Optimal. Leaf size=117 \[ \frac{3}{4} d^4 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}+\frac{3 d^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.0547876, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {692, 621, 206} \[ \frac{3}{4} d^4 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}+\frac{3 d^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^4}{\sqrt{a+b x+c x^2}} \, dx &=\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}+\frac{1}{4} \left (3 \left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx\\ &=\frac{3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}+\frac{1}{8} \left (3 \left (b^2-4 a c\right )^2 d^4\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=\frac{3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}+\frac{1}{4} \left (3 \left (b^2-4 a c\right )^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )\\ &=\frac{3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}+\frac{3 \left (b^2-4 a c\right )^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.106997, size = 100, normalized size = 0.85 \[ d^4 \left (\frac{1}{4} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (2 c x^2-3 a\right )+5 b^2+8 b c x\right )+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{8 \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 242, normalized size = 2.1 \begin{align*} 4\,{d}^{4}{c}^{3}{x}^{3}\sqrt{c{x}^{2}+bx+a}+6\,{d}^{4}{c}^{2}b{x}^{2}\sqrt{c{x}^{2}+bx+a}+{\frac{9\,{d}^{4}{b}^{2}cx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{d}^{4}{b}^{3}}{4}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{d}^{4}{b}^{4}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-3\,{d}^{4}\sqrt{c}{b}^{2}a\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) -3\,{d}^{4}cba\sqrt{c{x}^{2}+bx+a}-6\,{d}^{4}{c}^{2}ax\sqrt{c{x}^{2}+bx+a}+6\,{d}^{4}{c}^{3/2}{a}^{2}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \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 [A] time = 3.3668, size = 710, normalized size = 6.07 \begin{align*} \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{c} d^{4} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left (16 \, c^{4} d^{4} x^{3} + 24 \, b c^{3} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{4} x +{\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{16 \, c}, -\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-c} d^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left (16 \, c^{4} d^{4} x^{3} + 24 \, b c^{3} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{4} x +{\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{8 \, c}\right ] \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*} d^{4} \left (\int \frac{b^{4}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{16 c^{4} x^{4}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{32 b c^{3} x^{3}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{24 b^{2} c^{2} x^{2}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{8 b^{3} c x}{\sqrt{a + b x + c x^{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17606, size = 215, normalized size = 1.84 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c^{3} d^{4} x + 3 \, b c^{2} d^{4}\right )} x + \frac{3 \,{\left (3 \, b^{2} c^{4} d^{4} - 4 \, a c^{5} d^{4}\right )}}{c^{3}}\right )} x + \frac{5 \, b^{3} c^{3} d^{4} - 12 \, a b c^{4} d^{4}}{c^{3}}\right )} - \frac{3 \,{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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