Optimal. Leaf size=101 \[ -\frac {3 d^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac {3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac {d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac {d^5 (b+2 c x)^{12}}{1536 c^4} \]
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Rubi [A] time = 0.24, 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^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac {3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac {d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac {d^5 (b+2 c x)^{12}}{1536 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)^5 \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)^5}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^7}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^9}{64 c^3 d^4}+\frac {(b d+2 c d x)^{11}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 d^5 (b+2 c x)^6}{768 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^5 (b+2 c x)^8}{1024 c^4}-\frac {3 \left (b^2-4 a c\right ) d^5 (b+2 c x)^{10}}{1280 c^4}+\frac {d^5 (b+2 c x)^{12}}{1536 c^4}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 224, normalized size = 2.22 \begin {gather*} \frac {1}{60} d^5 x (b+c x) \left (20 a^3 \left (3 b^4+12 b^3 c x+28 b^2 c^2 x^2+32 b c^3 x^3+16 c^4 x^4\right )+30 a^2 x \left (3 b^5+19 b^4 c x+56 b^3 c^2 x^2+88 b^2 c^3 x^3+72 b c^4 x^4+24 c^5 x^5\right )+12 a x^2 (b+c x)^2 \left (5 b^4+30 b^3 c x+78 b^2 c^2 x^2+96 b c^3 x^3+48 c^4 x^4\right )+x^3 (b+c x)^3 \left (15 b^4+96 b^3 c x+256 b^2 c^2 x^2+320 b c^3 x^3+160 c^4 x^4\right )\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)^5 \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 = 0.34, size = 424, normalized size = 4.20 \begin {gather*} \frac {8}{3} x^{12} d^{5} c^{8} + 16 x^{11} d^{5} c^{7} b + \frac {208}{5} x^{10} d^{5} c^{6} b^{2} + \frac {48}{5} x^{10} d^{5} c^{7} a + \frac {184}{3} x^{9} d^{5} c^{5} b^{3} + 48 x^{9} d^{5} c^{6} b a + \frac {225}{4} x^{8} d^{5} c^{4} b^{4} + 102 x^{8} d^{5} c^{5} b^{2} a + 12 x^{8} d^{5} c^{6} a^{2} + 33 x^{7} d^{5} c^{3} b^{5} + 120 x^{7} d^{5} c^{4} b^{3} a + 48 x^{7} d^{5} c^{5} b a^{2} + \frac {73}{6} x^{6} d^{5} c^{2} b^{6} + 85 x^{6} d^{5} c^{3} b^{4} a + 80 x^{6} d^{5} c^{4} b^{2} a^{2} + \frac {16}{3} x^{6} d^{5} c^{5} a^{3} + \frac {13}{5} x^{5} d^{5} c b^{7} + \frac {183}{5} x^{5} d^{5} c^{2} b^{5} a + 72 x^{5} d^{5} c^{3} b^{3} a^{2} + 16 x^{5} d^{5} c^{4} b a^{3} + \frac {1}{4} x^{4} d^{5} b^{8} + 9 x^{4} d^{5} c b^{6} a + \frac {75}{2} x^{4} d^{5} c^{2} b^{4} a^{2} + 20 x^{4} d^{5} c^{3} b^{2} a^{3} + x^{3} d^{5} b^{7} a + 11 x^{3} d^{5} c b^{5} a^{2} + \frac {40}{3} x^{3} d^{5} c^{2} b^{3} a^{3} + \frac {3}{2} x^{2} d^{5} b^{6} a^{2} + 5 x^{2} d^{5} c b^{4} a^{3} + x d^{5} b^{5} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 424, normalized size = 4.20 \begin {gather*} \frac {8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac {208}{5} \, b^{2} c^{6} d^{5} x^{10} + \frac {48}{5} \, a c^{7} d^{5} x^{10} + \frac {184}{3} \, b^{3} c^{5} d^{5} x^{9} + 48 \, a b c^{6} d^{5} x^{9} + \frac {225}{4} \, b^{4} c^{4} d^{5} x^{8} + 102 \, a b^{2} c^{5} d^{5} x^{8} + 12 \, a^{2} c^{6} d^{5} x^{8} + 33 \, b^{5} c^{3} d^{5} x^{7} + 120 \, a b^{3} c^{4} d^{5} x^{7} + 48 \, a^{2} b c^{5} d^{5} x^{7} + \frac {73}{6} \, b^{6} c^{2} d^{5} x^{6} + 85 \, a b^{4} c^{3} d^{5} x^{6} + 80 \, a^{2} b^{2} c^{4} d^{5} x^{6} + \frac {16}{3} \, a^{3} c^{5} d^{5} x^{6} + \frac {13}{5} \, b^{7} c d^{5} x^{5} + \frac {183}{5} \, a b^{5} c^{2} d^{5} x^{5} + 72 \, a^{2} b^{3} c^{3} d^{5} x^{5} + 16 \, a^{3} b c^{4} d^{5} x^{5} + \frac {1}{4} \, b^{8} d^{5} x^{4} + 9 \, a b^{6} c d^{5} x^{4} + \frac {75}{2} \, a^{2} b^{4} c^{2} d^{5} x^{4} + 20 \, a^{3} b^{2} c^{3} d^{5} x^{4} + a b^{7} d^{5} x^{3} + 11 \, a^{2} b^{5} c d^{5} x^{3} + \frac {40}{3} \, a^{3} b^{3} c^{2} d^{5} x^{3} + \frac {3}{2} \, a^{2} b^{6} d^{5} x^{2} + 5 \, a^{3} b^{4} c d^{5} x^{2} + a^{3} b^{5} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 810, normalized size = 8.02 \begin {gather*} \frac {8 c^{8} d^{5} x^{12}}{3}+16 b \,c^{7} d^{5} x^{11}+a^{3} b^{5} d^{5} x +\frac {\left (320 b^{2} c^{6} d^{5}+32 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c^{5} d^{5}\right ) x^{10}}{10}+\frac {\left (280 b^{3} c^{5} d^{5}+80 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b \,c^{4} d^{5}+32 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c^{5} d^{5}\right ) x^{9}}{9}+\frac {\left (130 b^{4} c^{4} d^{5}+80 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{2} c^{3} d^{5}+80 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b \,c^{4} d^{5}+32 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c^{5} d^{5}\right ) x^{8}}{8}+\frac {\left (96 a^{2} b \,c^{5} d^{5}+31 b^{5} c^{3} d^{5}+40 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{3} c^{2} d^{5}+80 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{2} c^{3} d^{5}+80 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b \,c^{4} d^{5}\right ) x^{7}}{7}+\frac {\left (32 a^{3} c^{5} d^{5}+240 a^{2} b^{2} c^{4} d^{5}+3 b^{6} c^{2} d^{5}+10 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{4} c \,d^{5}+40 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{3} c^{2} d^{5}+80 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{2} c^{3} d^{5}\right ) x^{6}}{6}+\frac {\left (80 a^{3} b \,c^{4} d^{5}+240 a^{2} b^{3} c^{3} d^{5}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b^{5} d^{5}+10 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{4} c \,d^{5}+40 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{3} c^{2} d^{5}\right ) x^{5}}{5}+\frac {\left (80 a^{3} b^{2} c^{3} d^{5}+120 a^{2} b^{4} c^{2} d^{5}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b^{5} d^{5}+10 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{4} c \,d^{5}\right ) x^{4}}{4}+\frac {\left (40 a^{3} b^{3} c^{2} d^{5}+30 a^{2} b^{5} c \,d^{5}+\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b^{5} d^{5}\right ) x^{3}}{3}+\frac {\left (10 b^{4} d^{5} c \,a^{3}+3 b^{6} d^{5} 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 = 1.33, size = 342, normalized size = 3.39 \begin {gather*} \frac {8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac {16}{5} \, {\left (13 \, b^{2} c^{6} + 3 \, a c^{7}\right )} d^{5} x^{10} + \frac {8}{3} \, {\left (23 \, b^{3} c^{5} + 18 \, a b c^{6}\right )} d^{5} x^{9} + a^{3} b^{5} d^{5} x + \frac {3}{4} \, {\left (75 \, b^{4} c^{4} + 136 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{5} x^{8} + 3 \, {\left (11 \, b^{5} c^{3} + 40 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{5} x^{7} + \frac {1}{6} \, {\left (73 \, b^{6} c^{2} + 510 \, a b^{4} c^{3} + 480 \, a^{2} b^{2} c^{4} + 32 \, a^{3} c^{5}\right )} d^{5} x^{6} + \frac {1}{5} \, {\left (13 \, b^{7} c + 183 \, a b^{5} c^{2} + 360 \, a^{2} b^{3} c^{3} + 80 \, a^{3} b c^{4}\right )} d^{5} x^{5} + \frac {1}{4} \, {\left (b^{8} + 36 \, a b^{6} c + 150 \, a^{2} b^{4} c^{2} + 80 \, a^{3} b^{2} c^{3}\right )} d^{5} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{7} + 33 \, a^{2} b^{5} c + 40 \, a^{3} b^{3} c^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{6} + 10 \, a^{3} b^{4} c\right )} d^{5} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 326, normalized size = 3.23 \begin {gather*} \frac {8\,c^8\,d^5\,x^{12}}{3}+\frac {3\,c^4\,d^5\,x^8\,\left (16\,a^2\,c^2+136\,a\,b^2\,c+75\,b^4\right )}{4}+a^3\,b^5\,d^5\,x+16\,b\,c^7\,d^5\,x^{11}+\frac {b^2\,d^5\,x^4\,\left (80\,a^3\,c^3+150\,a^2\,b^2\,c^2+36\,a\,b^4\,c+b^6\right )}{4}+\frac {16\,c^6\,d^5\,x^{10}\,\left (13\,b^2+3\,a\,c\right )}{5}+\frac {c^2\,d^5\,x^6\,\left (32\,a^3\,c^3+480\,a^2\,b^2\,c^2+510\,a\,b^4\,c+73\,b^6\right )}{6}+\frac {a^2\,b^4\,d^5\,x^2\,\left (3\,b^2+10\,a\,c\right )}{2}+\frac {a\,b^3\,d^5\,x^3\,\left (40\,a^2\,c^2+33\,a\,b^2\,c+3\,b^4\right )}{3}+3\,b\,c^3\,d^5\,x^7\,\left (16\,a^2\,c^2+40\,a\,b^2\,c+11\,b^4\right )+\frac {b\,c\,d^5\,x^5\,\left (80\,a^3\,c^3+360\,a^2\,b^2\,c^2+183\,a\,b^4\,c+13\,b^6\right )}{5}+\frac {8\,b\,c^5\,d^5\,x^9\,\left (23\,b^2+18\,a\,c\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 428, normalized size = 4.24 \begin {gather*} a^{3} b^{5} d^{5} x + 16 b c^{7} d^{5} x^{11} + \frac {8 c^{8} d^{5} x^{12}}{3} + x^{10} \left (\frac {48 a c^{7} d^{5}}{5} + \frac {208 b^{2} c^{6} d^{5}}{5}\right ) + x^{9} \left (48 a b c^{6} d^{5} + \frac {184 b^{3} c^{5} d^{5}}{3}\right ) + x^{8} \left (12 a^{2} c^{6} d^{5} + 102 a b^{2} c^{5} d^{5} + \frac {225 b^{4} c^{4} d^{5}}{4}\right ) + x^{7} \left (48 a^{2} b c^{5} d^{5} + 120 a b^{3} c^{4} d^{5} + 33 b^{5} c^{3} d^{5}\right ) + x^{6} \left (\frac {16 a^{3} c^{5} d^{5}}{3} + 80 a^{2} b^{2} c^{4} d^{5} + 85 a b^{4} c^{3} d^{5} + \frac {73 b^{6} c^{2} d^{5}}{6}\right ) + x^{5} \left (16 a^{3} b c^{4} d^{5} + 72 a^{2} b^{3} c^{3} d^{5} + \frac {183 a b^{5} c^{2} d^{5}}{5} + \frac {13 b^{7} c d^{5}}{5}\right ) + x^{4} \left (20 a^{3} b^{2} c^{3} d^{5} + \frac {75 a^{2} b^{4} c^{2} d^{5}}{2} + 9 a b^{6} c d^{5} + \frac {b^{8} d^{5}}{4}\right ) + x^{3} \left (\frac {40 a^{3} b^{3} c^{2} d^{5}}{3} + 11 a^{2} b^{5} c d^{5} + a b^{7} d^{5}\right ) + x^{2} \left (5 a^{3} b^{4} c d^{5} + \frac {3 a^{2} b^{6} d^{5}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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