Optimal. Leaf size=38 \[ \frac {b (c+d x)^9}{9 d^2}-\frac {(c+d x)^8 (b c-a d)}{8 d^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} \frac {b (c+d x)^9}{9 d^2}-\frac {(c+d x)^8 (b c-a d)}{8 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^7 \, dx &=\int \left (\frac {(-b c+a d) (c+d x)^7}{d}+\frac {b (c+d x)^8}{d}\right ) \, dx\\ &=-\frac {(b c-a d) (c+d x)^8}{8 d^2}+\frac {b (c+d x)^9}{9 d^2}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 151, normalized size = 3.97 \begin {gather*} \frac {1}{2} c^6 x^2 (7 a d+b c)+\frac {7}{3} c^5 d x^3 (3 a d+b c)+\frac {7}{4} c^4 d^2 x^4 (5 a d+3 b c)+7 c^3 d^3 x^5 (a d+b c)+\frac {7}{6} c^2 d^4 x^6 (3 a d+5 b c)+\frac {1}{8} d^6 x^8 (a d+7 b c)+c d^5 x^7 (a d+3 b c)+a c^7 x+\frac {1}{9} b d^7 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) (c+d x)^7 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.57, size = 169, normalized size = 4.45 \begin {gather*} \frac {1}{9} x^{9} d^{7} b + \frac {7}{8} x^{8} d^{6} c b + \frac {1}{8} x^{8} d^{7} a + 3 x^{7} d^{5} c^{2} b + x^{7} d^{6} c a + \frac {35}{6} x^{6} d^{4} c^{3} b + \frac {7}{2} x^{6} d^{5} c^{2} a + 7 x^{5} d^{3} c^{4} b + 7 x^{5} d^{4} c^{3} a + \frac {21}{4} x^{4} d^{2} c^{5} b + \frac {35}{4} x^{4} d^{3} c^{4} a + \frac {7}{3} x^{3} d c^{6} b + 7 x^{3} d^{2} c^{5} a + \frac {1}{2} x^{2} c^{7} b + \frac {7}{2} x^{2} d c^{6} a + x c^{7} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 169, normalized size = 4.45 \begin {gather*} \frac {1}{9} \, b d^{7} x^{9} + \frac {7}{8} \, b c d^{6} x^{8} + \frac {1}{8} \, a d^{7} x^{8} + 3 \, b c^{2} d^{5} x^{7} + a c d^{6} x^{7} + \frac {35}{6} \, b c^{3} d^{4} x^{6} + \frac {7}{2} \, a c^{2} d^{5} x^{6} + 7 \, b c^{4} d^{3} x^{5} + 7 \, a c^{3} d^{4} x^{5} + \frac {21}{4} \, b c^{5} d^{2} x^{4} + \frac {35}{4} \, a c^{4} d^{3} x^{4} + \frac {7}{3} \, b c^{6} d x^{3} + 7 \, a c^{5} d^{2} x^{3} + \frac {1}{2} \, b c^{7} x^{2} + \frac {7}{2} \, a c^{6} d x^{2} + a c^{7} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 169, normalized size = 4.45 \begin {gather*} \frac {b \,d^{7} x^{9}}{9}+a \,c^{7} x +\frac {\left (a \,d^{7}+7 b c \,d^{6}\right ) x^{8}}{8}+\frac {\left (7 a c \,d^{6}+21 b \,c^{2} d^{5}\right ) x^{7}}{7}+\frac {\left (21 a \,c^{2} d^{5}+35 b \,c^{3} d^{4}\right ) x^{6}}{6}+\frac {\left (35 a \,c^{3} d^{4}+35 b \,c^{4} d^{3}\right ) x^{5}}{5}+\frac {\left (35 a \,c^{4} d^{3}+21 b \,c^{5} d^{2}\right ) x^{4}}{4}+\frac {\left (21 a \,c^{5} d^{2}+7 b \,c^{6} d \right ) x^{3}}{3}+\frac {\left (7 a \,c^{6} d +b \,c^{7}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 163, normalized size = 4.29 \begin {gather*} \frac {1}{9} \, b d^{7} x^{9} + a c^{7} x + \frac {1}{8} \, {\left (7 \, b c d^{6} + a d^{7}\right )} x^{8} + {\left (3 \, b c^{2} d^{5} + a c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b c^{3} d^{4} + 3 \, a c^{2} d^{5}\right )} x^{6} + 7 \, {\left (b c^{4} d^{3} + a c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (3 \, b c^{5} d^{2} + 5 \, a c^{4} d^{3}\right )} x^{4} + \frac {7}{3} \, {\left (b c^{6} d + 3 \, a c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{7} + 7 \, a c^{6} d\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 = 143, normalized size = 3.76 \begin {gather*} x^2\,\left (\frac {b\,c^7}{2}+\frac {7\,a\,d\,c^6}{2}\right )+x^8\,\left (\frac {a\,d^7}{8}+\frac {7\,b\,c\,d^6}{8}\right )+\frac {b\,d^7\,x^9}{9}+a\,c^7\,x+\frac {7\,c^5\,d\,x^3\,\left (3\,a\,d+b\,c\right )}{3}+c\,d^5\,x^7\,\left (a\,d+3\,b\,c\right )+7\,c^3\,d^3\,x^5\,\left (a\,d+b\,c\right )+\frac {7\,c^4\,d^2\,x^4\,\left (5\,a\,d+3\,b\,c\right )}{4}+\frac {7\,c^2\,d^4\,x^6\,\left (3\,a\,d+5\,b\,c\right )}{6} \end {gather*}
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 \begin {gather*} a c^{7} x + \frac {b d^{7} x^{9}}{9} + x^{8} \left (\frac {a d^{7}}{8} + \frac {7 b c d^{6}}{8}\right ) + x^{7} \left (a c d^{6} + 3 b c^{2} d^{5}\right ) + x^{6} \left (\frac {7 a c^{2} d^{5}}{2} + \frac {35 b c^{3} d^{4}}{6}\right ) + x^{5} \left (7 a c^{3} d^{4} + 7 b c^{4} d^{3}\right ) + x^{4} \left (\frac {35 a c^{4} d^{3}}{4} + \frac {21 b c^{5} d^{2}}{4}\right ) + x^{3} \left (7 a c^{5} d^{2} + \frac {7 b c^{6} d}{3}\right ) + x^{2} \left (\frac {7 a c^{6} d}{2} + \frac {b c^{7}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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