Optimal. Leaf size=73 \[ \frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \]
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Rubi [A]
time = 0.09, 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^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac {d^4 (b+2 c x)^9}{288 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)^4 \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)^4}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^6}{8 c^2 d^2}+\frac {(b d+2 c d x)^8}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(73)=146\).
time = 0.02, size = 179, normalized size = 2.45 \begin {gather*} d^4 \left (a^2 b^4 x+a b^3 \left (b^2+4 a c\right ) x^2+\frac {1}{3} b^2 \left (b^4+18 a b^2 c+24 a^2 c^2\right ) x^3+\frac {1}{2} b c \left (5 b^4+32 a b^2 c+16 a^2 c^2\right ) x^4+\frac {1}{5} c^2 \left (41 b^4+112 a b^2 c+16 a^2 c^2\right ) x^5+\frac {4}{3} b c^3 \left (11 b^2+12 a c\right ) x^6+\frac {8}{7} c^4 \left (13 b^2+4 a c\right ) x^7+8 b c^5 x^8+\frac {16 c^6 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. \(299\) vs.
\(2(67)=134\).
time = 0.65, size = 300, normalized size = 4.11
method | result | size |
gosper | \(\frac {x \left (1120 c^{6} x^{8}+5040 b \,c^{5} x^{7}+2880 x^{6} c^{5} a +9360 x^{6} b^{2} c^{4}+10080 x^{5} a b \,c^{4}+9240 x^{5} b^{3} c^{3}+2016 x^{4} a^{2} c^{4}+14112 x^{4} a \,b^{2} c^{3}+5166 x^{4} b^{4} c^{2}+5040 x^{3} a^{2} b \,c^{3}+10080 x^{3} a \,b^{3} c^{2}+1575 x^{3} b^{5} c +5040 x^{2} a^{2} b^{2} c^{2}+3780 x^{2} a \,b^{4} c +210 x^{2} b^{6}+2520 a^{2} b^{3} c x +630 a \,b^{5} x +630 a^{2} b^{4}\right ) d^{4}}{630}\) | \(190\) |
norman | \(\left (\frac {32}{7} a \,c^{5} d^{4}+\frac {104}{7} b^{2} d^{4} c^{4}\right ) x^{7}+\left (16 c^{4} d^{4} a b +\frac {44}{3} b^{3} c^{3} d^{4}\right ) x^{6}+\left (\frac {16}{5} c^{4} d^{4} a^{2}+\frac {112}{5} a \,b^{2} c^{3} d^{4}+\frac {41}{5} b^{4} d^{4} c^{2}\right ) x^{5}+\left (8 b \,c^{3} d^{4} a^{2}+16 b^{3} d^{4} c^{2} a +\frac {5}{2} b^{5} d^{4} c \right ) x^{4}+\left (8 b^{2} d^{4} c^{2} a^{2}+6 b^{4} d^{4} c a +\frac {1}{3} b^{6} d^{4}\right ) x^{3}+\left (4 b^{3} d^{4} c \,a^{2}+b^{5} d^{4} a \right ) x^{2}+b^{4} d^{4} a^{2} x +\frac {16 c^{6} d^{4} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}\) | \(226\) |
risch | \(\frac {16}{9} c^{6} d^{4} x^{9}+8 b \,c^{5} d^{4} x^{8}+\frac {32}{7} d^{4} x^{7} c^{5} a +\frac {104}{7} d^{4} x^{7} b^{2} c^{4}+16 d^{4} x^{6} a b \,c^{4}+\frac {44}{3} d^{4} x^{6} b^{3} c^{3}+\frac {16}{5} d^{4} x^{5} a^{2} c^{4}+\frac {112}{5} b^{2} d^{4} a \,c^{3} x^{5}+\frac {41}{5} d^{4} x^{5} b^{4} c^{2}+8 d^{4} a^{2} b \,c^{3} x^{4}+16 d^{4} x^{4} a \,b^{3} c^{2}+\frac {5}{2} d^{4} x^{4} b^{5} c +8 d^{4} x^{3} a^{2} b^{2} c^{2}+6 d^{4} a \,b^{4} c \,x^{3}+\frac {1}{3} d^{4} x^{3} b^{6}+4 d^{4} a^{2} b^{3} c \,x^{2}+d^{4} a \,b^{5} x^{2}+b^{4} d^{4} a^{2} x\) | \(241\) |
default | \(\frac {16 c^{6} d^{4} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}+\frac {\left (88 b^{2} d^{4} c^{4}+16 c^{4} d^{4} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (56 b^{3} c^{3} d^{4}+32 b \,c^{3} d^{4} \left (2 a c +b^{2}\right )+32 c^{4} d^{4} a b \right ) x^{6}}{6}+\frac {\left (17 b^{4} d^{4} c^{2}+24 b^{2} d^{4} c^{2} \left (2 a c +b^{2}\right )+64 a \,b^{2} c^{3} d^{4}+16 c^{4} d^{4} a^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{5} d^{4} c +8 b^{3} d^{4} c \left (2 a c +b^{2}\right )+48 b^{3} d^{4} c^{2} a +32 b \,c^{3} d^{4} a^{2}\right ) x^{4}}{4}+\frac {\left (b^{4} d^{4} \left (2 a c +b^{2}\right )+16 b^{4} d^{4} c a +24 b^{2} d^{4} c^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c \,a^{2}+2 b^{5} d^{4} a \right ) x^{2}}{2}+b^{4} d^{4} a^{2} x\) | \(300\) |
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. 201 vs.
\(2 (67) = 134\).
time = 0.28, size = 201, normalized size = 2.75 \begin {gather*} \frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} 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. 201 vs.
\(2 (67) = 134\).
time = 2.92, size = 201, normalized size = 2.75 \begin {gather*} \frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} 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. 248 vs.
\(2 (68) = 136\).
time = 0.03, size = 248, normalized size = 3.40 \begin {gather*} a^{2} b^{4} d^{4} x + 8 b c^{5} d^{4} x^{8} + \frac {16 c^{6} d^{4} x^{9}}{9} + x^{7} \cdot \left (\frac {32 a c^{5} d^{4}}{7} + \frac {104 b^{2} c^{4} d^{4}}{7}\right ) + x^{6} \cdot \left (16 a b c^{4} d^{4} + \frac {44 b^{3} c^{3} d^{4}}{3}\right ) + x^{5} \cdot \left (\frac {16 a^{2} c^{4} d^{4}}{5} + \frac {112 a b^{2} c^{3} d^{4}}{5} + \frac {41 b^{4} c^{2} d^{4}}{5}\right ) + x^{4} \cdot \left (8 a^{2} b c^{3} d^{4} + 16 a b^{3} c^{2} d^{4} + \frac {5 b^{5} c d^{4}}{2}\right ) + x^{3} \cdot \left (8 a^{2} b^{2} c^{2} d^{4} + 6 a b^{4} c d^{4} + \frac {b^{6} d^{4}}{3}\right ) + x^{2} \cdot \left (4 a^{2} b^{3} c d^{4} + a b^{5} d^{4}\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. 240 vs.
\(2 (67) = 134\).
time = 0.83, size = 240, normalized size = 3.29 \begin {gather*} \frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {104}{7} \, b^{2} c^{4} d^{4} x^{7} + \frac {32}{7} \, a c^{5} d^{4} x^{7} + \frac {44}{3} \, b^{3} c^{3} d^{4} x^{6} + 16 \, a b c^{4} d^{4} x^{6} + \frac {41}{5} \, b^{4} c^{2} d^{4} x^{5} + \frac {112}{5} \, a b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{2} c^{4} d^{4} x^{5} + \frac {5}{2} \, b^{5} c d^{4} x^{4} + 16 \, a b^{3} c^{2} d^{4} x^{4} + 8 \, a^{2} b c^{3} d^{4} x^{4} + \frac {1}{3} \, b^{6} d^{4} x^{3} + 6 \, a b^{4} c d^{4} x^{3} + 8 \, a^{2} b^{2} c^{2} d^{4} x^{3} + a b^{5} d^{4} x^{2} + 4 \, a^{2} b^{3} c d^{4} x^{2} + a^{2} b^{4} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.46, size = 190, normalized size = 2.60 \begin {gather*} \frac {16\,c^6\,d^4\,x^9}{9}+\frac {c^2\,d^4\,x^5\,\left (16\,a^2\,c^2+112\,a\,b^2\,c+41\,b^4\right )}{5}+a^2\,b^4\,d^4\,x+8\,b\,c^5\,d^4\,x^8+\frac {8\,c^4\,d^4\,x^7\,\left (13\,b^2+4\,a\,c\right )}{7}+\frac {b^2\,d^4\,x^3\,\left (24\,a^2\,c^2+18\,a\,b^2\,c+b^4\right )}{3}+\frac {b\,c\,d^4\,x^4\,\left (16\,a^2\,c^2+32\,a\,b^2\,c+5\,b^4\right )}{2}+a\,b^3\,d^4\,x^2\,\left (b^2+4\,a\,c\right )+\frac {4\,b\,c^3\,d^4\,x^6\,\left (11\,b^2+12\,a\,c\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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