{3} d^{2}+6 \left (a \,e^{2}+2 b d e \right ) a^{5} b +15 \left (2 a d e +b \,d^{2}\right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (15 a^{5} b^{2} d^{2}+\left (a \,e^{2}+2 b d e \right ) a^{6}+6 \left (2 a d e +b \,d^{2}\right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b \,d^{2}+\left (2 a d e +b \,d^{2}\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 = 0.56, size = 273, normalized size = 4.20 \begin {gather*} \frac {1}{10} \, b^{7} e^{2} x^{10} + a^{7} d^{2} x + \frac {1}{9} \, {\left (2 \, b^{7} d e + 7 \, a b^{6} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{2} + 14 \, a b^{6} d e + 21 \, a^{2} b^{5} e^{2}\right )} x^{8} + {\left (a b^{6} d^{2} + 6 \, a^{2} b^{5} d e + 5 \, a^{3} b^{4} e^{2}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d^{2} + 10 \, a^{3} b^{4} d e + 5 \, a^{4} b^{3} e^{2}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{2} + 10 \, a^{4} b^{3} d e + 3 \, a^{5} b^{2} e^{2}\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d^{2} + 6 \, a^{5} b^{2} d e + a^{6} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{2} + 14 \, a^{6} b d e + a^{7} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{2} + 2 \, a^{7} d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.09, size = 249, normalized size = 3.83 \begin {gather*} x^3\,\left (\frac {a^7\,e^2}{3}+\frac {14\,a^6\,b\,d\,e}{3}+7\,a^5\,b^2\,d^2\right )+x^8\,\left (\frac {21\,a^2\,b^5\,e^2}{8}+\frac {7\,a\,b^6\,d\,e}{4}+\frac {b^7\,d^2}{8}\right )+a^7\,d^2\,x+\frac {b^7\,e^2\,x^{10}}{10}+\frac {a^6\,d\,x^2\,\left (2\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e\,x^9\,\left (7\,a\,e+2\,b\,d\right )}{9}+\frac {7\,a^4\,b\,x^4\,\left (a^2\,e^2+6\,a\,b\,d\,e+5\,b^2\,d^2\right )}{4}+a\,b^4\,x^7\,\left (5\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )+\frac {7\,a^3\,b^2\,x^5\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{5}+\frac {7\,a^2\,b^3\,x^6\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 303, normalized size = 4.66 \begin {gather*} a^{7} d^{2} x + \frac {b^{7} e^{2} x^{10}}{10} + x^{9} \left (\frac {7 a b^{6} e^{2}}{9} + \frac {2 b^{7} d e}{9}\right ) + x^{8} \left (\frac {21 a^{2} b^{5} e^{2}}{8} + \frac {7 a b^{6} d e}{4} + \frac {b^{7} d^{2}}{8}\right ) + x^{7} \left (5 a^{3} b^{4} e^{2} + 6 a^{2} b^{5} d e + a b^{6} d^{2}\right ) + x^{6} \left (\frac {35 a^{4} b^{3} e^{2}}{6} + \frac {35 a^{3} b^{4} d e}{3} + \frac {7 a^{2} b^{5} d^{2}}{2}\right ) + x^{5} \left (\frac {21 a^{5} b^{2} e^{2}}{5} + 14 a^{4} b^{3} d e + 7 a^{3} b^{4} d^{2}\right ) + x^{4} \left (\frac {7 a^{6} b e^{2}}{4} + \frac {21 a^{5} b^{2} d e}{2} + \frac {35 a^{4} b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{7} e^{2}}{3} + \frac {14 a^{6} b d e}{3} + 7 a^{5} b^{2} d^{2}\right ) + x^{2} \left (a^{7} d e + \frac {7 a^{6} b d^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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