Optimal. Leaf size=192 \[ \frac{5 c^2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}-\frac{5 c^2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}+\frac{5 c (b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.129688, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {687, 694, 329, 298, 203, 206} \[ \frac{5 c^2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}-\frac{5 c^2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}+\frac{5 c (b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 687
Rule 694
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}-\frac{(5 c) \int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{5 c (b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{\left (5 c^2\right ) \int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac{(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{5 c (b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{(5 c) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{4 \left (b^2-4 a c\right )^2 d}\\ &=-\frac{(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{5 c (b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{(5 c) \operatorname{Subst}\left (\int \frac{x^2}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )}{2 \left (b^2-4 a c\right )^2 d}\\ &=-\frac{(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{5 c (b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}-\frac{\left (5 c^2 d\right ) \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 )^2}+\frac{\left (5 c^2 d\right ) \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 )^2}\\ &=-\frac{(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{5 c (b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{5 c^2 \sqrt{d} \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 )^{9/4}}-\frac{5 c^2 \sqrt{d} \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 )^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.045127, size = 59, normalized size = 0.31 \[ -\frac{64 c^2 (d (b+2 c x))^{3/2} \, _2F_1\left (\frac{3}{4},3;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 d \left (b^2-4 a c\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.194, size = 419, normalized size = 2.2 \begin{align*} 8\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{3/2}}{ \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 ) ^{2}}}+10\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{3/2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{2} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{4}\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{9}{4}}}}+{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{9}{4}}}}-{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{9}{4}}}} \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 = 2.11057, size = 4201, normalized size = 21.88 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27762, size = 871, normalized size = 4.54 \begin{align*} -\frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} \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 )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} \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 )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} + \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} \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 )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} \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 )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} - \frac{2 \,{\left (9 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{3} - 36 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{3} d^{3} - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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