Optimal. Leaf size=881 \[ -\frac {\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac {\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac {2 \sqrt [4]{c x^2+b x+a}}{e} \]
[Out]
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Rubi [A] time = 2.64, antiderivative size = 881, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 17, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {734, 843, 623, 220, 749, 748, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac {\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac {\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac {2 \sqrt [4]{c x^2+b x+a}}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 108
Rule 205
Rule 208
Rule 212
Rule 220
Rule 401
Rule 409
Rule 444
Rule 537
Rule 623
Rule 734
Rule 747
Rule 748
Rule 749
Rule 843
Rule 1213
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x+c x^2}}{d+e x} \, dx &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\int \frac {b d-2 a e+(2 c d-b e) x}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{2 e}\\ &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {(2 c d-b e) \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{2 e^2}-\frac {(e (b d-2 a e)-d (2 c d-b e)) \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{2 e^2}\\ &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\left (2 (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 )}{e^2 (b+2 c x)}-\frac {\left ((e (b d-2 a e)-d (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 ) \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}{2 e^2 \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac {\left (\sqrt {2} (e (b d-2 a e)-d (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 (-\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 )}{e^2 \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}+\frac {\left (\sqrt {2} (e (b d-2 a e)-d (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 {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 )}{e \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (\sqrt {2} c (2 c d-b e) (e (b d-2 a e)-d (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 ) e^2 \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}+\frac {\left ((e (b d-2 a e)-d (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}{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 )}{\sqrt {2} e \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (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}} \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 )}{\sqrt {2} \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 ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac {\left (2 \sqrt {2} c^2 (e (b d-2 a e)-d (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}{-\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 \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (2 \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}} \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 ) e^2 \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 {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac {\left (\left (b^2-4 a c\right ) (e (b d-2 a e)-d (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}{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} \sqrt {c} e \sqrt {c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (\left (b^2-4 a c\right ) (e (b d-2 a e)-d (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}{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} \sqrt {c} e \sqrt {c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}-\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}} \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 ) e^2 \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 (\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}} \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 ) e^2 \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 {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \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} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \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} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\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}} \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 ) e^2 \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 (\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}} \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 ) e^2 \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 {2 \sqrt [4]{a+b x+c x^2}}{e}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \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} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \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} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt [4]{b^2-4 a c} (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 )}{\sqrt {2} \sqrt [4]{c} e^2 (b+2 c x)}-\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 e^2 (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 e^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 1.87, size = 613, normalized size = 0.70 \[ \frac {-\frac {\left (4 a c-b^2\right )^{3/4} \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} (b+2 c x) \sqrt [4]{e (a e-b d)+c d^2} \left (\tan ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )\right )+\sqrt [4]{4 a c-b^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2+e (a e-b d)}};\left .\sin ^{-1}\left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )+\sqrt [4]{4 a c-b^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2+e (a e-b d)}};\left .\sin ^{-1}\left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )\right )}{\sqrt {2} c (b+2 c x)}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} (b e-2 c d) F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{c}+2 e (a+x (b+c x))}{e^2 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}{e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{1/4}}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [4]{a + b x + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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