Optimal. Leaf size=92 \[ \frac {3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac {3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac {(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac {d^3 (a+b x)^9}{9 b^4} \]
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Rubi [A] time = 0.16, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac {3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac {(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac {d^3 (a+b x)^9}{9 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^5 (c+d x)^3 \, dx &=\int \left (\frac {(b c-a d)^3 (a+b x)^5}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^6}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^7}{b^3}+\frac {d^3 (a+b x)^8}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^6}{6 b^4}+\frac {3 d (b c-a d)^2 (a+b x)^7}{7 b^4}+\frac {3 d^2 (b c-a d) (a+b x)^8}{8 b^4}+\frac {d^3 (a+b x)^9}{9 b^4}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 235, normalized size = 2.55 \[ \frac {1}{504} x \left (126 a^5 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+126 a^4 b x \left (10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right )+84 a^3 b^2 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )+36 a^2 b^3 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+9 a b^4 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+b^5 x^5 \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 303, normalized size = 3.29 \[ \frac {1}{9} x^{9} d^{3} b^{5} + \frac {3}{8} x^{8} d^{2} c b^{5} + \frac {5}{8} x^{8} d^{3} b^{4} a + \frac {3}{7} x^{7} d c^{2} b^{5} + \frac {15}{7} x^{7} d^{2} c b^{4} a + \frac {10}{7} x^{7} d^{3} b^{3} a^{2} + \frac {1}{6} x^{6} c^{3} b^{5} + \frac {5}{2} x^{6} d c^{2} b^{4} a + 5 x^{6} d^{2} c b^{3} a^{2} + \frac {5}{3} x^{6} d^{3} b^{2} a^{3} + x^{5} c^{3} b^{4} a + 6 x^{5} d c^{2} b^{3} a^{2} + 6 x^{5} d^{2} c b^{2} a^{3} + x^{5} d^{3} b a^{4} + \frac {5}{2} x^{4} c^{3} b^{3} a^{2} + \frac {15}{2} x^{4} d c^{2} b^{2} a^{3} + \frac {15}{4} x^{4} d^{2} c b a^{4} + \frac {1}{4} x^{4} d^{3} a^{5} + \frac {10}{3} x^{3} c^{3} b^{2} a^{3} + 5 x^{3} d c^{2} b a^{4} + x^{3} d^{2} c a^{5} + \frac {5}{2} x^{2} c^{3} b a^{4} + \frac {3}{2} x^{2} d c^{2} a^{5} + x c^{3} a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.01, size = 303, normalized size = 3.29 \[ \frac {1}{9} \, b^{5} d^{3} x^{9} + \frac {3}{8} \, b^{5} c d^{2} x^{8} + \frac {5}{8} \, a b^{4} d^{3} x^{8} + \frac {3}{7} \, b^{5} c^{2} d x^{7} + \frac {15}{7} \, a b^{4} c d^{2} x^{7} + \frac {10}{7} \, a^{2} b^{3} d^{3} x^{7} + \frac {1}{6} \, b^{5} c^{3} x^{6} + \frac {5}{2} \, a b^{4} c^{2} d x^{6} + 5 \, a^{2} b^{3} c d^{2} x^{6} + \frac {5}{3} \, a^{3} b^{2} d^{3} x^{6} + a b^{4} c^{3} x^{5} + 6 \, a^{2} b^{3} c^{2} d x^{5} + 6 \, a^{3} b^{2} c d^{2} x^{5} + a^{4} b d^{3} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{3} x^{4} + \frac {15}{2} \, a^{3} b^{2} c^{2} d x^{4} + \frac {15}{4} \, a^{4} b c d^{2} x^{4} + \frac {1}{4} \, a^{5} d^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{3} x^{3} + 5 \, a^{4} b c^{2} d x^{3} + a^{5} c d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{3} x^{2} + \frac {3}{2} \, a^{5} c^{2} d x^{2} + a^{5} c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 281, normalized size = 3.05 \[ \frac {b^{5} d^{3} x^{9}}{9}+a^{5} c^{3} x +\frac {\left (5 a \,b^{4} d^{3}+3 b^{5} c \,d^{2}\right ) x^{8}}{8}+\frac {\left (10 a^{2} b^{3} d^{3}+15 a \,b^{4} c \,d^{2}+3 b^{5} c^{2} d \right ) x^{7}}{7}+\frac {\left (10 a^{3} b^{2} d^{3}+30 a^{2} b^{3} c \,d^{2}+15 a \,b^{4} c^{2} d +b^{5} c^{3}\right ) x^{6}}{6}+\frac {\left (5 a^{4} b \,d^{3}+30 a^{3} b^{2} c \,d^{2}+30 a^{2} b^{3} c^{2} d +5 a \,b^{4} c^{3}\right ) x^{5}}{5}+\frac {\left (a^{5} d^{3}+15 a^{4} b c \,d^{2}+30 a^{3} b^{2} c^{2} d +10 a^{2} b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{5} c \,d^{2}+15 a^{4} b \,c^{2} d +10 a^{3} b^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{5} c^{2} d +5 a^{4} b \,c^{3}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 277, normalized size = 3.01 \[ \frac {1}{9} \, b^{5} d^{3} x^{9} + a^{5} c^{3} x + \frac {1}{8} \, {\left (3 \, b^{5} c d^{2} + 5 \, a b^{4} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, b^{5} c^{2} d + 15 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{3} + 15 \, a b^{4} c^{2} d + 30 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{6} + {\left (a b^{4} c^{3} + 6 \, a^{2} b^{3} c^{2} d + 6 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, a^{2} b^{3} c^{3} + 30 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{3} + 15 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{3} + 3 \, a^{5} c^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 261, normalized size = 2.84 \[ x^5\,\left (a^4\,b\,d^3+6\,a^3\,b^2\,c\,d^2+6\,a^2\,b^3\,c^2\,d+a\,b^4\,c^3\right )+x^4\,\left (\frac {a^5\,d^3}{4}+\frac {15\,a^4\,b\,c\,d^2}{4}+\frac {15\,a^3\,b^2\,c^2\,d}{2}+\frac {5\,a^2\,b^3\,c^3}{2}\right )+x^6\,\left (\frac {5\,a^3\,b^2\,d^3}{3}+5\,a^2\,b^3\,c\,d^2+\frac {5\,a\,b^4\,c^2\,d}{2}+\frac {b^5\,c^3}{6}\right )+a^5\,c^3\,x+\frac {b^5\,d^3\,x^9}{9}+\frac {a^4\,c^2\,x^2\,\left (3\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d^2\,x^8\,\left (5\,a\,d+3\,b\,c\right )}{8}+\frac {a^3\,c\,x^3\,\left (3\,a^2\,d^2+15\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3}+\frac {b^3\,d\,x^7\,\left (10\,a^2\,d^2+15\,a\,b\,c\,d+3\,b^2\,c^2\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 308, normalized size = 3.35 \[ a^{5} c^{3} x + \frac {b^{5} d^{3} x^{9}}{9} + x^{8} \left (\frac {5 a b^{4} d^{3}}{8} + \frac {3 b^{5} c d^{2}}{8}\right ) + x^{7} \left (\frac {10 a^{2} b^{3} d^{3}}{7} + \frac {15 a b^{4} c d^{2}}{7} + \frac {3 b^{5} c^{2} d}{7}\right ) + x^{6} \left (\frac {5 a^{3} b^{2} d^{3}}{3} + 5 a^{2} b^{3} c d^{2} + \frac {5 a b^{4} c^{2} d}{2} + \frac {b^{5} c^{3}}{6}\right ) + x^{5} \left (a^{4} b d^{3} + 6 a^{3} b^{2} c d^{2} + 6 a^{2} b^{3} c^{2} d + a b^{4} c^{3}\right ) + x^{4} \left (\frac {a^{5} d^{3}}{4} + \frac {15 a^{4} b c d^{2}}{4} + \frac {15 a^{3} b^{2} c^{2} d}{2} + \frac {5 a^{2} b^{3} c^{3}}{2}\right ) + x^{3} \left (a^{5} c d^{2} + 5 a^{4} b c^{2} d + \frac {10 a^{3} b^{2} c^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{5} c^{2} d}{2} + \frac {5 a^{4} b c^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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