Optimal. Leaf size=73 \[ -\frac {d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac {d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac {d^5 (b+2 c x)^{10}}{320 c^3} \]
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Rubi [A] time = 0.17, 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 = {683} \begin {gather*} -\frac {d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac {d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac {d^5 (b+2 c x)^{10}}{320 c^3} \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 )^2 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^5}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^7}{8 c^2 d^2}+\frac {(b d+2 c d x)^9}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 d^5 (b+2 c x)^6}{192 c^3}-\frac {\left (b^2-4 a c\right ) d^5 (b+2 c x)^8}{128 c^3}+\frac {d^5 (b+2 c x)^{10}}{320 c^3}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 168, normalized size = 2.30 \begin {gather*} \frac {1}{15} d^5 x (b+c x) \left (5 a^2 \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 )+5 a 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 )+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 )\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 )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.34, size = 286, normalized size = 3.92 \begin {gather*} \frac {16}{5} x^{10} d^{5} c^{7} + 16 x^{9} d^{5} c^{6} b + 34 x^{8} d^{5} c^{5} b^{2} + 8 x^{8} d^{5} c^{6} a + 40 x^{7} d^{5} c^{4} b^{3} + 32 x^{7} d^{5} c^{5} b a + \frac {85}{3} x^{6} d^{5} c^{3} b^{4} + \frac {160}{3} x^{6} d^{5} c^{4} b^{2} a + \frac {16}{3} x^{6} d^{5} c^{5} a^{2} + \frac {61}{5} x^{5} d^{5} c^{2} b^{5} + 48 x^{5} d^{5} c^{3} b^{3} a + 16 x^{5} d^{5} c^{4} b a^{2} + 3 x^{4} d^{5} c b^{6} + 25 x^{4} d^{5} c^{2} b^{4} a + 20 x^{4} d^{5} c^{3} b^{2} a^{2} + \frac {1}{3} x^{3} d^{5} b^{7} + \frac {22}{3} x^{3} d^{5} c b^{5} a + \frac {40}{3} x^{3} d^{5} c^{2} b^{3} a^{2} + x^{2} d^{5} b^{6} a + 5 x^{2} d^{5} c b^{4} a^{2} + x d^{5} b^{5} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 286, normalized size = 3.92 \begin {gather*} \frac {16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 34 \, b^{2} c^{5} d^{5} x^{8} + 8 \, a c^{6} d^{5} x^{8} + 40 \, b^{3} c^{4} d^{5} x^{7} + 32 \, a b c^{5} d^{5} x^{7} + \frac {85}{3} \, b^{4} c^{3} d^{5} x^{6} + \frac {160}{3} \, a b^{2} c^{4} d^{5} x^{6} + \frac {16}{3} \, a^{2} c^{5} d^{5} x^{6} + \frac {61}{5} \, b^{5} c^{2} d^{5} x^{5} + 48 \, a b^{3} c^{3} d^{5} x^{5} + 16 \, a^{2} b c^{4} d^{5} x^{5} + 3 \, b^{6} c d^{5} x^{4} + 25 \, a b^{4} c^{2} d^{5} x^{4} + 20 \, a^{2} b^{2} c^{3} d^{5} x^{4} + \frac {1}{3} \, b^{7} d^{5} x^{3} + \frac {22}{3} \, a b^{5} c d^{5} x^{3} + \frac {40}{3} \, a^{2} b^{3} c^{2} d^{5} x^{3} + a b^{6} d^{5} x^{2} + 5 \, a^{2} b^{4} c d^{5} x^{2} + a^{2} b^{5} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 362, normalized size = 4.96 \begin {gather*} \frac {16 c^{7} d^{5} x^{10}}{5}+16 b \,c^{6} d^{5} x^{9}+a^{2} b^{5} d^{5} x +\frac {\left (240 b^{2} c^{5} d^{5}+32 \left (2 a c +b^{2}\right ) c^{5} d^{5}\right ) x^{8}}{8}+\frac {\left (64 a b \,c^{5} d^{5}+200 b^{3} c^{4} d^{5}+80 \left (2 a c +b^{2}\right ) b \,c^{4} d^{5}\right ) x^{7}}{7}+\frac {\left (32 a^{2} c^{5} d^{5}+160 a \,b^{2} c^{4} d^{5}+90 b^{4} c^{3} d^{5}+80 \left (2 a c +b^{2}\right ) b^{2} c^{3} d^{5}\right ) x^{6}}{6}+\frac {\left (80 a^{2} b \,c^{4} d^{5}+160 a \,b^{3} c^{3} d^{5}+21 b^{5} c^{2} d^{5}+40 \left (2 a c +b^{2}\right ) b^{3} c^{2} d^{5}\right ) x^{5}}{5}+\frac {\left (80 a^{2} b^{2} c^{3} d^{5}+80 a \,b^{4} c^{2} d^{5}+2 b^{6} c \,d^{5}+10 \left (2 a c +b^{2}\right ) b^{4} c \,d^{5}\right ) x^{4}}{4}+\frac {\left (40 a^{2} b^{3} c^{2} d^{5}+20 a \,b^{5} c \,d^{5}+\left (2 a c +b^{2}\right ) b^{5} d^{5}\right ) x^{3}}{3}+\frac {\left (10 b^{4} d^{5} c \,a^{2}+2 b^{6} d^{5} a \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 237, normalized size = 3.25 \begin {gather*} \frac {16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 2 \, {\left (17 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{5} x^{8} + a^{2} b^{5} d^{5} x + 8 \, {\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} d^{5} x^{7} + \frac {1}{3} \, {\left (85 \, b^{4} c^{3} + 160 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{5} x^{6} + \frac {1}{5} \, {\left (61 \, b^{5} c^{2} + 240 \, a b^{3} c^{3} + 80 \, a^{2} b c^{4}\right )} d^{5} x^{5} + {\left (3 \, b^{6} c + 25 \, a b^{4} c^{2} + 20 \, a^{2} b^{2} c^{3}\right )} d^{5} x^{4} + \frac {1}{3} \, {\left (b^{7} + 22 \, a b^{5} c + 40 \, a^{2} b^{3} c^{2}\right )} d^{5} x^{3} + {\left (a b^{6} + 5 \, a^{2} 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 = 0.48, size = 224, normalized size = 3.07 \begin {gather*} \frac {16\,c^7\,d^5\,x^{10}}{5}+\frac {c^3\,d^5\,x^6\,\left (16\,a^2\,c^2+160\,a\,b^2\,c+85\,b^4\right )}{3}+a^2\,b^5\,d^5\,x+16\,b\,c^6\,d^5\,x^9+2\,c^5\,d^5\,x^8\,\left (17\,b^2+4\,a\,c\right )+\frac {b^3\,d^5\,x^3\,\left (40\,a^2\,c^2+22\,a\,b^2\,c+b^4\right )}{3}+a\,b^4\,d^5\,x^2\,\left (b^2+5\,a\,c\right )+b^2\,c\,d^5\,x^4\,\left (20\,a^2\,c^2+25\,a\,b^2\,c+3\,b^4\right )+\frac {b\,c^2\,d^5\,x^5\,\left (80\,a^2\,c^2+240\,a\,b^2\,c+61\,b^4\right )}{5}+8\,b\,c^4\,d^5\,x^7\,\left (5\,b^2+4\,a\,c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 291, normalized size = 3.99 \begin {gather*} a^{2} b^{5} d^{5} x + 16 b c^{6} d^{5} x^{9} + \frac {16 c^{7} d^{5} x^{10}}{5} + x^{8} \left (8 a c^{6} d^{5} + 34 b^{2} c^{5} d^{5}\right ) + x^{7} \left (32 a b c^{5} d^{5} + 40 b^{3} c^{4} d^{5}\right ) + x^{6} \left (\frac {16 a^{2} c^{5} d^{5}}{3} + \frac {160 a b^{2} c^{4} d^{5}}{3} + \frac {85 b^{4} c^{3} d^{5}}{3}\right ) + x^{5} \left (16 a^{2} b c^{4} d^{5} + 48 a b^{3} c^{3} d^{5} + \frac {61 b^{5} c^{2} d^{5}}{5}\right ) + x^{4} \left (20 a^{2} b^{2} c^{3} d^{5} + 25 a b^{4} c^{2} d^{5} + 3 b^{6} c d^{5}\right ) + x^{3} \left (\frac {40 a^{2} b^{3} c^{2} d^{5}}{3} + \frac {22 a b^{5} c d^{5}}{3} + \frac {b^{7} d^{5}}{3}\right ) + x^{2} \left (5 a^{2} b^{4} c d^{5} + a b^{6} d^{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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