Optimal. Leaf size=92 \[ \frac {3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac {d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac {(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac {d^3 (a+b x)^8}{8 b^4} \]
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Rubi [A] time = 0.12, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {626, 43} \begin {gather*} \frac {3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac {d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac {(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac {d^3 (a+b x)^8}{8 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int (a+b x) \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx &=\int (a+b x)^4 (c+d x)^3 \, dx\\ &=\int \left (\frac {(b c-a d)^3 (a+b x)^4}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^5}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^6}{b^3}+\frac {d^3 (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^5}{5 b^4}+\frac {d (b c-a d)^2 (a+b x)^6}{2 b^4}+\frac {3 d^2 (b c-a d) (a+b x)^7}{7 b^4}+\frac {d^3 (a+b x)^8}{8 b^4}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 217, normalized size = 2.36 \begin {gather*} a^4 c^3 x+\frac {1}{2} a^3 c^2 x^2 (3 a d+4 b c)+\frac {1}{2} b^2 d x^6 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c x^3 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac {1}{5} b x^5 \left (4 a^3 d^3+18 a^2 b c d^2+12 a b^2 c^2 d+b^3 c^3\right )+\frac {1}{4} a x^4 \left (a^3 d^3+12 a^2 b c d^2+18 a b^2 c^2 d+4 b^3 c^3\right )+\frac {1}{7} b^3 d^2 x^7 (4 a d+3 b c)+\frac {1}{8} b^4 d^3 x^8 \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) \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.35, size = 245, normalized size = 2.66 \begin {gather*} \frac {1}{8} x^{8} d^{3} b^{4} + \frac {3}{7} x^{7} d^{2} c b^{4} + \frac {4}{7} x^{7} d^{3} b^{3} a + \frac {1}{2} x^{6} d c^{2} b^{4} + 2 x^{6} d^{2} c b^{3} a + x^{6} d^{3} b^{2} a^{2} + \frac {1}{5} x^{5} c^{3} b^{4} + \frac {12}{5} x^{5} d c^{2} b^{3} a + \frac {18}{5} x^{5} d^{2} c b^{2} a^{2} + \frac {4}{5} x^{5} d^{3} b a^{3} + x^{4} c^{3} b^{3} a + \frac {9}{2} x^{4} d c^{2} b^{2} a^{2} + 3 x^{4} d^{2} c b a^{3} + \frac {1}{4} x^{4} d^{3} a^{4} + 2 x^{3} c^{3} b^{2} a^{2} + 4 x^{3} d c^{2} b a^{3} + x^{3} d^{2} c a^{4} + 2 x^{2} c^{3} b a^{3} + \frac {3}{2} x^{2} d c^{2} a^{4} + x c^{3} a^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 245, normalized size = 2.66 \begin {gather*} \frac {1}{8} \, b^{4} d^{3} x^{8} + \frac {3}{7} \, b^{4} c d^{2} x^{7} + \frac {4}{7} \, a b^{3} d^{3} x^{7} + \frac {1}{2} \, b^{4} c^{2} d x^{6} + 2 \, a b^{3} c d^{2} x^{6} + a^{2} b^{2} d^{3} x^{6} + \frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {12}{5} \, a b^{3} c^{2} d x^{5} + \frac {18}{5} \, a^{2} b^{2} c d^{2} x^{5} + \frac {4}{5} \, a^{3} b d^{3} x^{5} + a b^{3} c^{3} x^{4} + \frac {9}{2} \, a^{2} b^{2} c^{2} d x^{4} + 3 \, a^{3} b c d^{2} x^{4} + \frac {1}{4} \, a^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{3} x^{3} + 4 \, a^{3} b c^{2} d x^{3} + a^{4} c d^{2} x^{3} + 2 \, a^{3} b c^{3} x^{2} + \frac {3}{2} \, a^{4} c^{2} d x^{2} + a^{4} c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 391, normalized size = 4.25 \begin {gather*} \frac {b^{4} d^{3} x^{8}}{8}+a^{4} c^{3} x +\frac {\left (a \,b^{3} d^{3}+3 \left (a d +b c \right ) b^{3} d^{2}\right ) x^{7}}{7}+\frac {\left (3 \left (a d +b c \right ) a \,b^{2} d^{2}+\left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b d \right ) b \right ) x^{6}}{6}+\frac {\left (\left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b d \right ) a +\left (4 \left (a d +b c \right ) a b c d +\left (a d +b c \right ) \left (2 a b c d +\left (a d +b c \right )^{2}\right )\right ) b \right ) x^{5}}{5}+\frac {\left (\left (4 \left (a d +b c \right ) a b c d +\left (a d +b c \right ) \left (2 a b c d +\left (a d +b c \right )^{2}\right )\right ) a +\left (a^{2} b \,c^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a c +2 \left (a d +b c \right )^{2} a c \right ) b \right ) x^{4}}{4}+\frac {\left (3 \left (a d +b c \right ) a^{2} b \,c^{2}+\left (a^{2} b \,c^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a c +2 \left (a d +b c \right )^{2} a c \right ) a \right ) x^{3}}{3}+\frac {\left (a^{3} b \,c^{3}+3 \left (a d +b c \right ) a^{3} c^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.11, size = 225, normalized size = 2.45 \begin {gather*} \frac {1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 208, normalized size = 2.26 \begin {gather*} x^4\,\left (\frac {a^4\,d^3}{4}+3\,a^3\,b\,c\,d^2+\frac {9\,a^2\,b^2\,c^2\,d}{2}+a\,b^3\,c^3\right )+x^5\,\left (\frac {4\,a^3\,b\,d^3}{5}+\frac {18\,a^2\,b^2\,c\,d^2}{5}+\frac {12\,a\,b^3\,c^2\,d}{5}+\frac {b^4\,c^3}{5}\right )+a^4\,c^3\,x+\frac {b^4\,d^3\,x^8}{8}+\frac {a^3\,c^2\,x^2\,\left (3\,a\,d+4\,b\,c\right )}{2}+\frac {b^3\,d^2\,x^7\,\left (4\,a\,d+3\,b\,c\right )}{7}+a^2\,c\,x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )+\frac {b^2\,d\,x^6\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 243, normalized size = 2.64 \begin {gather*} a^{4} c^{3} x + \frac {b^{4} d^{3} x^{8}}{8} + x^{7} \left (\frac {4 a b^{3} d^{3}}{7} + \frac {3 b^{4} c d^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} d^{3} + 2 a b^{3} c d^{2} + \frac {b^{4} c^{2} d}{2}\right ) + x^{5} \left (\frac {4 a^{3} b d^{3}}{5} + \frac {18 a^{2} b^{2} c d^{2}}{5} + \frac {12 a b^{3} c^{2} d}{5} + \frac {b^{4} c^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} d^{3}}{4} + 3 a^{3} b c d^{2} + \frac {9 a^{2} b^{2} c^{2} d}{2} + a b^{3} c^{3}\right ) + x^{3} \left (a^{4} c d^{2} + 4 a^{3} b c^{2} d + 2 a^{2} b^{2} c^{3}\right ) + x^{2} \left (\frac {3 a^{4} c^{2} d}{2} + 2 a^{3} b c^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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