Optimal. Leaf size=101 \[ -\frac {\left (b^2-4 a c\right )^3 d^2 (b+2 c x)^3}{384 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^5}{640 c^4}-\frac {3 \left (b^2-4 a c\right ) d^2 (b+2 c x)^7}{896 c^4}+\frac {d^2 (b+2 c x)^9}{1152 c^4} \]
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Rubi [A]
time = 0.09, 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 = {697}
\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 697
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 = 0.02, size = 179, normalized size = 1.77 \begin {gather*} d^2 \left (a^3 b^2 x+\frac {1}{2} a^2 b \left (3 b^2+4 a c\right ) x^2+\frac {1}{3} a \left (3 b^4+15 a b^2 c+4 a^2 c^2\right ) x^3+\frac {1}{4} b \left (b^4+18 a b^2 c+24 a^2 c^2\right ) x^4+\frac {1}{5} c \left (7 b^4+39 a b^2 c+12 a^2 c^2\right ) x^5+\frac {1}{6} b c^2 \left (19 b^2+36 a c\right ) x^6+\frac {1}{7} c^3 \left (25 b^2+12 a c\right ) x^7+2 b c^4 x^8+\frac {4 c^5 x^9}{9}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs.
\(2(93)=186\).
time = 0.65, size = 396, normalized size = 3.92
method | result | size |
gosper | \(\frac {x \left (560 c^{5} x^{8}+2520 b \,c^{4} x^{7}+2160 x^{6} a \,c^{4}+4500 x^{6} b^{2} c^{3}+7560 x^{5} a b \,c^{3}+3990 x^{5} b^{3} c^{2}+3024 a^{2} c^{3} x^{4}+9828 x^{4} b^{2} c^{2} a +1764 b^{4} c \,x^{4}+7560 a^{2} b \,c^{2} x^{3}+5670 x^{3} a \,b^{3} c +315 b^{5} x^{3}+1680 x^{2} c^{2} a^{3}+6300 x^{2} c \,a^{2} b^{2}+1260 a \,b^{4} x^{2}+2520 a^{3} b c x +1890 a^{2} b^{3} x +1260 a^{3} b^{2}\right ) d^{2}}{1260}\) | \(185\) |
norman | \(\left (\frac {12}{7} a \,c^{4} d^{2}+\frac {25}{7} b^{2} d^{2} c^{3}\right ) x^{7}+\left (6 a b \,c^{3} d^{2}+\frac {19}{6} b^{3} d^{2} c^{2}\right ) x^{6}+\left (2 a^{3} b c \,d^{2}+\frac {3}{2} a^{2} b^{3} d^{2}\right ) x^{2}+\left (\frac {12}{5} a^{2} c^{3} d^{2}+\frac {39}{5} a \,c^{2} d^{2} b^{2}+\frac {7}{5} b^{4} c \,d^{2}\right ) x^{5}+\left (\frac {4}{3} c^{2} d^{2} a^{3}+5 b^{2} c \,d^{2} a^{2}+a \,b^{4} d^{2}\right ) x^{3}+\left (6 a^{2} b \,c^{2} d^{2}+\frac {9}{2} a \,b^{3} c \,d^{2}+\frac {1}{4} b^{5} d^{2}\right ) x^{4}+b^{2} d^{2} a^{3} x +\frac {4 c^{5} d^{2} x^{9}}{9}+2 c^{4} b \,d^{2} x^{8}\) | \(221\) |
risch | \(\frac {4}{9} c^{5} d^{2} x^{9}+2 c^{4} b \,d^{2} x^{8}+\frac {12}{7} d^{2} x^{7} a \,c^{4}+\frac {25}{7} d^{2} x^{7} b^{2} c^{3}+6 d^{2} x^{6} a b \,c^{3}+\frac {19}{6} d^{2} b^{3} c^{2} x^{6}+\frac {12}{5} d^{2} a^{2} c^{3} x^{5}+\frac {39}{5} d^{2} x^{5} b^{2} c^{2} a +\frac {7}{5} d^{2} x^{5} c \,b^{4}+6 d^{2} x^{4} a^{2} b \,c^{2}+\frac {9}{2} d^{2} x^{4} a \,b^{3} c +\frac {1}{4} d^{2} x^{4} b^{5}+\frac {4}{3} d^{2} x^{3} c^{2} a^{3}+5 a^{2} b^{2} c \,d^{2} x^{3}+d^{2} a \,b^{4} x^{3}+2 d^{2} a^{3} b c \,x^{2}+\frac {3}{2} d^{2} a^{2} x^{2} b^{3}+b^{2} d^{2} a^{3} x\) | \(236\) |
default | \(\frac {4 c^{5} d^{2} x^{9}}{9}+2 c^{4} b \,d^{2} x^{8}+\frac {\left (13 b^{2} d^{2} c^{3}+4 c^{2} d^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 b^{3} d^{2} c^{2}+4 b c \,d^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+4 c^{2} d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (b^{2} d^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+4 b c \,d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+4 c^{2} d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{5}}{5}+\frac {\left (b^{2} d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+4 b c \,d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+12 a^{2} b \,c^{2} d^{2}\right ) x^{4}}{4}+\frac {\left (b^{2} d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+12 b^{2} c \,d^{2} a^{2}+4 c^{2} d^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (4 a^{3} b c \,d^{2}+3 a^{2} b^{3} d^{2}\right ) x^{2}}{2}+b^{2} d^{2} a^{3} x\) | \(396\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs.
\(2 (93) = 186\).
time = 0.28, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs.
\(2 (93) = 186\).
time = 3.15, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 246 vs.
\(2 (100) = 200\).
time = 0.03, 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} \cdot \left (\frac {12 a c^{4} d^{2}}{7} + \frac {25 b^{2} c^{3} d^{2}}{7}\right ) + x^{6} \cdot \left (6 a b c^{3} d^{2} + \frac {19 b^{3} c^{2} d^{2}}{6}\right ) + x^{5} \cdot \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} \cdot \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} \cdot \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} \cdot \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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 235 vs.
\(2 (93) = 186\).
time = 1.09, 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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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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