Optimal. Leaf size=1209 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.51291, antiderivative size = 1209, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 18, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818, Rules used = {734, 843, 623, 305, 220, 1196, 749, 748, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ -\frac{(2 c d-b e) \sqrt [4]{c x^2+b x+a} (b+2 c x)}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt [4]{4 a c-b^2} \left (c d^2-b e d+a e^2\right )^{3/4} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \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 e d+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{4 a c-b^2} \left (c d^2-b e d+a e^2\right )^{3/4} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \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 e d+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{c x^2+b x+a}}+\frac{2 \left (c x^2+b x+a\right )^{3/4}}{3 e}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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^3 \sqrt [4]{c x^2+b x+a} (b+2 c x)}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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^3 \sqrt [4]{c x^2+b x+a} (b+2 c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 734
Rule 843
Rule 623
Rule 305
Rule 220
Rule 1196
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{\left (a+b x+c x^2\right )^{3/4}}{d+e x} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{\int \frac{b d-2 a e+(2 c d-b e) x}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{2 e}\\ &=\frac{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{(2 c d-b e) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{2 e^2}-\frac{(e (b d-2 a e)-d (2 c d-b e)) \int \frac{1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{2 e^2}\\ &=\frac{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{\left (2 (2 c d-b e) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{e^2 (b+2 c x)}-\frac{\left ((e (b d-2 a e)-d (2 c d-b e)) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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}{2 e^2 \sqrt [4]{a+b x+c x^2}}\\ &=\frac{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{\left (\sqrt{b^2-4 a c} (2 c d-b e) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\sqrt{c} e^2 (b+2 c x)}+\frac{\left (\sqrt{b^2-4 a c} (2 c d-b e) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{2 \sqrt{c} x^2}{\sqrt{b^2-4 a c}}}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\sqrt{c} e^2 (b+2 c x)}-\frac{\left ((e (b d-2 a e)-d (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}{\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{2} e^2 \sqrt [4]{a+b x+c x^2}}\\ &=\frac{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}+\frac{\left ((e (b d-2 a e)-d (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{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{2} e \sqrt [4]{a+b x+c x^2}}+\frac{\left (c (2 c d-b e) (e (b d-2 a e)-d (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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^2 \sqrt [4]{a+b x+c x^2}}\\ &=\frac{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}+\frac{\left ((e (b d-2 a e)-d (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}{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 )}{2 \sqrt{2} e \sqrt [4]{a+b x+c x^2}}+\frac{\left (\sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (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 ) e^2 \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{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (\sqrt{2} c^2 (e (b d-2 a e)-d (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{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 ) e \sqrt [4]{a+b x+c x^2}}+\frac{\left (c \sqrt{-b^2+4 a c} (2 c d-b e) (e (b d-2 a e)-d (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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^3 \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 (c \sqrt{-b^2+4 a c} (2 c d-b e) (e (b d-2 a e)-d (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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^3 \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{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (\left (-b^2+4 a c\right )^{3/2} (e (b d-2 a e)-d (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}{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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^2 \sqrt [4]{a+b x+c x^2}}+\frac{\left (\left (-b^2+4 a c\right )^{3/2} (e (b d-2 a e)-d (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}{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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^2 \sqrt [4]{a+b x+c x^2}}+\frac{\left (c \sqrt{-b^2+4 a c} (2 c d-b e) (e (b d-2 a e)-d (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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^3 \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 (c \sqrt{-b^2+4 a c} (2 c d-b e) (e (b d-2 a e)-d (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 )}{\sqrt{2} \left (b^2-4 a c\right ) e^3 \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{2 \left (a+b x+c x^2\right )^{3/4}}{3 e}-\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\sqrt [4]{-b^2+4 a c} \left (c d^2-b d e+a e^2\right )^{3/4} \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} e^{5/2} \sqrt [4]{a+b x+c x^2}}-\frac{\sqrt [4]{-b^2+4 a c} \left (c d^2-b d e+a e^2\right )^{3/4} \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} e^{5/2} \sqrt [4]{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}+\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{c d^2-b d e+a e^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}} \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^3 (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{c d^2-b d e+a e^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}} \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^3 (b+2 c x) \sqrt [4]{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.42403, size = 180, normalized size = 0.15 \[ \frac{4 \sqrt{2} (a+x (b+c x))^{3/4} F_1\left (-\frac{3}{2};-\frac{3}{4},-\frac{3}{4};-\frac{1}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{3 e \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.278, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}}\, 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{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}}{e x + d}\,{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{\left (a + b x + c x^{2}\right )^{\frac{3}{4}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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