Optimal. Leaf size=73 \[ \frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 73, 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 {d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \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 )^2 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^2}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{8 c^2 d^2}+\frac {(b d+2 c d x)^6}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 111, normalized size = 1.52 \begin {gather*} d^2 \left (a^2 b^2 x+a b \left (b^2+2 a c\right ) x^2+\frac {1}{3} \left (b^4+10 a b^2 c+4 a^2 c^2\right ) x^3+\frac {1}{2} b c \left (3 b^2+8 a c\right ) x^4+\frac {1}{5} c^2 \left (13 b^2+8 a c\right ) x^5+2 b c^3 x^6+\frac {4 c^4 x^7}{7}\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. \(175\) vs.
\(2(67)=134\).
time = 0.76, size = 176, normalized size = 2.41
method | result | size |
gosper | \(\frac {x \left (120 c^{4} x^{6}+420 b \,c^{3} x^{5}+336 x^{4} c^{3} a +546 b^{2} c^{2} x^{4}+840 x^{3} a b \,c^{2}+315 x^{3} b^{3} c +280 a^{2} c^{2} x^{2}+700 a \,b^{2} c \,x^{2}+70 b^{4} x^{2}+420 a^{2} b c x +210 a \,b^{3} x +210 a^{2} b^{2}\right ) d^{2}}{210}\) | \(116\) |
norman | \(\left (\frac {8}{5} a \,c^{3} d^{2}+\frac {13}{5} b^{2} c^{2} d^{2}\right ) x^{5}+\left (4 a b \,c^{2} d^{2}+\frac {3}{2} b^{3} c \,d^{2}\right ) x^{4}+\left (\frac {4}{3} a^{2} c^{2} d^{2}+\frac {10}{3} a \,b^{2} c \,d^{2}+\frac {1}{3} b^{4} d^{2}\right ) x^{3}+\left (2 a^{2} b c \,d^{2}+b^{3} d^{2} a \right ) x^{2}+b^{2} d^{2} a^{2} x +\frac {4 c^{4} d^{2} x^{7}}{7}+2 b \,c^{3} d^{2} x^{6}\) | \(142\) |
risch | \(\frac {4}{7} c^{4} d^{2} x^{7}+2 b \,c^{3} d^{2} x^{6}+\frac {8}{5} d^{2} x^{5} c^{3} a +\frac {13}{5} d^{2} x^{5} b^{2} c^{2}+4 d^{2} x^{4} a b \,c^{2}+\frac {3}{2} d^{2} x^{4} b^{3} c +\frac {4}{3} a^{2} c^{2} d^{2} x^{3}+\frac {10}{3} a \,b^{2} c \,d^{2} x^{3}+\frac {1}{3} d^{2} x^{3} b^{4}+2 a^{2} b c \,d^{2} x^{2}+d^{2} a \,b^{3} x^{2}+b^{2} d^{2} a^{2} x\) | \(149\) |
default | \(\frac {4 c^{4} d^{2} x^{7}}{7}+2 b \,c^{3} d^{2} x^{6}+\frac {\left (9 b^{2} c^{2} d^{2}+4 c^{2} d^{2} \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 b^{3} c \,d^{2}+4 b c \,d^{2} \left (2 a c +b^{2}\right )+8 a b \,c^{2} d^{2}\right ) x^{4}}{4}+\frac {\left (b^{2} d^{2} \left (2 a c +b^{2}\right )+8 a \,b^{2} c \,d^{2}+4 a^{2} c^{2} d^{2}\right ) x^{3}}{3}+\frac {\left (4 a^{2} b c \,d^{2}+2 b^{3} d^{2} a \right ) x^{2}}{2}+b^{2} d^{2} a^{2} x\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 127, normalized size = 1.74 \begin {gather*} \frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {1}{5} \, {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac {1}{2} \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} + {\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.38, size = 127, normalized size = 1.74 \begin {gather*} \frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {1}{5} \, {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac {1}{2} \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} + {\left (a b^{3} + 2 \, a^{2} 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. 156 vs.
\(2 (68) = 136\).
time = 0.02, size = 156, normalized size = 2.14 \begin {gather*} a^{2} b^{2} d^{2} x + 2 b c^{3} d^{2} x^{6} + \frac {4 c^{4} d^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {8 a c^{3} d^{2}}{5} + \frac {13 b^{2} c^{2} d^{2}}{5}\right ) + x^{4} \cdot \left (4 a b c^{2} d^{2} + \frac {3 b^{3} c d^{2}}{2}\right ) + x^{3} \cdot \left (\frac {4 a^{2} c^{2} d^{2}}{3} + \frac {10 a b^{2} c d^{2}}{3} + \frac {b^{4} d^{2}}{3}\right ) + x^{2} \cdot \left (2 a^{2} b c d^{2} + a b^{3} d^{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. 148 vs.
\(2 (67) = 134\).
time = 1.03, size = 148, normalized size = 2.03 \begin {gather*} \frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {13}{5} \, b^{2} c^{2} d^{2} x^{5} + \frac {8}{5} \, a c^{3} d^{2} x^{5} + \frac {3}{2} \, b^{3} c d^{2} x^{4} + 4 \, a b c^{2} d^{2} x^{4} + \frac {1}{3} \, b^{4} d^{2} x^{3} + \frac {10}{3} \, a b^{2} c d^{2} x^{3} + \frac {4}{3} \, a^{2} c^{2} d^{2} x^{3} + a b^{3} d^{2} x^{2} + 2 \, a^{2} b c d^{2} x^{2} + a^{2} b^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 120, normalized size = 1.64 \begin {gather*} \frac {4\,c^4\,d^2\,x^7}{7}+\frac {d^2\,x^3\,\left (4\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{3}+a^2\,b^2\,d^2\,x+2\,b\,c^3\,d^2\,x^6+\frac {c^2\,d^2\,x^5\,\left (13\,b^2+8\,a\,c\right )}{5}+a\,b\,d^2\,x^2\,\left (b^2+2\,a\,c\right )+\frac {b\,c\,d^2\,x^4\,\left (3\,b^2+8\,a\,c\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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