Optimal. Leaf size=92 \[ -\frac {3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac {3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac {b^3 (d+e x)^9}{9 e^4} \]
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Rubi [A] time = 0.15, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} -\frac {3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac {3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac {b^3 (d+e x)^9}{9 e^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)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^5 \, dx\\ &=\int \left (\frac {(-b d+a e)^3 (d+e x)^5}{e^3}+\frac {3 b (b d-a e)^2 (d+e x)^6}{e^3}-\frac {3 b^2 (b d-a e) (d+e x)^7}{e^3}+\frac {b^3 (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {(b d-a e)^3 (d+e x)^6}{6 e^4}+\frac {3 b (b d-a e)^2 (d+e x)^7}{7 e^4}-\frac {3 b^2 (b d-a e) (d+e x)^8}{8 e^4}+\frac {b^3 (d+e x)^9}{9 e^4}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 267, normalized size = 2.90 \begin {gather*} a^3 d^5 x+\frac {1}{7} b e^3 x^7 \left (3 a^2 e^2+15 a b d e+10 b^2 d^2\right )+\frac {1}{3} a d^3 x^3 \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )+\frac {1}{2} a^2 d^4 x^2 (5 a e+3 b d)+\frac {1}{6} e^2 x^6 \left (a^3 e^3+15 a^2 b d e^2+30 a b^2 d^2 e+10 b^3 d^3\right )+d e x^5 \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{4} d^2 x^4 \left (10 a^3 e^3+30 a^2 b d e^2+15 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{8} b^2 e^4 x^8 (3 a e+5 b d)+\frac {1}{9} b^3 e^5 x^9 \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 ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 303, normalized size = 3.29 \begin {gather*} \frac {1}{9} x^{9} e^{5} b^{3} + \frac {5}{8} x^{8} e^{4} d b^{3} + \frac {3}{8} x^{8} e^{5} b^{2} a + \frac {10}{7} x^{7} e^{3} d^{2} b^{3} + \frac {15}{7} x^{7} e^{4} d b^{2} a + \frac {3}{7} x^{7} e^{5} b a^{2} + \frac {5}{3} x^{6} e^{2} d^{3} b^{3} + 5 x^{6} e^{3} d^{2} b^{2} a + \frac {5}{2} x^{6} e^{4} d b a^{2} + \frac {1}{6} x^{6} e^{5} a^{3} + x^{5} e d^{4} b^{3} + 6 x^{5} e^{2} d^{3} b^{2} a + 6 x^{5} e^{3} d^{2} b a^{2} + x^{5} e^{4} d a^{3} + \frac {1}{4} x^{4} d^{5} b^{3} + \frac {15}{4} x^{4} e d^{4} b^{2} a + \frac {15}{2} x^{4} e^{2} d^{3} b a^{2} + \frac {5}{2} x^{4} e^{3} d^{2} a^{3} + x^{3} d^{5} b^{2} a + 5 x^{3} e d^{4} b a^{2} + \frac {10}{3} x^{3} e^{2} d^{3} a^{3} + \frac {3}{2} x^{2} d^{5} b a^{2} + \frac {5}{2} x^{2} e d^{4} a^{3} + x d^{5} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 291, normalized size = 3.16 \begin {gather*} \frac {1}{9} \, b^{3} x^{9} e^{5} + \frac {5}{8} \, b^{3} d x^{8} e^{4} + \frac {10}{7} \, b^{3} d^{2} x^{7} e^{3} + \frac {5}{3} \, b^{3} d^{3} x^{6} e^{2} + b^{3} d^{4} x^{5} e + \frac {1}{4} \, b^{3} d^{5} x^{4} + \frac {3}{8} \, a b^{2} x^{8} e^{5} + \frac {15}{7} \, a b^{2} d x^{7} e^{4} + 5 \, a b^{2} d^{2} x^{6} e^{3} + 6 \, a b^{2} d^{3} x^{5} e^{2} + \frac {15}{4} \, a b^{2} d^{4} x^{4} e + a b^{2} d^{5} x^{3} + \frac {3}{7} \, a^{2} b x^{7} e^{5} + \frac {5}{2} \, a^{2} b d x^{6} e^{4} + 6 \, a^{2} b d^{2} x^{5} e^{3} + \frac {15}{2} \, a^{2} b d^{3} x^{4} e^{2} + 5 \, a^{2} b d^{4} x^{3} e + \frac {3}{2} \, a^{2} b d^{5} x^{2} + \frac {1}{6} \, a^{3} x^{6} e^{5} + a^{3} d x^{5} e^{4} + \frac {5}{2} \, a^{3} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{3} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{3} d^{4} x^{2} e + a^{3} 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 = 394, normalized size = 4.28 \begin {gather*} \frac {b^{3} e^{5} x^{9}}{9}+a^{3} d^{5} x +\frac {\left (2 a \,b^{2} e^{5}+\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{2}\right ) x^{8}}{8}+\frac {\left (a^{2} b \,e^{5}+2 \left (a \,e^{5}+5 b d \,e^{4}\right ) a b +\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{2}\right ) x^{7}}{7}+\frac {\left (\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2}+2 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a b +\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) b^{2}\right ) x^{6}}{6}+\frac {\left (\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2}+2 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a b +\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{2}+2 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a b +\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (a \,b^{2} d^{5}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{2}+2 \left (5 a \,d^{4} e +b \,d^{5}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a^{2} b \,d^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2}\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 = 277, normalized size = 3.01 \begin {gather*} \frac {1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} + {\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{5} + 5 \, a^{3} 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 = 0.11, size = 261, normalized size = 2.84 \begin {gather*} x^5\,\left (a^3\,d\,e^4+6\,a^2\,b\,d^2\,e^3+6\,a\,b^2\,d^3\,e^2+b^3\,d^4\,e\right )+x^4\,\left (\frac {5\,a^3\,d^2\,e^3}{2}+\frac {15\,a^2\,b\,d^3\,e^2}{2}+\frac {15\,a\,b^2\,d^4\,e}{4}+\frac {b^3\,d^5}{4}\right )+x^6\,\left (\frac {a^3\,e^5}{6}+\frac {5\,a^2\,b\,d\,e^4}{2}+5\,a\,b^2\,d^2\,e^3+\frac {5\,b^3\,d^3\,e^2}{3}\right )+a^3\,d^5\,x+\frac {b^3\,e^5\,x^9}{9}+\frac {a^2\,d^4\,x^2\,\left (5\,a\,e+3\,b\,d\right )}{2}+\frac {b^2\,e^4\,x^8\,\left (3\,a\,e+5\,b\,d\right )}{8}+\frac {a\,d^3\,x^3\,\left (10\,a^2\,e^2+15\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b\,e^3\,x^7\,\left (3\,a^2\,e^2+15\,a\,b\,d\,e+10\,b^2\,d^2\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 308, normalized size = 3.35 \begin {gather*} a^{3} d^{5} x + \frac {b^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac {3 a b^{2} e^{5}}{8} + \frac {5 b^{3} d e^{4}}{8}\right ) + x^{7} \left (\frac {3 a^{2} b e^{5}}{7} + \frac {15 a b^{2} d e^{4}}{7} + \frac {10 b^{3} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{5}}{6} + \frac {5 a^{2} b d e^{4}}{2} + 5 a b^{2} d^{2} e^{3} + \frac {5 b^{3} d^{3} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} b d^{2} e^{3} + 6 a b^{2} d^{3} e^{2} + b^{3} d^{4} e\right ) + x^{4} \left (\frac {5 a^{3} d^{2} e^{3}}{2} + \frac {15 a^{2} b d^{3} e^{2}}{2} + \frac {15 a b^{2} d^{4} e}{4} + \frac {b^{3} d^{5}}{4}\right ) + x^{3} \left (\frac {10 a^{3} d^{3} e^{2}}{3} + 5 a^{2} b d^{4} e + a b^{2} d^{5}\right ) + x^{2} \left (\frac {5 a^{3} d^{4} e}{2} + \frac {3 a^{2} b d^{5}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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