Optimal. Leaf size=709 \[ -\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 1.57429, antiderivative size = 709, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {749, 748, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 749
Rule 748
Rule 747
Rule 401
Rule 108
Rule 409
Rule 1213
Rule 537
Rule 444
Rule 63
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx &=\frac{\left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \int \frac{1}{(d+e x) \left (-\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}\right )^{3/4}} \, dx}{\left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{\left (2 \sqrt{2} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{c (2 c d-b e)}{b^2-4 a c}+e x\right ) \left (1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4}} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac{\left (2 \sqrt{2} e \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \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 (a+b x+c x^2\right )^{3/4}}-\frac{\left (2 \sqrt{2} c (2 c d-b e) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \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 ) \left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac{\left (\sqrt{2} e \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \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 )}{\left (a+b x+c x^2\right )^{3/4}}-\frac{\left (\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{\left (b^2-4 a c\right ) x}{c^2}} \left (1-\frac{\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \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 )}{\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 ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{\left (4 \sqrt{2} c^2 e \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-\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 ) \left (a+b x+c x^2\right )^{3/4}}+\frac{\left (4 \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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\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{(b+2 c x)^2}{b^2-4 a c}}\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 ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{\left (\sqrt{2} \left (b^2-4 a c\right ) e \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\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{c} \sqrt{c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}+\frac{\left (\sqrt{2} \left (b^2-4 a c\right ) e \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\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{c} \sqrt{c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}+\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac{(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}}+\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac{(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt{e} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt{e} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}+\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (1-\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac{(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}}+\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (1+\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac{(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt{e} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt{e} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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} c \left (c d^2-b d e+a e^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\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}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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} c \left (c d^2-b d e+a e^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 1.74005, size = 533, normalized size = 0.75 \[ \frac{\sqrt [4]{a+x (b+c x)} \left (-\sqrt{2} \sqrt [4]{c} \sqrt{e} (b+2 c x) \sqrt [4]{e (a e-b d)+c d^2} \left (\tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )+\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )\right )+\sqrt [4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left .-\sin ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )+\sqrt [4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left .-\sin ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )\right )}{\sqrt{2} \sqrt [4]{4 a c-b^2} (b+2 c x) \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.242, 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{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{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 ) \left (a + b x + c x^{2}\right )^{\frac{3}{4}}}\, 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{3}{4}}{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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