Optimal. Leaf size=73 \[ 16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac {4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac {d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 73, 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 = {686, 628} \[ 16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac {4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac {d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 628
Rule 686
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}+\left (4 c d^2\right ) \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {4 c d^5 (b+2 c x)^2}{a+b x+c x^2}+\left (16 c^2 d^4\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=-\frac {d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {4 c d^5 (b+2 c x)^2}{a+b x+c x^2}+16 c^2 d^5 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.89 \[ d^5 \left (16 c^2 \log (a+x (b+c x))-\frac {\left (b^2-4 a c\right ) \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )}{2 (a+x (b+c x))^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.28, size = 182, normalized size = 2.49 \[ -\frac {16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x + {\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5} - 32 \, {\left (c^{4} d^{5} x^{4} + 2 \, b c^{3} d^{5} x^{3} + 2 \, a b c^{2} d^{5} x + a^{2} c^{2} d^{5} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 110, normalized size = 1.51 \[ 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac {b^{4} d^{5} + 8 \, a b^{2} c d^{5} - 48 \, a^{2} c^{2} d^{5} + 16 \, {\left (b^{2} c^{2} d^{5} - 4 \, a c^{3} d^{5}\right )} x^{2} + 16 \, {\left (b^{3} c d^{5} - 4 \, a b c^{2} d^{5}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 181, normalized size = 2.48 \[ \frac {32 a \,c^{3} d^{5} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {8 b^{2} c^{2} d^{5} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {32 a b \,c^{2} d^{5} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {8 b^{3} c \,d^{5} x}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {24 a^{2} c^{2} d^{5}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {4 a \,b^{2} c \,d^{5}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {b^{4} d^{5}}{2 \left (c \,x^{2}+b x +a \right )^{2}}+16 c^{2} d^{5} \ln \left (c \,x^{2}+b x +a \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 124, normalized size = 1.70 \[ 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac {16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x + {\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 136, normalized size = 1.86 \[ \frac {x^2\,\left (32\,a\,c^3\,d^5-8\,b^2\,c^2\,d^5\right )-\frac {b^4\,d^5}{2}+8\,b\,x\,\left (4\,a\,c^2\,d^5-b^2\,c\,d^5\right )+24\,a^2\,c^2\,d^5-4\,a\,b^2\,c\,d^5}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+16\,c^2\,d^5\,\ln \left (c\,x^2+b\,x+a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.78, size = 141, normalized size = 1.93 \[ 16 c^{2} d^{5} \log {\left (a + b x + c x^{2} \right )} + \frac {48 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} - b^{4} d^{5} + x^{2} \left (64 a c^{3} d^{5} - 16 b^{2} c^{2} d^{5}\right ) + x \left (64 a b c^{2} d^{5} - 16 b^{3} c d^{5}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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