Optimal. Leaf size=123 \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt{b^2-4 a c}}-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4} \]
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Rubi [A] time = 0.0731791, antiderivative size = 123, 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 = {684, 688, 205} \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt{b^2-4 a c}}-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4} \]
Antiderivative was successfully verified.
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Rule 684
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^5} \, dx &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac{3 \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^3} \, dx}{16 c d^2}\\ &=-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac{3 \int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{128 c^2 d^4}\\ &=-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )}{32 c d^4}\\ &=-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{5/2} \sqrt{b^2-4 a c} d^5}\\ \end{align*}
Mathematica [A] time = 0.245584, size = 162, normalized size = 1.32 \[ \frac{-2 c \left (8 a^2 c+a \left (3 b^2+28 b c x+28 c^2 x^2\right )+x \left (23 b^2 c x+3 b^3+40 b c^2 x^2+20 c^3 x^3\right )\right )-3 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left (2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )}{128 c^3 d^5 (b+2 c x)^4 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.194, size = 622, normalized size = 5.1 \begin{align*} -{\frac{1}{32\,{c}^{4}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-4}}-{\frac{1}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{16\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a}{32\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{b}^{2}}{128\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{a}^{2}}{8\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{3\,{b}^{2}a}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{3\,{b}^{4}}{128\,{c}^{3}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \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 = 12.4489, size = 1320, normalized size = 10.73 \begin{align*} \left [-\frac{3 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \,{\left (3 \, b^{4} c - 4 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 20 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + 20 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{256 \,{\left (16 \,{\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{5} x^{4} + 32 \,{\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{5} x^{3} + 24 \,{\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{5} x^{2} + 8 \,{\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{5} x +{\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{5}\right )}}, -\frac{3 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt{b^{2} c - 4 \, a c^{2}} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \,{\left (3 \, b^{4} c - 4 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 20 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + 20 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{128 \,{\left (16 \,{\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{5} x^{4} + 32 \,{\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{5} x^{3} + 24 \,{\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{5} x^{2} + 8 \,{\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{5} x +{\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{5}\right )}}\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*} \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.79571, size = 911, normalized size = 7.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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