Optimal. Leaf size=66 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]
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Rubi [A] time = 0.0790939, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {982, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]
Antiderivative was successfully verified.
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Rule 982
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx &=-\left ((2 b) \operatorname{Subst}\left (\int \frac{1}{b \left (b^2-4 c d\right )-(a b-b d) x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}}\\ \end{align*}
Mathematica [B] time = 0.225642, size = 161, normalized size = 2.44 \[ \frac{\tanh ^{-1}\left (\frac{4 a c-2 c x \sqrt{b^2-4 c d}-b \left (\sqrt{b^2-4 c d}+b\right )}{4 c \sqrt{a-d} \sqrt{a+x (b+c x)}}\right )+\tanh ^{-1}\left (\frac{-2 c \left (2 a+x \sqrt{b^2-4 c d}\right )-b \sqrt{b^2-4 c d}+b^2}{4 c \sqrt{a-d} \sqrt{a+x (b+c x)}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.327, size = 307, normalized size = 4.7 \begin{align*}{\ln \left ({ \left ( 2\,a-2\,d-\sqrt{{b}^{2}-4\,cd} \left ( x+{\frac{1}{2\,c} \left ( \sqrt{{b}^{2}-4\,cd}+b \right ) } \right ) +2\,\sqrt{a-d}\sqrt{ \left ( x+1/2\,{\frac{\sqrt{{b}^{2}-4\,cd}+b}{c}} \right ) ^{2}c-\sqrt{{b}^{2}-4\,cd} \left ( x+1/2\,{\frac{\sqrt{{b}^{2}-4\,cd}+b}{c}} \right ) +a-d} \right ) \left ( x+{\frac{1}{2\,c} \left ( \sqrt{{b}^{2}-4\,cd}+b \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2}-4\,cd}}}{\frac{1}{\sqrt{a-d}}}}-{\ln \left ({ \left ( 2\,a-2\,d+\sqrt{{b}^{2}-4\,cd} \left ( x-{\frac{1}{2\,c} \left ( -b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) +2\,\sqrt{a-d}\sqrt{ \left ( x-1/2\,{\frac{-b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) ^{2}c+\sqrt{{b}^{2}-4\,cd} \left ( x-1/2\,{\frac{-b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) +a-d} \right ) \left ( x-{\frac{1}{2\,c} \left ( -b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2}-4\,cd}}}{\frac{1}{\sqrt{a-d}}}} \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 = 1.45844, size = 1755, normalized size = 26.59 \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{a + b x + c x^{2}} \left (b x + c x^{2} + d\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 14.6685, size = 620, normalized size = 9.39 \begin{align*} \frac{\log \left ({\left | -3 \, a b^{2} c + 4 \, a^{2} c^{2} + 2 \, b^{2} c d -{\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d + 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} c^{\frac{3}{2}}\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt{c} -{\left (b^{3} \sqrt{c} + 4 \, a b c^{\frac{3}{2}} - 8 \, b c^{\frac{3}{2}} d + 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b c\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \right |}\right )}{\sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} - \frac{\log \left ({\left | -3 \, a b^{2} c + 4 \, a^{2} c^{2} + 2 \, b^{2} c d -{\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d - 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} c^{\frac{3}{2}}\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} + \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt{c} -{\left (b^{3} \sqrt{c} + 4 \, a b c^{\frac{3}{2}} - 8 \, b c^{\frac{3}{2}} d - 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b c\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \right |}\right )}{\sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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