Optimal. Leaf size=140 \[ \frac{60 c^2}{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{60 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{7/2}}+\frac{5 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0993551, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {687, 693, 618, 206} \[ \frac{60 c^2}{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{60 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{7/2}}+\frac{5 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx &=-\frac{1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}-\frac{(5 c) \int \frac{1}{(b d+2 c d x)^2 \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^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac{5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac{\left (30 c^2\right ) \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2}\\ &=\frac{60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac{1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac{5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac{\left (30 c^2\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=\frac{60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac{1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac{5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{\left (60 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^3 d^2}\\ &=\frac{60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac{1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac{5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{60 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^2}\\ \end{align*}
Mathematica [A] time = 0.134535, size = 119, normalized size = 0.85 \[ \frac{\frac{120 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{14 c (b+2 c x)}{a+x (b+c x)}+\frac{64 c^2}{b+2 c x}}{2 d^2 \left (b^2-4 a c\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.162, size = 273, normalized size = 2. \begin{align*} -14\,{\frac{{c}^{3}{x}^{3}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-21\,{\frac{b{c}^{2}{x}^{2}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-18\,{\frac{a{c}^{2}x}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{b}^{2}cx}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-9\,{\frac{abc}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{{b}^{3}}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-60\,{\frac{{c}^{2}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{7/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-32\,{\frac{{c}^{2}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 2\,cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.36787, size = 2534, normalized size = 18.1 \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 = 15.2761, size = 801, normalized size = 5.72 \begin{align*} \frac{30 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} \log{\left (x + \frac{- 7680 a^{4} c^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 7680 a^{3} b^{2} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 2880 a^{2} b^{4} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 480 a b^{6} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 30 b^{8} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} - \frac{30 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} \log{\left (x + \frac{7680 a^{4} c^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 7680 a^{3} b^{2} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 2880 a^{2} b^{4} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 480 a b^{6} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b^{8} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} - \frac{64 a^{2} c^{2} + 18 a b^{2} c - b^{4} + 240 b c^{3} x^{3} + 120 c^{4} x^{4} + x^{2} \left (200 a c^{3} + 130 b^{2} c^{2}\right ) + x \left (200 a b c^{2} + 10 b^{3} c\right )}{128 a^{5} b c^{3} d^{2} - 96 a^{4} b^{3} c^{2} d^{2} + 24 a^{3} b^{5} c d^{2} - 2 a^{2} b^{7} d^{2} + x^{5} \left (256 a^{3} c^{6} d^{2} - 192 a^{2} b^{2} c^{5} d^{2} + 48 a b^{4} c^{4} d^{2} - 4 b^{6} c^{3} d^{2}\right ) + x^{4} \left (640 a^{3} b c^{5} d^{2} - 480 a^{2} b^{3} c^{4} d^{2} + 120 a b^{5} c^{3} d^{2} - 10 b^{7} c^{2} d^{2}\right ) + x^{3} \left (512 a^{4} c^{5} d^{2} + 128 a^{3} b^{2} c^{4} d^{2} - 288 a^{2} b^{4} c^{3} d^{2} + 88 a b^{6} c^{2} d^{2} - 8 b^{8} c d^{2}\right ) + x^{2} \left (768 a^{4} b c^{4} d^{2} - 448 a^{3} b^{3} c^{3} d^{2} + 48 a^{2} b^{5} c^{2} d^{2} + 12 a b^{7} c d^{2} - 2 b^{9} d^{2}\right ) + x \left (256 a^{5} c^{4} d^{2} + 64 a^{4} b^{2} c^{3} d^{2} - 144 a^{3} b^{4} c^{2} d^{2} + 44 a^{2} b^{6} c d^{2} - 4 a b^{8} d^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18273, size = 408, normalized size = 2.91 \begin{align*} \frac{32 \, c^{8} d^{11}}{{\left (b^{6} c^{6} d^{12} - 12 \, a b^{4} c^{7} d^{12} + 48 \, a^{2} b^{2} c^{8} d^{12} - 64 \, a^{3} c^{9} d^{12}\right )}{\left (2 \, c d x + b d\right )}} - \frac{60 \, c^{2} \arctan \left (-\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c} d^{2}} - \frac{4 \,{\left (\frac{9 \, b^{2} c^{2} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac{36 \, a c^{3} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac{7 \, c^{2}}{{\left (2 \, c d x + b d\right )} d}\right )}}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\left (\frac{b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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