Optimal. Leaf size=170 \[ -\frac{5 c^2 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c^2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.124467, antiderivative size = 170, 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 = {686, 694, 329, 212, 206, 203} \[ -\frac{5 c^2 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c^2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 686
Rule 694
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{1}{2} \left (5 c d^2\right ) \int \frac{(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac{1}{2} \left (5 c^2 d^4\right ) \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac{1}{4} \left (5 c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )\\ &=-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac{1}{2} \left (5 c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )\\ &=-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac{\left (5 c^2 d^4\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 )}{\sqrt{b^2-4 a c}}-\frac{\left (5 c^2 d^4\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 )}{\sqrt{b^2-4 a c}}\\ &=-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac{5 c^2 d^{7/2} \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 )^{3/4}}-\frac{5 c^2 d^{7/2} \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 )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.280626, size = 196, normalized size = 1.15 \[ \frac{(d (b+2 c x))^{7/2} \left (-64 \left (b^2-4 a c\right )^{3/4} (b+2 c x)^{5/2}+40 \left (b^2-4 a c\right )^{7/4} \sqrt{b+2 c x}+20 c (a+x (b+c x)) \left (2 \left (b^2-4 a c\right )^{3/4} \sqrt{b+2 c x}-12 c (a+x (b+c x)) \left (\tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+\tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )\right )\right )}{48 \left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2} (a+x (b+c x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.224, size = 435, normalized size = 2.6 \begin{align*} -18\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{5/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-40\,{\frac{{c}^{3}{d}^{7}a\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+10\,{\frac{{c}^{2}{d}^{7}{b}^{2}\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\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{3}{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{3}{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{3}{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 = 1.75763, size = 1530, normalized size = 9. \begin{align*} \frac{20 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \arctan \left (-\frac{\left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{3}{4}}{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} \sqrt{2 \, c d x + b d} d^{3} - \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{3}{4}} \sqrt{2 \, c^{5} d^{7} x + b c^{4} d^{7} + \sqrt{\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}}{c^{8} d^{14}}\right ) - 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt{2 \, c d x + b d} c^{2} d^{3} + 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) + 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt{2 \, c d x + b d} c^{2} d^{3} - 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) -{\left (9 \, c^{2} d^{3} x^{2} + 9 \, b c d^{3} x +{\left (b^{2} + 5 \, a c\right )} d^{3}\right )} \sqrt{2 \, c d x + b d}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \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 = 1.30434, size = 689, normalized size = 4.05 \begin{align*} -\frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3} \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^{2} - 4 \, \sqrt{2} a c} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3} \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^{2} - 4 \, \sqrt{2} a c} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3} \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^{2} - 4 \, \sqrt{2} a c\right )}} + \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3} \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^{2} - 4 \, \sqrt{2} a c\right )}} + \frac{2 \,{\left (5 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{7} - 20 \, \sqrt{2 \, c d x + b d} a c^{3} d^{7} - 9 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{5}\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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