Optimal. Leaf size=733 \[ \frac{\sqrt [4]{4 a c-b^2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{\sqrt [4]{c} \sqrt{e} \sqrt [4]{a+b x+c x^2} \sqrt [4]{a e^2-b d e+c d^2}}-\frac{\sqrt [4]{4 a c-b^2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{\sqrt [4]{c} \sqrt{e} \sqrt [4]{a+b x+c x^2} \sqrt [4]{a e^2-b d e+c d^2}}-\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e (b+2 c x) \sqrt [4]{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e (b+2 c x) \sqrt [4]{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}} \]
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Rubi [A] time = 1.55996, antiderivative size = 733, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {749, 748, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac{\sqrt [4]{4 a c-b^2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{\sqrt [4]{c} \sqrt{e} \sqrt [4]{a+b x+c x^2} \sqrt [4]{a e^2-b d e+c d^2}}-\frac{\sqrt [4]{4 a c-b^2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{\sqrt [4]{c} \sqrt{e} \sqrt [4]{a+b x+c x^2} \sqrt [4]{a e^2-b d e+c d^2}}-\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e (b+2 c x) \sqrt [4]{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e (b+2 c x) \sqrt [4]{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}} \]
Antiderivative was successfully verified.
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Rule 749
Rule 748
Rule 746
Rule 399
Rule 490
Rule 1213
Rule 537
Rule 444
Rule 63
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx &=\frac{\sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac{1}{(d+e x) \sqrt [4]{-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}}} \, dx}{\sqrt [4]{a+b x+c x^2}}\\ &=\frac{\left (\sqrt{2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{c (2 c d-b e)}{b^2-4 a c}+e x\right ) \sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}}} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{\sqrt [4]{a+b x+c x^2}}\\ &=-\frac{\left (\sqrt{2} e \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{\sqrt [4]{a+b x+c x^2}}-\frac{\left (\sqrt{2} c (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{\left (e \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x}{c^2}} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2\right )}{\sqrt{2} \sqrt [4]{a+b x+c x^2}}-\frac{\left (2 \sqrt{2} c (2 c d-b e) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4} \left (e^2-\frac{(2 c d-b e)^2}{b^2-4 a c}-e^2 x^4\right )} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}\\ &=\frac{\left (2 \sqrt{2} c^2 e \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{-\frac{c^2 e^2}{b^2-4 a c}+\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}+\frac{c^2 e^2 x^4}{b^2-4 a c}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac{\left (\sqrt{2} c \sqrt{-b^2+4 a c} (2 c d-b e) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}-\sqrt{-b^2+4 a c} e x^2\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac{\left (\sqrt{2} c \sqrt{-b^2+4 a c} (2 c d-b e) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}+\sqrt{-b^2+4 a c} e x^2\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}\\ &=\frac{\left (\sqrt{2} \left (-b^2+4 a c\right )^{3/2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}-\sqrt{-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac{\left (\sqrt{2} \left (-b^2+4 a c\right )^{3/2} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}+\sqrt{-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac{\left (\sqrt{2} c \sqrt{-b^2+4 a c} (2 c d-b e) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}-\sqrt{-b^2+4 a c} e x^2\right )} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac{\left (\sqrt{2} c \sqrt{-b^2+4 a c} (2 c d-b e) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}+\sqrt{-b^2+4 a c} e x^2\right )} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}\\ &=\frac{\sqrt [4]{-b^2+4 a c} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \sqrt{e} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}-\frac{\sqrt [4]{-b^2+4 a c} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \sqrt{e} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \Pi \left (-\frac{\sqrt{-b^2+4 a c} e}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} \sqrt{-b^2+4 a c} e \sqrt{c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \Pi \left (\frac{\sqrt{-b^2+4 a c} e}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} \sqrt{-b^2+4 a c} e \sqrt{c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.326815, size = 178, normalized size = 0.24 \[ -\frac{\sqrt{2} \sqrt [4]{\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} F_1\left (\frac{1}{2};\frac{1}{4},\frac{1}{4};\frac{3}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{e \sqrt [4]{a+x (b+c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.245, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right ) \sqrt [4]{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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