Optimal. Leaf size=65 \[ \frac {2 e (a+b x)^7 (b d-a e)}{7 b^3}+\frac {(a+b x)^6 (b d-a e)^2}{6 b^3}+\frac {e^2 (a+b x)^8}{8 b^3} \]
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Rubi [A] time = 0.11, 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)^7 (b d-a e)}{7 b^3}+\frac {(a+b x)^6 (b d-a e)^2}{6 b^3}+\frac {e^2 (a+b x)^8}{8 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 )^2 \, dx &=\int (a+b x)^5 (d+e x)^2 \, dx\\ &=\int \left (\frac {(b d-a e)^2 (a+b x)^5}{b^2}+\frac {2 e (b d-a e) (a+b x)^6}{b^2}+\frac {e^2 (a+b x)^7}{b^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (a+b x)^6}{6 b^3}+\frac {2 e (b d-a e) (a+b x)^7}{7 b^3}+\frac {e^2 (a+b x)^8}{8 b^3}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 189, normalized size = 2.91 \begin {gather*} a^5 d^2 x+\frac {1}{2} a^4 d x^2 (2 a e+5 b d)+a b^2 x^5 \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+\frac {5}{4} a^2 b x^4 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+\frac {1}{6} b^3 x^6 \left (10 a^2 e^2+10 a b d e+b^2 d^2\right )+\frac {1}{3} a^3 x^3 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )+\frac {1}{7} b^4 e x^7 (5 a e+2 b d)+\frac {1}{8} b^5 e^2 x^8 \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 )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.36, size = 212, normalized size = 3.26 \begin {gather*} \frac {1}{8} x^{8} e^{2} b^{5} + \frac {2}{7} x^{7} e d b^{5} + \frac {5}{7} x^{7} e^{2} b^{4} a + \frac {1}{6} x^{6} d^{2} b^{5} + \frac {5}{3} x^{6} e d b^{4} a + \frac {5}{3} x^{6} e^{2} b^{3} a^{2} + x^{5} d^{2} b^{4} a + 4 x^{5} e d b^{3} a^{2} + 2 x^{5} e^{2} b^{2} a^{3} + \frac {5}{2} x^{4} d^{2} b^{3} a^{2} + 5 x^{4} e d b^{2} a^{3} + \frac {5}{4} x^{4} e^{2} b a^{4} + \frac {10}{3} x^{3} d^{2} b^{2} a^{3} + \frac {10}{3} x^{3} e d b a^{4} + \frac {1}{3} x^{3} e^{2} a^{5} + \frac {5}{2} x^{2} d^{2} b a^{4} + x^{2} e d a^{5} + x d^{2} a^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 212, normalized size = 3.26 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} e^{2} + \frac {2}{7} \, b^{5} d x^{7} e + \frac {1}{6} \, b^{5} d^{2} x^{6} + \frac {5}{7} \, a b^{4} x^{7} e^{2} + \frac {5}{3} \, a b^{4} d x^{6} e + a b^{4} d^{2} x^{5} + \frac {5}{3} \, a^{2} b^{3} x^{6} e^{2} + 4 \, a^{2} b^{3} d x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{2} x^{4} + 2 \, a^{3} b^{2} x^{5} e^{2} + 5 \, a^{3} b^{2} d x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{2} x^{3} + \frac {5}{4} \, a^{4} b x^{4} e^{2} + \frac {10}{3} \, a^{4} b d x^{3} e + \frac {5}{2} \, a^{4} b d^{2} x^{2} + \frac {1}{3} \, a^{5} x^{3} e^{2} + a^{5} d x^{2} e + a^{5} 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 = 301, normalized size = 4.63 \begin {gather*} \frac {b^{5} e^{2} x^{8}}{8}+a^{5} d^{2} x +\frac {\left (4 a \,b^{4} e^{2}+\left (a \,e^{2}+2 b d e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{3} e^{2}+4 \left (a \,e^{2}+2 b d e \right ) a \,b^{3}+\left (2 a d e +b \,d^{2}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (4 a^{3} b^{2} e^{2}+a \,b^{4} d^{2}+6 \left (a \,e^{2}+2 b d e \right ) a^{2} b^{2}+4 \left (2 a d e +b \,d^{2}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (a^{4} b \,e^{2}+4 a^{2} b^{3} d^{2}+4 \left (a \,e^{2}+2 b d e \right ) a^{3} b +6 \left (2 a d e +b \,d^{2}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 a^{3} b^{2} d^{2}+\left (a \,e^{2}+2 b d e \right ) a^{4}+4 \left (2 a d e +b \,d^{2}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{2}+\left (2 a d e +b \,d^{2}\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 = 0.63, size = 197, normalized size = 3.03 \begin {gather*} \frac {1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} + {\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 181, normalized size = 2.78 \begin {gather*} x^3\,\left (\frac {a^5\,e^2}{3}+\frac {10\,a^4\,b\,d\,e}{3}+\frac {10\,a^3\,b^2\,d^2}{3}\right )+x^6\,\left (\frac {5\,a^2\,b^3\,e^2}{3}+\frac {5\,a\,b^4\,d\,e}{3}+\frac {b^5\,d^2}{6}\right )+a^5\,d^2\,x+\frac {b^5\,e^2\,x^8}{8}+\frac {a^4\,d\,x^2\,\left (2\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e\,x^7\,\left (5\,a\,e+2\,b\,d\right )}{7}+\frac {5\,a^2\,b\,x^4\,\left (a^2\,e^2+4\,a\,b\,d\,e+2\,b^2\,d^2\right )}{4}+a\,b^2\,x^5\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 218, normalized size = 3.35 \begin {gather*} a^{5} d^{2} x + \frac {b^{5} e^{2} x^{8}}{8} + x^{7} \left (\frac {5 a b^{4} e^{2}}{7} + \frac {2 b^{5} d e}{7}\right ) + x^{6} \left (\frac {5 a^{2} b^{3} e^{2}}{3} + \frac {5 a b^{4} d e}{3} + \frac {b^{5} d^{2}}{6}\right ) + x^{5} \left (2 a^{3} b^{2} e^{2} + 4 a^{2} b^{3} d e + a b^{4} d^{2}\right ) + x^{4} \left (\frac {5 a^{4} b e^{2}}{4} + 5 a^{3} b^{2} d e + \frac {5 a^{2} b^{3} d^{2}}{2}\right ) + x^{3} \left (\frac {a^{5} e^{2}}{3} + \frac {10 a^{4} b d e}{3} + \frac {10 a^{3} b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{5} d e + \frac {5 a^{4} b d^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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