Optimal. Leaf size=101 \[ -\frac {3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac {3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac {d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac {d^2 (b+2 c x)^9}{1152 c^4} \]
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Rubi [A] time = 0.13, 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 {3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac {3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac {d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac {d^2 (b+2 c x)^9}{1152 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)^2 \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)^2}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^6}{64 c^3 d^4}+\frac {(b d+2 c d x)^8}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 d^2 (b+2 c x)^3}{384 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^5}{640 c^4}-\frac {3 \left (b^2-4 a c\right ) d^2 (b+2 c x)^7}{896 c^4}+\frac {d^2 (b+2 c x)^9}{1152 c^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 179, normalized size = 1.77 \begin {gather*} d^2 \left (a^3 b^2 x+\frac {1}{2} a^2 b x^2 \left (4 a c+3 b^2\right )+\frac {1}{5} c x^5 \left (12 a^2 c^2+39 a b^2 c+7 b^4\right )+\frac {1}{4} b x^4 \left (24 a^2 c^2+18 a b^2 c+b^4\right )+\frac {1}{3} a x^3 \left (4 a^2 c^2+15 a b^2 c+3 b^4\right )+\frac {1}{7} c^3 x^7 \left (12 a c+25 b^2\right )+\frac {1}{6} b c^2 x^6 \left (36 a c+19 b^2\right )+2 b c^4 x^8+\frac {4 c^5 x^9}{9}\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)^2 \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 = 0.35, size = 235, normalized size = 2.33 \begin {gather*} \frac {4}{9} x^{9} d^{2} c^{5} + 2 x^{8} d^{2} c^{4} b + \frac {25}{7} x^{7} d^{2} c^{3} b^{2} + \frac {12}{7} x^{7} d^{2} c^{4} a + \frac {19}{6} x^{6} d^{2} c^{2} b^{3} + 6 x^{6} d^{2} c^{3} b a + \frac {7}{5} x^{5} d^{2} c b^{4} + \frac {39}{5} x^{5} d^{2} c^{2} b^{2} a + \frac {12}{5} x^{5} d^{2} c^{3} a^{2} + \frac {1}{4} x^{4} d^{2} b^{5} + \frac {9}{2} x^{4} d^{2} c b^{3} a + 6 x^{4} d^{2} c^{2} b a^{2} + x^{3} d^{2} b^{4} a + 5 x^{3} d^{2} c b^{2} a^{2} + \frac {4}{3} x^{3} d^{2} c^{2} a^{3} + \frac {3}{2} x^{2} d^{2} b^{3} a^{2} + 2 x^{2} d^{2} c b a^{3} + x d^{2} b^{2} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 235, normalized size = 2.33 \begin {gather*} \frac {4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac {25}{7} \, b^{2} c^{3} d^{2} x^{7} + \frac {12}{7} \, a c^{4} d^{2} x^{7} + \frac {19}{6} \, b^{3} c^{2} d^{2} x^{6} + 6 \, a b c^{3} d^{2} x^{6} + \frac {7}{5} \, b^{4} c d^{2} x^{5} + \frac {39}{5} \, a b^{2} c^{2} d^{2} x^{5} + \frac {12}{5} \, a^{2} c^{3} d^{2} x^{5} + \frac {1}{4} \, b^{5} d^{2} x^{4} + \frac {9}{2} \, a b^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{2} d^{2} x^{4} + a b^{4} d^{2} x^{3} + 5 \, a^{2} b^{2} c d^{2} x^{3} + \frac {4}{3} \, a^{3} c^{2} d^{2} x^{3} + \frac {3}{2} \, a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b c d^{2} x^{2} + a^{3} b^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 396, normalized size = 3.92 \begin {gather*} \frac {4 c^{5} d^{2} x^{9}}{9}+2 b \,c^{4} d^{2} x^{8}+a^{3} b^{2} d^{2} x +\frac {\left (13 b^{2} c^{3} d^{2}+4 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c^{2} d^{2}\right ) x^{7}}{7}+\frac {\left (3 b^{3} c^{2} d^{2}+4 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b c \,d^{2}+4 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c^{2} d^{2}\right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{2} d^{2}+4 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b c \,d^{2}+4 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c^{2} d^{2}\right ) x^{5}}{5}+\frac {\left (12 a^{2} b \,c^{2} d^{2}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{2} d^{2}+4 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b c \,d^{2}\right ) x^{4}}{4}+\frac {\left (4 a^{3} c^{2} d^{2}+12 a^{2} b^{2} c \,d^{2}+\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{2} d^{2}\right ) x^{3}}{3}+\frac {\left (4 b c \,d^{2} a^{3}+3 b^{3} d^{2} 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 = 1.42, size = 198, normalized size = 1.96 \begin {gather*} \frac {4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac {1}{7} \, {\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{7} + \frac {1}{6} \, {\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{6} + a^{3} b^{2} d^{2} x + \frac {1}{5} \, {\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{5} + \frac {1}{4} \, {\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 188, normalized size = 1.86 \begin {gather*} \frac {4\,c^5\,d^2\,x^9}{9}+a^3\,b^2\,d^2\,x+2\,b\,c^4\,d^2\,x^8+\frac {b\,d^2\,x^4\,\left (24\,a^2\,c^2+18\,a\,b^2\,c+b^4\right )}{4}+\frac {c^3\,d^2\,x^7\,\left (25\,b^2+12\,a\,c\right )}{7}+\frac {a\,d^2\,x^3\,\left (4\,a^2\,c^2+15\,a\,b^2\,c+3\,b^4\right )}{3}+\frac {c\,d^2\,x^5\,\left (12\,a^2\,c^2+39\,a\,b^2\,c+7\,b^4\right )}{5}+\frac {a^2\,b\,d^2\,x^2\,\left (3\,b^2+4\,a\,c\right )}{2}+\frac {b\,c^2\,d^2\,x^6\,\left (19\,b^2+36\,a\,c\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 246, normalized size = 2.44 \begin {gather*} a^{3} b^{2} d^{2} x + 2 b c^{4} d^{2} x^{8} + \frac {4 c^{5} d^{2} x^{9}}{9} + x^{7} \left (\frac {12 a c^{4} d^{2}}{7} + \frac {25 b^{2} c^{3} d^{2}}{7}\right ) + x^{6} \left (6 a b c^{3} d^{2} + \frac {19 b^{3} c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac {12 a^{2} c^{3} d^{2}}{5} + \frac {39 a b^{2} c^{2} d^{2}}{5} + \frac {7 b^{4} c d^{2}}{5}\right ) + x^{4} \left (6 a^{2} b c^{2} d^{2} + \frac {9 a b^{3} c d^{2}}{2} + \frac {b^{5} d^{2}}{4}\right ) + x^{3} \left (\frac {4 a^{3} c^{2} d^{2}}{3} + 5 a^{2} b^{2} c d^{2} + a b^{4} d^{2}\right ) + x^{2} \left (2 a^{3} b c d^{2} + \frac {3 a^{2} b^{3} d^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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