Optimal. Leaf size=38 \[ \frac {(a+b x)^8 (b d-a e)}{8 b^2}+\frac {e (a+b x)^9}{9 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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} \[ \frac {(a+b x)^8 (b d-a e)}{8 b^2}+\frac {e (a+b x)^9}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x) \, dx\\ &=\int \left (\frac {(b d-a e) (a+b x)^7}{b}+\frac {e (a+b x)^8}{b}\right ) \, dx\\ &=\frac {(b d-a e) (a+b x)^8}{8 b^2}+\frac {e (a+b x)^9}{9 b^2}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 151, normalized size = 3.97 \[ a^7 d x+\frac {1}{2} a^6 x^2 (a e+7 b d)+\frac {7}{3} a^5 b x^3 (a e+3 b d)+\frac {7}{4} a^4 b^2 x^4 (3 a e+5 b d)+7 a^3 b^3 x^5 (a e+b d)+\frac {7}{6} a^2 b^4 x^6 (5 a e+3 b d)+\frac {1}{8} b^6 x^8 (7 a e+b d)+a b^5 x^7 (3 a e+b d)+\frac {1}{9} b^7 e x^9 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 169, normalized size = 4.45 \[ \frac {1}{9} x^{9} e b^{7} + \frac {1}{8} x^{8} d b^{7} + \frac {7}{8} x^{8} e b^{6} a + x^{7} d b^{6} a + 3 x^{7} e b^{5} a^{2} + \frac {7}{2} x^{6} d b^{5} a^{2} + \frac {35}{6} x^{6} e b^{4} a^{3} + 7 x^{5} d b^{4} a^{3} + 7 x^{5} e b^{3} a^{4} + \frac {35}{4} x^{4} d b^{3} a^{4} + \frac {21}{4} x^{4} e b^{2} a^{5} + 7 x^{3} d b^{2} a^{5} + \frac {7}{3} x^{3} e b a^{6} + \frac {7}{2} x^{2} d b a^{6} + \frac {1}{2} x^{2} e a^{7} + x d a^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 177, normalized size = 4.66 \[ \frac {1}{9} \, b^{7} x^{9} e + \frac {1}{8} \, b^{7} d x^{8} + \frac {7}{8} \, a b^{6} x^{8} e + a b^{6} d x^{7} + 3 \, a^{2} b^{5} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d x^{6} + \frac {35}{6} \, a^{3} b^{4} x^{6} e + 7 \, a^{3} b^{4} d x^{5} + 7 \, a^{4} b^{3} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d x^{4} + \frac {21}{4} \, a^{5} b^{2} x^{4} e + 7 \, a^{5} b^{2} d x^{3} + \frac {7}{3} \, a^{6} b x^{3} e + \frac {7}{2} \, a^{6} b d x^{2} + \frac {1}{2} \, a^{7} x^{2} e + a^{7} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 250, normalized size = 6.58 \[ \frac {b^{7} e \,x^{9}}{9}+a^{7} d x +\frac {\left (6 a \,b^{6} e +\left (a e +b d \right ) b^{6}\right ) x^{8}}{8}+\frac {\left (15 a^{2} b^{5} e +a \,b^{6} d +6 \left (a e +b d \right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (20 a^{3} b^{4} e +6 a^{2} b^{5} d +15 \left (a e +b d \right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (15 a^{4} b^{3} e +15 a^{3} b^{4} d +20 \left (a e +b d \right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (6 a^{5} b^{2} e +20 a^{4} b^{3} d +15 \left (a e +b d \right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (a^{6} b e +15 a^{5} b^{2} d +6 \left (a e +b d \right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b d +\left (a e +b d \right ) a^{6}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 163, normalized size = 4.29 \[ \frac {1}{9} \, b^{7} e x^{9} + a^{7} d x + \frac {1}{8} \, {\left (b^{7} d + 7 \, a b^{6} e\right )} x^{8} + {\left (a b^{6} d + 3 \, a^{2} b^{5} e\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d + 5 \, a^{3} b^{4} e\right )} x^{6} + 7 \, {\left (a^{3} b^{4} d + a^{4} b^{3} e\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d + 3 \, a^{5} b^{2} e\right )} x^{4} + \frac {7}{3} \, {\left (3 \, a^{5} b^{2} d + a^{6} b e\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d + a^{7} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 143, normalized size = 3.76 \[ x^2\,\left (\frac {e\,a^7}{2}+\frac {7\,b\,d\,a^6}{2}\right )+x^8\,\left (\frac {d\,b^7}{8}+\frac {7\,a\,e\,b^6}{8}\right )+\frac {b^7\,e\,x^9}{9}+a^7\,d\,x+\frac {7\,a^5\,b\,x^3\,\left (a\,e+3\,b\,d\right )}{3}+a\,b^5\,x^7\,\left (3\,a\,e+b\,d\right )+7\,a^3\,b^3\,x^5\,\left (a\,e+b\,d\right )+\frac {7\,a^4\,b^2\,x^4\,\left (3\,a\,e+5\,b\,d\right )}{4}+\frac {7\,a^2\,b^4\,x^6\,\left (5\,a\,e+3\,b\,d\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 178, normalized size = 4.68 \[ a^{7} d x + \frac {b^{7} e x^{9}}{9} + x^{8} \left (\frac {7 a b^{6} e}{8} + \frac {b^{7} d}{8}\right ) + x^{7} \left (3 a^{2} b^{5} e + a b^{6} d\right ) + x^{6} \left (\frac {35 a^{3} b^{4} e}{6} + \frac {7 a^{2} b^{5} d}{2}\right ) + x^{5} \left (7 a^{4} b^{3} e + 7 a^{3} b^{4} d\right ) + x^{4} \left (\frac {21 a^{5} b^{2} e}{4} + \frac {35 a^{4} b^{3} d}{4}\right ) + x^{3} \left (\frac {7 a^{6} b e}{3} + 7 a^{5} b^{2} d\right ) + x^{2} \left (\frac {a^{7} e}{2} + \frac {7 a^{6} b d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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