Optimal. Leaf size=92 \[ \frac {3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac {e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac {(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4} \]
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Rubi [A] time = 0.22, antiderivative size = 92, 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 {3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac {e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac {(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4} \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)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^3 \, dx\\ &=\int \left (\frac {(b d-a e)^3 (a+b x)^7}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^8}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^9}{b^3}+\frac {e^3 (a+b x)^{10}}{b^3}\right ) \, dx\\ &=\frac {(b d-a e)^3 (a+b x)^8}{8 b^4}+\frac {e (b d-a e)^2 (a+b x)^9}{3 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^{10}}{10 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 360, normalized size = 3.91 \begin {gather*} a^7 d^3 x+\frac {1}{2} a^6 d^2 x^2 (3 a e+7 b d)+\frac {1}{3} b^5 e x^9 \left (7 a^2 e^2+7 a b d e+b^2 d^2\right )+a^5 d x^3 \left (a^2 e^2+7 a b d e+7 b^2 d^2\right )+a b^3 x^7 \left (5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+\frac {7}{2} a^2 b^2 x^6 \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 {7}{5} a^3 b x^5 \left (a^3 e^3+9 a^2 b d e^2+15 a b^2 d^2 e+5 b^3 d^3\right )+\frac {1}{8} b^4 x^8 \left (35 a^3 e^3+63 a^2 b d e^2+21 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{4} a^4 x^4 \left (a^3 e^3+21 a^2 b d e^2+63 a b^2 d^2 e+35 b^3 d^3\right )+\frac {1}{10} b^6 e^2 x^{10} (7 a e+3 b d)+\frac {1}{11} b^7 e^3 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)^3 \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 = 0.44, size = 420, normalized size = 4.57 \begin {gather*} \frac {1}{11} x^{11} e^{3} b^{7} + \frac {3}{10} x^{10} e^{2} d b^{7} + \frac {7}{10} x^{10} e^{3} b^{6} a + \frac {1}{3} x^{9} e d^{2} b^{7} + \frac {7}{3} x^{9} e^{2} d b^{6} a + \frac {7}{3} x^{9} e^{3} b^{5} a^{2} + \frac {1}{8} x^{8} d^{3} b^{7} + \frac {21}{8} x^{8} e d^{2} b^{6} a + \frac {63}{8} x^{8} e^{2} d b^{5} a^{2} + \frac {35}{8} x^{8} e^{3} b^{4} a^{3} + x^{7} d^{3} b^{6} a + 9 x^{7} e d^{2} b^{5} a^{2} + 15 x^{7} e^{2} d b^{4} a^{3} + 5 x^{7} e^{3} b^{3} a^{4} + \frac {7}{2} x^{6} d^{3} b^{5} a^{2} + \frac {35}{2} x^{6} e d^{2} b^{4} a^{3} + \frac {35}{2} x^{6} e^{2} d b^{3} a^{4} + \frac {7}{2} x^{6} e^{3} b^{2} a^{5} + 7 x^{5} d^{3} b^{4} a^{3} + 21 x^{5} e d^{2} b^{3} a^{4} + \frac {63}{5} x^{5} e^{2} d b^{2} a^{5} + \frac {7}{5} x^{5} e^{3} b a^{6} + \frac {35}{4} x^{4} d^{3} b^{3} a^{4} + \frac {63}{4} x^{4} e d^{2} b^{2} a^{5} + \frac {21}{4} x^{4} e^{2} d b a^{6} + \frac {1}{4} x^{4} e^{3} a^{7} + 7 x^{3} d^{3} b^{2} a^{5} + 7 x^{3} e d^{2} b a^{6} + x^{3} e^{2} d a^{7} + \frac {7}{2} x^{2} d^{3} b a^{6} + \frac {3}{2} x^{2} e d^{2} a^{7} + x d^{3} a^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 412, normalized size = 4.48 \begin {gather*} \frac {1}{11} \, b^{7} x^{11} e^{3} + \frac {3}{10} \, b^{7} d x^{10} e^{2} + \frac {1}{3} \, b^{7} d^{2} x^{9} e + \frac {1}{8} \, b^{7} d^{3} x^{8} + \frac {7}{10} \, a b^{6} x^{10} e^{3} + \frac {7}{3} \, a b^{6} d x^{9} e^{2} + \frac {21}{8} \, a b^{6} d^{2} x^{8} e + a b^{6} d^{3} x^{7} + \frac {7}{3} \, a^{2} b^{5} x^{9} e^{3} + \frac {63}{8} \, a^{2} b^{5} d x^{8} e^{2} + 9 \, a^{2} b^{5} d^{2} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{3} x^{6} + \frac {35}{8} \, a^{3} b^{4} x^{8} e^{3} + 15 \, a^{3} b^{4} d x^{7} e^{2} + \frac {35}{2} \, a^{3} b^{4} d^{2} x^{6} e + 7 \, a^{3} b^{4} d^{3} x^{5} + 5 \, a^{4} b^{3} x^{7} e^{3} + \frac {35}{2} \, a^{4} b^{3} d x^{6} e^{2} + 21 \, a^{4} b^{3} d^{2} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{3} x^{4} + \frac {7}{2} \, a^{5} b^{2} x^{6} e^{3} + \frac {63}{5} \, a^{5} b^{2} d x^{5} e^{2} + \frac {63}{4} \, a^{5} b^{2} d^{2} x^{4} e + 7 \, a^{5} b^{2} d^{3} x^{3} + \frac {7}{5} \, a^{6} b x^{5} e^{3} + \frac {21}{4} \, a^{6} b d x^{4} e^{2} + 7 \, a^{6} b d^{2} x^{3} e + \frac {7}{2} \, a^{6} b d^{3} x^{2} + \frac {1}{4} \, a^{7} x^{4} e^{3} + a^{7} d x^{3} e^{2} + \frac {3}{2} \, a^{7} d^{2} x^{2} e + a^{7} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 616, normalized size = 6.70 \begin {gather*} \frac {b^{7} e^{3} x^{11}}{11}+a^{7} d^{3} x +\frac {\left (6 a \,b^{6} e^{3}+\left (a \,e^{3}+3 b d \,e^{2}\right ) b^{6}\right ) x^{10}}{10}+\frac {\left (15 a^{2} b^{5} e^{3}+6 \left (a \,e^{3}+3 b d \,e^{2}\right ) a \,b^{5}+\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) b^{6}\right ) x^{9}}{9}+\frac {\left (20 a^{3} b^{4} e^{3}+15 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{2} b^{4}+6 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a \,b^{5}+\left (3 a \,d^{2} e +b \,d^{3}\right ) b^{6}\right ) x^{8}}{8}+\frac {\left (15 a^{4} b^{3} e^{3}+a \,b^{6} d^{3}+20 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{3} b^{3}+15 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{2} b^{4}+6 \left (3 a \,d^{2} e +b \,d^{3}\right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (6 a^{5} b^{2} e^{3}+6 a^{2} b^{5} d^{3}+15 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{4} b^{2}+20 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{3} b^{3}+15 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (a^{6} b \,e^{3}+15 a^{3} b^{4} d^{3}+6 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{5} b +15 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{4} b^{2}+20 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (20 a^{4} b^{3} d^{3}+\left (a \,e^{3}+3 b d \,e^{2}\right ) a^{6}+6 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{5} b +15 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (15 a^{5} b^{2} d^{3}+\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{6}+6 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b \,d^{3}+\left (3 a \,d^{2} e +b \,d^{3}\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 = 0.58, size = 376, normalized size = 4.09 \begin {gather*} \frac {1}{11} \, b^{7} e^{3} x^{11} + a^{7} d^{3} x + \frac {1}{10} \, {\left (3 \, b^{7} d e^{2} + 7 \, a b^{6} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (b^{7} d^{2} e + 7 \, a b^{6} d e^{2} + 7 \, a^{2} b^{5} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{3} + 21 \, a b^{6} d^{2} e + 63 \, a^{2} b^{5} d e^{2} + 35 \, a^{3} b^{4} e^{3}\right )} x^{8} + {\left (a b^{6} d^{3} + 9 \, a^{2} b^{5} d^{2} e + 15 \, a^{3} b^{4} d e^{2} + 5 \, a^{4} b^{3} e^{3}\right )} x^{7} + \frac {7}{2} \, {\left (a^{2} b^{5} d^{3} + 5 \, a^{3} b^{4} d^{2} e + 5 \, a^{4} b^{3} d e^{2} + a^{5} b^{2} e^{3}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{3} + 15 \, a^{4} b^{3} d^{2} e + 9 \, a^{5} b^{2} d e^{2} + a^{6} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, a^{4} b^{3} d^{3} + 63 \, a^{5} b^{2} d^{2} e + 21 \, a^{6} b d e^{2} + a^{7} e^{3}\right )} x^{4} + {\left (7 \, a^{5} b^{2} d^{3} + 7 \, a^{6} b d^{2} e + a^{7} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{3} + 3 \, a^{7} d^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 356, normalized size = 3.87 \begin {gather*} x^7\,\left (5\,a^4\,b^3\,e^3+15\,a^3\,b^4\,d\,e^2+9\,a^2\,b^5\,d^2\,e+a\,b^6\,d^3\right )+x^5\,\left (\frac {7\,a^6\,b\,e^3}{5}+\frac {63\,a^5\,b^2\,d\,e^2}{5}+21\,a^4\,b^3\,d^2\,e+7\,a^3\,b^4\,d^3\right )+x^4\,\left (\frac {a^7\,e^3}{4}+\frac {21\,a^6\,b\,d\,e^2}{4}+\frac {63\,a^5\,b^2\,d^2\,e}{4}+\frac {35\,a^4\,b^3\,d^3}{4}\right )+x^8\,\left (\frac {35\,a^3\,b^4\,e^3}{8}+\frac {63\,a^2\,b^5\,d\,e^2}{8}+\frac {21\,a\,b^6\,d^2\,e}{8}+\frac {b^7\,d^3}{8}\right )+a^7\,d^3\,x+\frac {b^7\,e^3\,x^{11}}{11}+\frac {7\,a^2\,b^2\,x^6\,\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 {a^6\,d^2\,x^2\,\left (3\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^2\,x^{10}\,\left (7\,a\,e+3\,b\,d\right )}{10}+a^5\,d\,x^3\,\left (a^2\,e^2+7\,a\,b\,d\,e+7\,b^2\,d^2\right )+\frac {b^5\,e\,x^9\,\left (7\,a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 427, normalized size = 4.64 \begin {gather*} a^{7} d^{3} x + \frac {b^{7} e^{3} x^{11}}{11} + x^{10} \left (\frac {7 a b^{6} e^{3}}{10} + \frac {3 b^{7} d e^{2}}{10}\right ) + x^{9} \left (\frac {7 a^{2} b^{5} e^{3}}{3} + \frac {7 a b^{6} d e^{2}}{3} + \frac {b^{7} d^{2} e}{3}\right ) + x^{8} \left (\frac {35 a^{3} b^{4} e^{3}}{8} + \frac {63 a^{2} b^{5} d e^{2}}{8} + \frac {21 a b^{6} d^{2} e}{8} + \frac {b^{7} d^{3}}{8}\right ) + x^{7} \left (5 a^{4} b^{3} e^{3} + 15 a^{3} b^{4} d e^{2} + 9 a^{2} b^{5} d^{2} e + a b^{6} d^{3}\right ) + x^{6} \left (\frac {7 a^{5} b^{2} e^{3}}{2} + \frac {35 a^{4} b^{3} d e^{2}}{2} + \frac {35 a^{3} b^{4} d^{2} e}{2} + \frac {7 a^{2} b^{5} d^{3}}{2}\right ) + x^{5} \left (\frac {7 a^{6} b e^{3}}{5} + \frac {63 a^{5} b^{2} d e^{2}}{5} + 21 a^{4} b^{3} d^{2} e + 7 a^{3} b^{4} d^{3}\right ) + x^{4} \left (\frac {a^{7} e^{3}}{4} + \frac {21 a^{6} b d e^{2}}{4} + \frac {63 a^{5} b^{2} d^{2} e}{4} + \frac {35 a^{4} b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{7} d e^{2} + 7 a^{6} b d^{2} e + 7 a^{5} b^{2} d^{3}\right ) + x^{2} \left (\frac {3 a^{7} d^{2} e}{2} + \frac {7 a^{6} b d^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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