Optimal. Leaf size=154 \[ \frac{48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac{48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac{96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac{6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0881354, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {687, 693, 681, 31, 628} \[ \frac{48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac{48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac{96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac{6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 681
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx &=-\frac{1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}-\frac{(6 c) \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c}\\ &=-\frac{1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac{6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac{\left (48 c^2\right ) \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2}\\ &=\frac{48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac{1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac{6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac{\left (48 c^2\right ) \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=\frac{48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac{1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac{6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac{\left (48 c^2\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 d^4}-\frac{\left (192 c^3\right ) \int \frac{1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^4 d^3}\\ &=\frac{48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac{1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac{6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{96 c^2 \log (b+2 c x)}{\left (b^2-4 a c\right )^4 d^3}+\frac{48 c^2 \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^4 d^3}\\ \end{align*}
Mathematica [A] time = 0.144913, size = 111, normalized size = 0.72 \[ \frac{\frac{32 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac{16 c \left (b^2-4 a c\right )}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^2}{(a+x (b+c x))^2}+96 c^2 \log (a+x (b+c x))-192 c^2 \log (b+2 c x)}{2 d^3 \left (b^2-4 a c\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 332, normalized size = 2.2 \begin{align*} -32\,{\frac{a{x}^{2}{c}^{3}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+8\,{\frac{{b}^{2}{x}^{2}{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-32\,{\frac{ba{c}^{2}x}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+8\,{\frac{{b}^{3}cx}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-40\,{\frac{{a}^{2}{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+12\,{\frac{ac{b}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{b}^{4}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+48\,{\frac{{c}^{2}\ln \left ( c{x}^{2}+bx+a \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}}-96\,{\frac{{c}^{2}\ln \left ( 2\,cx+b \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}}-16\,{\frac{{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 2\,cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26, size = 747, normalized size = 4.85 \begin{align*} \frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2} + 36 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{2 \,{\left (4 \,{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \,{\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} +{\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \,{\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} +{\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \,{\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x +{\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} + \frac{48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} - \frac{96 \, c^{2} \log \left (2 \, c x + b\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33862, size = 1747, normalized size = 11.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 27.5422, size = 597, normalized size = 3.88 \begin{align*} - \frac{96 c^{2} \log{\left (\frac{b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac{48 c^{2} \log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} - \frac{32 a^{2} c^{2} + 20 a b^{2} c - b^{4} + 192 b c^{3} x^{3} + 96 c^{4} x^{4} + x^{2} \left (144 a c^{3} + 108 b^{2} c^{2}\right ) + x \left (144 a b c^{2} + 12 b^{3} c\right )}{128 a^{5} b^{2} c^{3} d^{3} - 96 a^{4} b^{4} c^{2} d^{3} + 24 a^{3} b^{6} c d^{3} - 2 a^{2} b^{8} d^{3} + x^{6} \left (512 a^{3} c^{7} d^{3} - 384 a^{2} b^{2} c^{6} d^{3} + 96 a b^{4} c^{5} d^{3} - 8 b^{6} c^{4} d^{3}\right ) + x^{5} \left (1536 a^{3} b c^{6} d^{3} - 1152 a^{2} b^{3} c^{5} d^{3} + 288 a b^{5} c^{4} d^{3} - 24 b^{7} c^{3} d^{3}\right ) + x^{4} \left (1024 a^{4} c^{6} d^{3} + 896 a^{3} b^{2} c^{5} d^{3} - 1056 a^{2} b^{4} c^{4} d^{3} + 296 a b^{6} c^{3} d^{3} - 26 b^{8} c^{2} d^{3}\right ) + x^{3} \left (2048 a^{4} b c^{5} d^{3} - 768 a^{3} b^{3} c^{4} d^{3} - 192 a^{2} b^{5} c^{3} d^{3} + 112 a b^{7} c^{2} d^{3} - 12 b^{9} c d^{3}\right ) + x^{2} \left (512 a^{5} c^{5} d^{3} + 896 a^{4} b^{2} c^{4} d^{3} - 736 a^{3} b^{4} c^{3} d^{3} + 136 a^{2} b^{6} c^{2} d^{3} + 4 a b^{8} c d^{3} - 2 b^{10} d^{3}\right ) + x \left (512 a^{5} b c^{4} d^{3} - 128 a^{4} b^{3} c^{3} d^{3} - 96 a^{3} b^{5} c^{2} d^{3} + 40 a^{2} b^{7} c d^{3} - 4 a b^{9} d^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2142, size = 408, normalized size = 2.65 \begin{align*} -\frac{96 \, c^{3} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{8} c d^{3} - 16 \, a b^{6} c^{2} d^{3} + 96 \, a^{2} b^{4} c^{3} d^{3} - 256 \, a^{3} b^{2} c^{4} d^{3} + 256 \, a^{4} c^{5} d^{3}} + \frac{48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}} + \frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 108 \, b^{2} c^{2} x^{2} + 144 \, a c^{3} x^{2} + 12 \, b^{3} c x + 144 \, a b c^{2} x - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2}}{2 \,{\left (b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}\right )}{\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + b^{2} x + 2 \, a c x + a b\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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