Optimal. Leaf size=65 \[ \frac {2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac {(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \]
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Rubi [A] time = 0.13, antiderivative size = 65, 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 = {626, 43} \[ \frac {2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac {(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^5 (c+d x)^2 \, dx\\ &=\int \left (\frac {(b c-a d)^2 (a+b x)^5}{b^2}+\frac {2 d (b c-a d) (a+b x)^6}{b^2}+\frac {d^2 (a+b x)^7}{b^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (a+b x)^6}{6 b^3}+\frac {2 d (b c-a d) (a+b x)^7}{7 b^3}+\frac {d^2 (a+b x)^8}{8 b^3}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 189, normalized size = 2.91 \[ a^5 c^2 x+\frac {1}{2} a^4 c x^2 (2 a d+5 b c)+a b^2 x^5 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {5}{4} a^2 b x^4 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac {1}{6} b^3 x^6 \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac {1}{3} a^3 x^3 \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )+\frac {1}{7} b^4 d x^7 (5 a d+2 b c)+\frac {1}{8} b^5 d^2 x^8 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 212, normalized size = 3.26 \[ \frac {1}{8} x^{8} d^{2} b^{5} + \frac {2}{7} x^{7} d c b^{5} + \frac {5}{7} x^{7} d^{2} b^{4} a + \frac {1}{6} x^{6} c^{2} b^{5} + \frac {5}{3} x^{6} d c b^{4} a + \frac {5}{3} x^{6} d^{2} b^{3} a^{2} + x^{5} c^{2} b^{4} a + 4 x^{5} d c b^{3} a^{2} + 2 x^{5} d^{2} b^{2} a^{3} + \frac {5}{2} x^{4} c^{2} b^{3} a^{2} + 5 x^{4} d c b^{2} a^{3} + \frac {5}{4} x^{4} d^{2} b a^{4} + \frac {10}{3} x^{3} c^{2} b^{2} a^{3} + \frac {10}{3} x^{3} d c b a^{4} + \frac {1}{3} x^{3} d^{2} a^{5} + \frac {5}{2} x^{2} c^{2} b a^{4} + x^{2} d c a^{5} + x c^{2} a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 212, normalized size = 3.26 \[ \frac {1}{8} \, b^{5} d^{2} x^{8} + \frac {2}{7} \, b^{5} c d x^{7} + \frac {5}{7} \, a b^{4} d^{2} x^{7} + \frac {1}{6} \, b^{5} c^{2} x^{6} + \frac {5}{3} \, a b^{4} c d x^{6} + \frac {5}{3} \, a^{2} b^{3} d^{2} x^{6} + a b^{4} c^{2} x^{5} + 4 \, a^{2} b^{3} c d x^{5} + 2 \, a^{3} b^{2} d^{2} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{2} x^{4} + 5 \, a^{3} b^{2} c d x^{4} + \frac {5}{4} \, a^{4} b d^{2} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{2} x^{3} + \frac {10}{3} \, a^{4} b c d x^{3} + \frac {1}{3} \, a^{5} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{2} x^{2} + a^{5} c d x^{2} + a^{5} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 315, normalized size = 4.85 \[ \frac {b^{5} d^{2} x^{8}}{8}+a^{5} c^{2} x +\frac {\left (3 a \,b^{4} d^{2}+2 \left (a d +b c \right ) b^{4} d \right ) x^{7}}{7}+\frac {\left (3 a^{2} b^{3} d^{2}+6 \left (a d +b c \right ) a \,b^{3} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b^{3}\right ) x^{6}}{6}+\frac {\left (a^{3} b^{2} d^{2}+6 \left (a d +b c \right ) a^{2} b^{2} d +2 \left (a d +b c \right ) a \,b^{3} c +3 \left (2 a b c d +\left (a d +b c \right )^{2}\right ) a \,b^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} b^{3} c^{2}+2 \left (a d +b c \right ) a^{3} b d +6 \left (a d +b c \right ) a^{2} b^{2} c +3 \left (2 a b c d +\left (a d +b c \right )^{2}\right ) a^{2} b \right ) x^{4}}{4}+\frac {\left (3 a^{3} b^{2} c^{2}+6 \left (a d +b c \right ) a^{3} b c +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{4} b \,c^{2}+2 \left (a d +b c \right ) a^{4} c \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 197, normalized size = 3.03 \[ \frac {1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} + {\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 181, normalized size = 2.78 \[ x^3\,\left (\frac {a^5\,d^2}{3}+\frac {10\,a^4\,b\,c\,d}{3}+\frac {10\,a^3\,b^2\,c^2}{3}\right )+x^6\,\left (\frac {5\,a^2\,b^3\,d^2}{3}+\frac {5\,a\,b^4\,c\,d}{3}+\frac {b^5\,c^2}{6}\right )+a^5\,c^2\,x+\frac {b^5\,d^2\,x^8}{8}+\frac {a^4\,c\,x^2\,\left (2\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d\,x^7\,\left (5\,a\,d+2\,b\,c\right )}{7}+\frac {5\,a^2\,b\,x^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )}{4}+a\,b^2\,x^5\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 218, normalized size = 3.35 \[ a^{5} c^{2} x + \frac {b^{5} d^{2} x^{8}}{8} + x^{7} \left (\frac {5 a b^{4} d^{2}}{7} + \frac {2 b^{5} c d}{7}\right ) + x^{6} \left (\frac {5 a^{2} b^{3} d^{2}}{3} + \frac {5 a b^{4} c d}{3} + \frac {b^{5} c^{2}}{6}\right ) + x^{5} \left (2 a^{3} b^{2} d^{2} + 4 a^{2} b^{3} c d + a b^{4} c^{2}\right ) + x^{4} \left (\frac {5 a^{4} b d^{2}}{4} + 5 a^{3} b^{2} c d + \frac {5 a^{2} b^{3} c^{2}}{2}\right ) + x^{3} \left (\frac {a^{5} d^{2}}{3} + \frac {10 a^{4} b c d}{3} + \frac {10 a^{3} b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{5} c d + \frac {5 a^{4} b c^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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