Optimal. Leaf size=61 \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac {1}{2} d^5 (b+2 c x)^4 \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {692, 628} \begin {gather*} d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac {1}{2} d^5 (b+2 c x)^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^5}{a+b x+c x^2} \, dx &=\frac {1}{2} d^5 (b+2 c x)^4+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right ) d^5 (b+2 c x)^2+\frac {1}{2} d^5 (b+2 c x)^4+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right ) d^5 (b+2 c x)^2+\frac {1}{2} d^5 (b+2 c x)^4+\left (b^2-4 a c\right )^2 d^5 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.89 \begin {gather*} d^5 \left (8 c x (b+c x) \left (c \left (c x^2-2 a\right )+b^2+b c x\right )+\left (b^2-4 a c\right )^2 \log (a+x (b+c x))\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 \frac {(b d+2 c d x)^5}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 99, normalized size = 1.62 \begin {gather*} 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \, {\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \, {\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 118, normalized size = 1.93 \begin {gather*} {\left (b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) + \frac {8 \, {\left (c^{8} d^{5} x^{4} + 2 \, b c^{7} d^{5} x^{3} + 2 \, b^{2} c^{6} d^{5} x^{2} - 2 \, a c^{7} d^{5} x^{2} + b^{3} c^{5} d^{5} x - 2 \, a b c^{6} d^{5} x\right )}}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 133, normalized size = 2.18 \begin {gather*} 8 c^{4} d^{5} x^{4}+16 b \,c^{3} d^{5} x^{3}-16 a \,c^{3} d^{5} x^{2}+16 b^{2} c^{2} d^{5} x^{2}+16 a^{2} c^{2} d^{5} \ln \left (c \,x^{2}+b x +a \right )-8 a \,b^{2} c \,d^{5} \ln \left (c \,x^{2}+b x +a \right )-16 a b \,c^{2} d^{5} x +b^{4} d^{5} \ln \left (c \,x^{2}+b x +a \right )+8 b^{3} c \,d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 99, normalized size = 1.62 \begin {gather*} 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \, {\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \, {\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 139, normalized size = 2.28 \begin {gather*} \ln \left (c\,x^2+b\,x+a\right )\,\left (16\,a^2\,c^2\,d^5-8\,a\,b^2\,c\,d^5+b^4\,d^5\right )-x^2\,\left (16\,a\,c^3\,d^5-16\,b^2\,c^2\,d^5\right )+x\,\left (40\,b^3\,c\,d^5+\frac {b\,\left (32\,a\,c^3\,d^5-32\,b^2\,c^2\,d^5\right )}{c}-48\,a\,b\,c^2\,d^5\right )+8\,c^4\,d^5\,x^4+16\,b\,c^3\,d^5\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 99, normalized size = 1.62 \begin {gather*} 16 b c^{3} d^{5} x^{3} + 8 c^{4} d^{5} x^{4} + d^{5} \left (4 a c - b^{2}\right )^{2} \log {\left (a + b x + c x^{2} \right )} + x^{2} \left (- 16 a c^{3} d^{5} + 16 b^{2} c^{2} d^{5}\right ) + x \left (- 16 a b c^{2} d^{5} + 8 b^{3} c d^{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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