Optimal. Leaf size=65 \[ 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {686, 692, 628} \begin {gather*} 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 686
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c d^2\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.98 \begin {gather*} d^5 \left (-\frac {\left (b^2-4 a c\right )^2}{a+x (b+c x)}+8 c \left (b^2-4 a c\right ) \log (a+x (b+c x))+16 b c^2 x+16 c^3 x^2\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}{\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 = 0.40, size = 165, normalized size = 2.54 \begin {gather*} \frac {16 \, c^{4} d^{5} x^{4} + 32 \, b c^{3} d^{5} x^{3} + 16 \, a b c^{2} d^{5} x + 16 \, {\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} - {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} + 8 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{5}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 100, normalized size = 1.54 \begin {gather*} 8 \, {\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}}{c x^{2} + b x + a} + \frac {16 \, {\left (c^{7} d^{5} x^{2} + 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 [A] time = 0.05, size = 128, normalized size = 1.97 \begin {gather*} 16 c^{3} d^{5} x^{2}-\frac {16 a^{2} c^{2} d^{5}}{c \,x^{2}+b x +a}+\frac {8 a \,b^{2} c \,d^{5}}{c \,x^{2}+b x +a}-32 a \,c^{2} d^{5} \ln \left (c \,x^{2}+b x +a \right )-\frac {b^{4} d^{5}}{c \,x^{2}+b x +a}+8 b^{2} c \,d^{5} \ln \left (c \,x^{2}+b x +a \right )+16 b \,c^{2} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 86, normalized size = 1.32 \begin {gather*} 16 \, c^{3} d^{5} x^{2} + 16 \, b c^{2} d^{5} x + 8 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5}}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 97, normalized size = 1.49 \begin {gather*} 16\,c^3\,d^5\,x^2-\frac {16\,a^2\,c^2\,d^5-8\,a\,b^2\,c\,d^5+b^4\,d^5}{c\,x^2+b\,x+a}-\ln \left (c\,x^2+b\,x+a\right )\,\left (32\,a\,c^2\,d^5-8\,b^2\,c\,d^5\right )+16\,b\,c^2\,d^5\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.64, size = 90, normalized size = 1.38 \begin {gather*} 16 b c^{2} d^{5} x + 16 c^{3} d^{5} x^{2} - 8 c d^{5} \left (4 a c - b^{2}\right ) \log {\left (a + b x + c x^{2} \right )} + \frac {- 16 a^{2} c^{2} d^{5} + 8 a b^{2} c d^{5} - b^{4} d^{5}}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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