Optimal. Leaf size=102 \[ 6 \sqrt{c} d^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0533681, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {686, 692, 621, 206} \[ 6 \sqrt{c} d^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 686
Rule 692
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}}+\left (12 c d^2\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}+\left (6 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right ) 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{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}+6 \sqrt{c} \left (b^2-4 a c\right ) d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.410818, size = 139, normalized size = 1.36 \[ d^4 \left (-\frac{6 c^{3/2} (a+x (b+c x))^{3/2} \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{\sqrt{4 a-\frac{b^2}{c}} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/2}}-\frac{2 (b+2 c x) \left (-2 c \left (3 a+c x^2\right )+b^2-2 b c x\right )}{\sqrt{a+x (b+c x)}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 340, normalized size = 3.3 \begin{align*} -6\,{\frac{c{d}^{4}{b}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+12\,{\frac{c{d}^{4}{b}^{3}a}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+24\,{\frac{{c}^{2}{d}^{4}{b}^{2}ax}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+8\,{\frac{{d}^{4}{c}^{3}{x}^{3}}{\sqrt{c{x}^{2}+bx+a}}}-3\,{\frac{{d}^{4}{b}^{5}}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-5\,{\frac{{d}^{4}{b}^{3}}{\sqrt{c{x}^{2}+bx+a}}}+12\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{\sqrt{c{x}^{2}+bx+a}}}-6\,{\frac{{d}^{4}{b}^{2}cx}{\sqrt{c{x}^{2}+bx+a}}}+6\,{d}^{4}\sqrt{c}{b}^{2}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) +12\,{\frac{c{d}^{4}ba}{\sqrt{c{x}^{2}+bx+a}}}+24\,{\frac{{d}^{4}a{c}^{2}x}{\sqrt{c{x}^{2}+bx+a}}}-24\,{d}^{4}{c}^{3/2}a\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 = 4.59705, size = 788, normalized size = 7.73 \begin{align*} \left [-\frac{3 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} +{\left (b^{3} - 4 \, a b c\right )} d^{4} x +{\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt{c} \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 ) - 2 \,{\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x -{\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{c x^{2} + b x + a}, -\frac{2 \,{\left (3 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} +{\left (b^{3} - 4 \, a b c\right )} d^{4} x +{\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt{-c} \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 ) -{\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x -{\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\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}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{16 c^{4} x^{4}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{32 b c^{3} x^{3}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{24 b^{2} c^{2} x^{2}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{8 b^{3} c x}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \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 [B] time = 1.21195, size = 336, normalized size = 3.29 \begin{align*} -\frac{6 \,{\left (b^{2} c d^{4} - 4 \, a 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 )}{\sqrt{c}} + \frac{2 \,{\left (2 \,{\left ({\left (\frac{2 \,{\left (b^{2} c^{5} d^{4} - 4 \, a c^{6} d^{4}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac{3 \,{\left (b^{3} c^{4} d^{4} - 4 \, a b c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac{6 \,{\left (a b^{2} c^{4} d^{4} - 4 \, a^{2} c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac{b^{5} c^{2} d^{4} - 10 \, a b^{3} c^{3} d^{4} + 24 \, a^{2} b c^{4} d^{4}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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