Optimal. Leaf size=38 \[ \frac {(a+b x)^7 (b d-a e)}{7 b^2}+\frac {e (a+b x)^8}{8 b^2} \]
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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac {(a+b x)^7 (b d-a e)}{7 b^2}+\frac {e (a+b x)^8}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x) \, dx\\ &=\int \left (\frac {(b d-a e) (a+b x)^6}{b}+\frac {e (a+b x)^7}{b}\right ) \, dx\\ &=\frac {(b d-a e) (a+b x)^7}{7 b^2}+\frac {e (a+b x)^8}{8 b^2}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 122, normalized size = 3.21 \[ \frac {1}{56} x \left (28 a^6 (2 d+e x)+56 a^5 b x (3 d+2 e x)+70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 145, normalized size = 3.82 \[ \frac {1}{8} x^{8} e b^{6} + \frac {1}{7} x^{7} d b^{6} + \frac {6}{7} x^{7} e b^{5} a + x^{6} d b^{5} a + \frac {5}{2} x^{6} e b^{4} a^{2} + 3 x^{5} d b^{4} a^{2} + 4 x^{5} e b^{3} a^{3} + 5 x^{4} d b^{3} a^{3} + \frac {15}{4} x^{4} e b^{2} a^{4} + 5 x^{3} d b^{2} a^{4} + 2 x^{3} e b a^{5} + 3 x^{2} d b a^{5} + \frac {1}{2} x^{2} e a^{6} + x d a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 152, normalized size = 4.00 \[ \frac {1}{8} \, b^{6} x^{8} e + \frac {1}{7} \, b^{6} d x^{7} + \frac {6}{7} \, a b^{5} x^{7} e + a b^{5} d x^{6} + \frac {5}{2} \, a^{2} b^{4} x^{6} e + 3 \, a^{2} b^{4} d x^{5} + 4 \, a^{3} b^{3} x^{5} e + 5 \, a^{3} b^{3} d x^{4} + \frac {15}{4} \, a^{4} b^{2} x^{4} e + 5 \, a^{4} b^{2} d x^{3} + 2 \, a^{5} b x^{3} e + 3 \, a^{5} b d x^{2} + \frac {1}{2} \, a^{6} x^{2} e + a^{6} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 145, normalized size = 3.82 \[ \frac {b^{6} e \,x^{8}}{8}+a^{6} d x +\frac {\left (6 e a \,b^{5}+d \,b^{6}\right ) x^{7}}{7}+\frac {\left (15 e \,a^{2} b^{4}+6 d a \,b^{5}\right ) x^{6}}{6}+\frac {\left (20 e \,a^{3} b^{3}+15 d \,a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (15 e \,a^{4} b^{2}+20 d \,a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (6 e \,a^{5} b +15 d \,a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (e \,a^{6}+6 d \,a^{5} b \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.30, size = 142, normalized size = 3.74 \[ \frac {1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac {1}{7} \, {\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} + {\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} + {\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d + a^{6} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 126, normalized size = 3.32 \[ x^2\,\left (\frac {e\,a^6}{2}+3\,b\,d\,a^5\right )+x^7\,\left (\frac {d\,b^6}{7}+\frac {6\,a\,e\,b^5}{7}\right )+\frac {b^6\,e\,x^8}{8}+a^6\,d\,x+a^4\,b\,x^3\,\left (2\,a\,e+5\,b\,d\right )+\frac {a\,b^4\,x^6\,\left (5\,a\,e+2\,b\,d\right )}{2}+\frac {5\,a^3\,b^2\,x^4\,\left (3\,a\,e+4\,b\,d\right )}{4}+a^2\,b^3\,x^5\,\left (4\,a\,e+3\,b\,d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 148, normalized size = 3.89 \[ a^{6} d x + \frac {b^{6} e x^{8}}{8} + x^{7} \left (\frac {6 a b^{5} e}{7} + \frac {b^{6} d}{7}\right ) + x^{6} \left (\frac {5 a^{2} b^{4} e}{2} + a b^{5} d\right ) + x^{5} \left (4 a^{3} b^{3} e + 3 a^{2} b^{4} d\right ) + x^{4} \left (\frac {15 a^{4} b^{2} e}{4} + 5 a^{3} b^{3} d\right ) + x^{3} \left (2 a^{5} b e + 5 a^{4} b^{2} d\right ) + x^{2} \left (\frac {a^{6} e}{2} + 3 a^{5} b d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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