Optimal. Leaf size=101 \[ -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac {3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4} \]
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Rubi [A] time = 0.19, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac {3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3 (b d+2 c d x)^4}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^6}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^8}{64 c^3 d^4}+\frac {(b d+2 c d x)^{10}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^5}{640 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^7}{896 c^4}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^9}{384 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 259, normalized size = 2.56 \begin {gather*} d^4 \left (a^3 b^4 x+\frac {1}{2} b c^2 x^6 \left (48 a^2 c^2+88 a b^2 c+17 b^4\right )+a b^2 x^3 \left (8 a^2 c^2+9 a b^2 c+b^4\right )+\frac {3}{7} c^3 x^7 \left (16 a^2 c^2+104 a b^2 c+43 b^4\right )+\frac {1}{2} a^2 b^3 x^2 \left (8 a c+3 b^2\right )+\frac {1}{5} c x^5 \left (16 a^3 c^3+168 a^2 b^2 c^2+123 a b^4 c+11 b^6\right )+\frac {1}{4} b x^4 \left (32 a^3 c^3+96 a^2 b^2 c^2+30 a b^4 c+b^6\right )+\frac {8}{3} c^5 x^9 \left (2 a c+7 b^2\right )+24 b c^4 x^8 \left (a c+b^2\right )+8 b c^6 x^{10}+\frac {16 c^7 x^{11}}{11}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.36, size = 361, normalized size = 3.57 \begin {gather*} \frac {16}{11} x^{11} d^{4} c^{7} + 8 x^{10} d^{4} c^{6} b + \frac {56}{3} x^{9} d^{4} c^{5} b^{2} + \frac {16}{3} x^{9} d^{4} c^{6} a + 24 x^{8} d^{4} c^{4} b^{3} + 24 x^{8} d^{4} c^{5} b a + \frac {129}{7} x^{7} d^{4} c^{3} b^{4} + \frac {312}{7} x^{7} d^{4} c^{4} b^{2} a + \frac {48}{7} x^{7} d^{4} c^{5} a^{2} + \frac {17}{2} x^{6} d^{4} c^{2} b^{5} + 44 x^{6} d^{4} c^{3} b^{3} a + 24 x^{6} d^{4} c^{4} b a^{2} + \frac {11}{5} x^{5} d^{4} c b^{6} + \frac {123}{5} x^{5} d^{4} c^{2} b^{4} a + \frac {168}{5} x^{5} d^{4} c^{3} b^{2} a^{2} + \frac {16}{5} x^{5} d^{4} c^{4} a^{3} + \frac {1}{4} x^{4} d^{4} b^{7} + \frac {15}{2} x^{4} d^{4} c b^{5} a + 24 x^{4} d^{4} c^{2} b^{3} a^{2} + 8 x^{4} d^{4} c^{3} b a^{3} + x^{3} d^{4} b^{6} a + 9 x^{3} d^{4} c b^{4} a^{2} + 8 x^{3} d^{4} c^{2} b^{2} a^{3} + \frac {3}{2} x^{2} d^{4} b^{5} a^{2} + 4 x^{2} d^{4} c b^{3} a^{3} + x d^{4} b^{4} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 361, normalized size = 3.57 \begin {gather*} \frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {56}{3} \, b^{2} c^{5} d^{4} x^{9} + \frac {16}{3} \, a c^{6} d^{4} x^{9} + 24 \, b^{3} c^{4} d^{4} x^{8} + 24 \, a b c^{5} d^{4} x^{8} + \frac {129}{7} \, b^{4} c^{3} d^{4} x^{7} + \frac {312}{7} \, a b^{2} c^{4} d^{4} x^{7} + \frac {48}{7} \, a^{2} c^{5} d^{4} x^{7} + \frac {17}{2} \, b^{5} c^{2} d^{4} x^{6} + 44 \, a b^{3} c^{3} d^{4} x^{6} + 24 \, a^{2} b c^{4} d^{4} x^{6} + \frac {11}{5} \, b^{6} c d^{4} x^{5} + \frac {123}{5} \, a b^{4} c^{2} d^{4} x^{5} + \frac {168}{5} \, a^{2} b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{3} c^{4} d^{4} x^{5} + \frac {1}{4} \, b^{7} d^{4} x^{4} + \frac {15}{2} \, a b^{5} c d^{4} x^{4} + 24 \, a^{2} b^{3} c^{2} d^{4} x^{4} + 8 \, a^{3} b c^{3} d^{4} x^{4} + a b^{6} d^{4} x^{3} + 9 \, a^{2} b^{4} c d^{4} x^{3} + 8 \, a^{3} b^{2} c^{2} d^{4} x^{3} + \frac {3}{2} \, a^{2} b^{5} d^{4} x^{2} + 4 \, a^{3} b^{3} c d^{4} x^{2} + a^{3} b^{4} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 672, normalized size = 6.65 \begin {gather*} \frac {16 c^{7} d^{4} x^{11}}{11}+8 b \,c^{6} d^{4} x^{10}+a^{3} b^{4} d^{4} x +\frac {\left (120 b^{2} c^{5} d^{4}+16 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c^{4} d^{4}\right ) x^{9}}{9}+\frac {\left (80 b^{3} c^{4} d^{4}+32 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b \,c^{3} d^{4}+16 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c^{4} d^{4}\right ) x^{8}}{8}+\frac {\left (25 b^{4} c^{3} d^{4}+24 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{2} c^{2} d^{4}+32 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b \,c^{3} d^{4}+16 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c^{4} d^{4}\right ) x^{7}}{7}+\frac {\left (48 a^{2} b \,c^{4} d^{4}+3 b^{5} c^{2} d^{4}+8 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{3} c \,d^{4}+24 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{2} c^{2} d^{4}+32 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b \,c^{3} d^{4}\right ) x^{6}}{6}+\frac {\left (16 a^{3} c^{4} d^{4}+96 a^{2} b^{2} c^{3} d^{4}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{4} d^{4}+8 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{3} c \,d^{4}+24 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{2} c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (32 a^{3} b \,c^{3} d^{4}+72 a^{2} b^{3} c^{2} d^{4}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{4} d^{4}+8 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{3} c \,d^{4}\right ) x^{4}}{4}+\frac {\left (24 a^{3} b^{2} c^{2} d^{4}+24 a^{2} b^{4} c \,d^{4}+\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{4} d^{4}\right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c \,a^{3}+3 b^{5} d^{4} a^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 290, normalized size = 2.87 \begin {gather*} \frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {8}{3} \, {\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \, {\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac {3}{7} \, {\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac {1}{2} \, {\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} + {\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{5} + 8 \, a^{3} b^{3} c\right )} d^{4} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 274, normalized size = 2.71 \begin {gather*} \frac {16\,c^7\,d^4\,x^{11}}{11}+\frac {3\,c^3\,d^4\,x^7\,\left (16\,a^2\,c^2+104\,a\,b^2\,c+43\,b^4\right )}{7}+a^3\,b^4\,d^4\,x+8\,b\,c^6\,d^4\,x^{10}+\frac {c\,d^4\,x^5\,\left (16\,a^3\,c^3+168\,a^2\,b^2\,c^2+123\,a\,b^4\,c+11\,b^6\right )}{5}+\frac {8\,c^5\,d^4\,x^9\,\left (7\,b^2+2\,a\,c\right )}{3}+\frac {b\,d^4\,x^4\,\left (32\,a^3\,c^3+96\,a^2\,b^2\,c^2+30\,a\,b^4\,c+b^6\right )}{4}+a\,b^2\,d^4\,x^3\,\left (8\,a^2\,c^2+9\,a\,b^2\,c+b^4\right )+\frac {a^2\,b^3\,d^4\,x^2\,\left (3\,b^2+8\,a\,c\right )}{2}+24\,b\,c^4\,d^4\,x^8\,\left (b^2+a\,c\right )+\frac {b\,c^2\,d^4\,x^6\,\left (48\,a^2\,c^2+88\,a\,b^2\,c+17\,b^4\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 371, normalized size = 3.67 \begin {gather*} a^{3} b^{4} d^{4} x + 8 b c^{6} d^{4} x^{10} + \frac {16 c^{7} d^{4} x^{11}}{11} + x^{9} \left (\frac {16 a c^{6} d^{4}}{3} + \frac {56 b^{2} c^{5} d^{4}}{3}\right ) + x^{8} \left (24 a b c^{5} d^{4} + 24 b^{3} c^{4} d^{4}\right ) + x^{7} \left (\frac {48 a^{2} c^{5} d^{4}}{7} + \frac {312 a b^{2} c^{4} d^{4}}{7} + \frac {129 b^{4} c^{3} d^{4}}{7}\right ) + x^{6} \left (24 a^{2} b c^{4} d^{4} + 44 a b^{3} c^{3} d^{4} + \frac {17 b^{5} c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {16 a^{3} c^{4} d^{4}}{5} + \frac {168 a^{2} b^{2} c^{3} d^{4}}{5} + \frac {123 a b^{4} c^{2} d^{4}}{5} + \frac {11 b^{6} c d^{4}}{5}\right ) + x^{4} \left (8 a^{3} b c^{3} d^{4} + 24 a^{2} b^{3} c^{2} d^{4} + \frac {15 a b^{5} c d^{4}}{2} + \frac {b^{7} d^{4}}{4}\right ) + x^{3} \left (8 a^{3} b^{2} c^{2} d^{4} + 9 a^{2} b^{4} c d^{4} + a b^{6} d^{4}\right ) + x^{2} \left (4 a^{3} b^{3} c d^{4} + \frac {3 a^{2} b^{5} d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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