Optimal. Leaf size=38 \[ \frac {(a+b x)^6 (b d-a e)}{6 b^2}+\frac {e (a+b x)^7}{7 b^2} \]
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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} \[ \frac {(a+b x)^6 (b d-a e)}{6 b^2}+\frac {e (a+b x)^7}{7 b^2} \]
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 )^2 \, dx &=\int (a+b x)^5 (d+e x) \, dx\\ &=\int \left (\frac {(b d-a e) (a+b x)^5}{b}+\frac {e (a+b x)^6}{b}\right ) \, dx\\ &=\frac {(b d-a e) (a+b x)^6}{6 b^2}+\frac {e (a+b x)^7}{7 b^2}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 109, normalized size = 2.87 \[ a^5 d x+\frac {1}{2} a^4 x^2 (a e+5 b d)+\frac {5}{3} a^3 b x^3 (a e+2 b d)+\frac {5}{2} a^2 b^2 x^4 (a e+b d)+\frac {1}{6} b^4 x^6 (5 a e+b d)+a b^3 x^5 (2 a e+b d)+\frac {1}{7} b^5 e x^7 \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 121, normalized size = 3.18 \[ \frac {1}{7} x^{7} e b^{5} + \frac {1}{6} x^{6} d b^{5} + \frac {5}{6} x^{6} e b^{4} a + x^{5} d b^{4} a + 2 x^{5} e b^{3} a^{2} + \frac {5}{2} x^{4} d b^{3} a^{2} + \frac {5}{2} x^{4} e b^{2} a^{3} + \frac {10}{3} x^{3} d b^{2} a^{3} + \frac {5}{3} x^{3} e b a^{4} + \frac {5}{2} x^{2} d b a^{4} + \frac {1}{2} x^{2} e a^{5} + x d a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 127, normalized size = 3.34 \[ \frac {1}{7} \, b^{5} x^{7} e + \frac {1}{6} \, b^{5} d x^{6} + \frac {5}{6} \, a b^{4} x^{6} e + a b^{4} d x^{5} + 2 \, a^{2} b^{3} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d x^{4} + \frac {5}{2} \, a^{3} b^{2} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d x^{3} + \frac {5}{3} \, a^{4} b x^{3} e + \frac {5}{2} \, a^{4} b d x^{2} + \frac {1}{2} \, a^{5} x^{2} e + a^{5} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 172, normalized size = 4.53 \[ \frac {b^{5} e \,x^{7}}{7}+a^{5} d x +\frac {\left (4 a \,b^{4} e +\left (a e +b d \right ) b^{4}\right ) x^{6}}{6}+\frac {\left (6 a^{2} b^{3} e +a \,b^{4} d +4 \left (a e +b d \right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 a^{3} b^{2} e +4 a^{2} b^{3} d +6 \left (a e +b d \right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (a^{4} b e +6 a^{3} b^{2} d +4 \left (a e +b d \right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b d +\left (a e +b d \right ) a^{4}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 115, normalized size = 3.03 \[ \frac {1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac {1}{6} \, {\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} + {\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.02, size = 103, normalized size = 2.71 \[ x^2\,\left (\frac {e\,a^5}{2}+\frac {5\,b\,d\,a^4}{2}\right )+x^6\,\left (\frac {d\,b^5}{6}+\frac {5\,a\,e\,b^4}{6}\right )+\frac {b^5\,e\,x^7}{7}+a^5\,d\,x+\frac {5\,a^3\,b\,x^3\,\left (a\,e+2\,b\,d\right )}{3}+a\,b^3\,x^5\,\left (2\,a\,e+b\,d\right )+\frac {5\,a^2\,b^2\,x^4\,\left (a\,e+b\,d\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 129, normalized size = 3.39 \[ a^{5} d x + \frac {b^{5} e x^{7}}{7} + x^{6} \left (\frac {5 a b^{4} e}{6} + \frac {b^{5} d}{6}\right ) + x^{5} \left (2 a^{2} b^{3} e + a b^{4} d\right ) + x^{4} \left (\frac {5 a^{3} b^{2} e}{2} + \frac {5 a^{2} b^{3} d}{2}\right ) + x^{3} \left (\frac {5 a^{4} b e}{3} + \frac {10 a^{3} b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{5} e}{2} + \frac {5 a^{4} b d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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