Optimal. Leaf size=97 \[ 48 c^2 d^7 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac {6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac {d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^7 (b+2 c x)^2 \]
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Rubi [A] time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, 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*} 48 c^2 d^7 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac {6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac {d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^7 (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)^7}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+\left (6 c d^2\right ) \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac {6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+\left (48 c^2 d^4\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=48 c^2 d^7 (b+2 c x)^2-\frac {d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac {6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+\left (48 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=48 c^2 d^7 (b+2 c x)^2-\frac {d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac {6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+48 c^2 \left (b^2-4 a c\right ) d^7 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 0.95 \begin {gather*} d^7 \left (48 c^2 \left (b^2-4 a c\right ) \log (a+x (b+c x))-\frac {12 c \left (b^2-4 a c\right )^2}{a+x (b+c x)}-\frac {\left (b^2-4 a c\right )^3}{2 (a+x (b+c x))^2}+64 b c^3 x+64 c^4 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)^7}{\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 = 0.39, size = 346, normalized size = 3.57 \begin {gather*} \frac {128 \, c^{6} d^{7} x^{6} + 384 \, b c^{5} d^{7} x^{5} + 128 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} d^{7} x^{4} + 128 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{7} x^{3} - 8 \, {\left (3 \, b^{4} c^{2} - 56 \, a b^{2} c^{3} + 32 \, a^{2} c^{4}\right )} d^{7} x^{2} - 8 \, {\left (3 \, b^{5} c - 24 \, a b^{3} c^{2} + 32 \, a^{2} b c^{3}\right )} d^{7} x - {\left (b^{6} + 12 \, a b^{4} c - 144 \, a^{2} b^{2} c^{2} + 320 \, a^{3} c^{3}\right )} d^{7} + 96 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + 2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{7} x^{3} + {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{7} x + {\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d^{7}\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 191, normalized size = 1.97 \begin {gather*} 48 \, {\left (b^{2} c^{2} d^{7} - 4 \, a c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{6} d^{7} + 12 \, a b^{4} c d^{7} - 144 \, a^{2} b^{2} c^{2} d^{7} + 320 \, a^{3} c^{3} d^{7} + 24 \, {\left (b^{4} c^{2} d^{7} - 8 \, a b^{2} c^{3} d^{7} + 16 \, a^{2} c^{4} d^{7}\right )} x^{2} + 24 \, {\left (b^{5} c d^{7} - 8 \, a b^{3} c^{2} d^{7} + 16 \, a^{2} b c^{3} d^{7}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {64 \, {\left (c^{10} d^{7} x^{2} + b c^{9} d^{7} x\right )}}{c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 307, normalized size = 3.16 \begin {gather*} -\frac {192 a^{2} c^{4} d^{7} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {96 a \,b^{2} c^{3} d^{7} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {12 b^{4} c^{2} d^{7} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {192 a^{2} b \,c^{3} d^{7} x}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {96 a \,b^{3} c^{2} d^{7} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {12 b^{5} c \,d^{7} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {160 a^{3} c^{3} d^{7}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {72 a^{2} b^{2} c^{2} d^{7}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 a \,b^{4} c \,d^{7}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {b^{6} d^{7}}{2 \left (c \,x^{2}+b x +a \right )^{2}}+64 c^{4} d^{7} x^{2}-192 a \,c^{3} d^{7} \ln \left (c \,x^{2}+b x +a \right )+48 b^{2} c^{2} d^{7} \ln \left (c \,x^{2}+b x +a \right )+64 b \,c^{3} d^{7} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 189, normalized size = 1.95 \begin {gather*} 64 \, c^{4} d^{7} x^{2} + 64 \, b c^{3} d^{7} x + 48 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac {24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (b^{6} + 12 \, a b^{4} c - 144 \, a^{2} b^{2} c^{2} + 320 \, a^{3} c^{3}\right )} d^{7}}{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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 214, normalized size = 2.21 \begin {gather*} 64\,c^4\,d^7\,x^2-\ln \left (c\,x^2+b\,x+a\right )\,\left (192\,a\,c^3\,d^7-48\,b^2\,c^2\,d^7\right )-\frac {\frac {b^6\,d^7}{2}+x^2\,\left (192\,a^2\,c^4\,d^7-96\,a\,b^2\,c^3\,d^7+12\,b^4\,c^2\,d^7\right )+12\,b\,x\,\left (16\,a^2\,c^3\,d^7-8\,a\,b^2\,c^2\,d^7+b^4\,c\,d^7\right )+160\,a^3\,c^3\,d^7-72\,a^2\,b^2\,c^2\,d^7+6\,a\,b^4\,c\,d^7}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+64\,b\,c^3\,d^7\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.99, size = 219, normalized size = 2.26 \begin {gather*} 64 b c^{3} d^{7} x + 64 c^{4} d^{7} x^{2} - 48 c^{2} d^{7} \left (4 a c - b^{2}\right ) \log {\left (a + b x + c x^{2} \right )} + \frac {- 320 a^{3} c^{3} d^{7} + 144 a^{2} b^{2} c^{2} d^{7} - 12 a b^{4} c d^{7} - b^{6} d^{7} + x^{2} \left (- 384 a^{2} c^{4} d^{7} + 192 a b^{2} c^{3} d^{7} - 24 b^{4} c^{2} d^{7}\right ) + x \left (- 384 a^{2} b c^{3} d^{7} + 192 a b^{3} c^{2} d^{7} - 24 b^{5} c d^{7}\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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