Optimal. Leaf size=143 \[ -\frac{\sqrt{b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{6 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}}+\frac{6 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}} \]
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Rubi [A] time = 0.102449, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {687, 694, 329, 212, 206, 203} \[ -\frac{\sqrt{b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{6 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}}+\frac{6 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 694
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^2} \, dx &=-\frac{\sqrt{b d+2 c d x}}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac{(3 c) \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c}\\ &=-\frac{\sqrt{b d+2 c d x}}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )}{2 \left (b^2-4 a c\right ) d}\\ &=-\frac{\sqrt{b d+2 c d x}}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right ) d}\\ &=-\frac{\sqrt{b d+2 c d x}}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}+\frac{(6 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d-x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{(6 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d+x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ &=-\frac{\sqrt{b d+2 c d x}}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}+\frac{6 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )}{\left (b^2-4 a c\right )^{7/4} \sqrt{d}}+\frac{6 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )}{\left (b^2-4 a c\right )^{7/4} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.163875, size = 167, normalized size = 1.17 \[ \frac{-\left (b^2-4 a c\right ) (b+2 c x)+6 c \sqrt [4]{b^2-4 a c} \sqrt{b+2 c x} (a+x (b+c x)) \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+6 c \sqrt [4]{b^2-4 a c} \sqrt{b+2 c x} (a+x (b+c x)) \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x)) \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.191, size = 345, normalized size = 2.4 \begin{align*} 4\,{\frac{c{d}^{3}\sqrt{2\,cdx+bd}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{3\,c{d}^{3}\sqrt{2}}{2}\ln \left ({ \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{7}{4}}}}+3\,{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{7/4}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-3\,{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{7/4}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \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 = 2.22683, size = 2684, normalized size = 18.77 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \left (b + 2 c x\right )} \left (a + b x + c x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16489, size = 684, normalized size = 4.78 \begin{align*} \frac{3 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} + \frac{3 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c \log \left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d - 8 \, \sqrt{2} a b^{2} c d + 16 \, \sqrt{2} a^{2} c^{2} d} - \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c \log \left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d - 8 \, \sqrt{2} a b^{2} c d + 16 \, \sqrt{2} a^{2} c^{2} d} + \frac{4 \, \sqrt{2 \, c d x + b d} c d}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}{\left (b^{2} - 4 \, a c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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