Optimal. Leaf size=1209 \[ \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 {\sqrt {-b^2+4 a c} (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} e^3 (b+2 c x) \sqrt [4]{a+b x+c x^2}}+\frac {\sqrt {-b^2+4 a c} (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} e^3 (b+2 c x) \sqrt [4]{a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.83, antiderivative size = 1209, normalized size of antiderivative = 1.00, 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 = {748,
857, 637, 311, 226, 1210, 763, 762, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214}
\begin {gather*} -\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}} \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 .\text {ArcSin}\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 .\text {ArcSin}\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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 211
Rule 214
Rule 226
Rule 304
Rule 311
Rule 408
Rule 455
Rule 504
Rule 551
Rule 637
Rule 748
Rule 760
Rule 762
Rule 763
Rule 857
Rule 1210
Rule 1227
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.33, size = 180, normalized size = 0.15 \begin {gather*} \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 (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}{e x +d}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{4}}}{d + e x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/4}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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