Optimal. Leaf size=1220 \[ \frac {3 \sqrt {4 a c-b^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 ) (2 c d-b e)^2}{4 \sqrt {2} \sqrt {c} e^3 \sqrt {c d^2-b e d+a e^2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}-\frac {3 \sqrt {4 a c-b^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 ) (2 c d-b e)^2}{4 \sqrt {2} \sqrt {c} e^3 \sqrt {c d^2-b e d+a e^2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}-\frac {3 \sqrt [4]{4 a c-b^2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} e^{5/2} \sqrt [4]{c d^2-b e d+a e^2} \sqrt [4]{c x^2+b x+a}}+\frac {3 \sqrt [4]{4 a c-b^2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} e^{5/2} \sqrt [4]{c d^2-b e d+a e^2} \sqrt [4]{c x^2+b x+a}}-\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}-\frac {\left (c x^2+b x+a\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{c x^2+b x+a}}{\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 )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.18, antiderivative size = 1220, 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 = {732, 843, 623, 305, 220, 1196, 749, 748, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac {3 \sqrt {4 a c-b^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 ) (2 c d-b e)^2}{4 \sqrt {2} \sqrt {c} e^3 \sqrt {c d^2-b e d+a e^2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}-\frac {3 \sqrt {4 a c-b^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 ) (2 c d-b e)^2}{4 \sqrt {2} \sqrt {c} e^3 \sqrt {c d^2-b e d+a e^2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}-\frac {3 \sqrt [4]{4 a c-b^2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} e^{5/2} \sqrt [4]{c d^2-b e d+a e^2} \sqrt [4]{c x^2+b x+a}}+\frac {3 \sqrt [4]{4 a c-b^2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} e^{5/2} \sqrt [4]{c d^2-b e d+a e^2} \sqrt [4]{c x^2+b x+a}}-\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}-\frac {\left (c x^2+b x+a\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{c x^2+b x+a}}{\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 )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 205
Rule 208
Rule 220
Rule 298
Rule 305
Rule 399
Rule 444
Rule 490
Rule 537
Rule 623
Rule 732
Rule 746
Rule 748
Rule 749
Rule 843
Rule 1196
Rule 1213
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/4}}{(d+e x)^2} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \int \frac {b+2 c x}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{4 e}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {(3 c) \int \frac {1}{\sqrt [4]{a+b x+c x^2}} \, dx}{2 e^2}-\frac {(3 (2 c d-b e)) \int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{4 e^2}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {\left (6 c \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 (3 (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}{4 e^2 \sqrt [4]{a+b x+c x^2}}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {\left (3 \sqrt {c} \sqrt {b^2-4 a c} \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 (3 \sqrt {c} \sqrt {b^2-4 a c} \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 )}{e^2 (b+2 c x)}-\frac {\left (3 (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 )}{2 \sqrt {2} e^2 \sqrt [4]{a+b x+c x^2}}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\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 {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {\left (3 (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 )}{2 \sqrt {2} e \sqrt [4]{a+b x+c x^2}}+\frac {\left (3 c (2 c d-b e)^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 [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 )}{2 \sqrt {2} \left (b^2-4 a c\right ) e^2 \sqrt [4]{a+b x+c x^2}}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\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 {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {\left (3 (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 )}{4 \sqrt {2} e \sqrt [4]{a+b x+c x^2}}+\frac {\left (3 c (2 c d-b e)^2 \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 )}{\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 ) \sqrt [4]{a+b x+c x^2}}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\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 {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}-\frac {\left (3 c^2 (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 )}{\sqrt {2} \left (b^2-4 a c\right ) e \sqrt [4]{a+b x+c x^2}}+\frac {\left (3 c \sqrt {-b^2+4 a c} (2 c d-b e)^2 \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 )}{2 \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 (3 c \sqrt {-b^2+4 a c} (2 c d-b e)^2 \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 )}{2 \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 (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\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 {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}-\frac {\left (3 \left (-b^2+4 a c\right )^{3/2} (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 )}{2 \sqrt {2} \left (b^2-4 a c\right ) e^2 \sqrt [4]{a+b x+c x^2}}+\frac {\left (3 \left (-b^2+4 a c\right )^{3/2} (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 )}{2 \sqrt {2} \left (b^2-4 a c\right ) e^2 \sqrt [4]{a+b x+c x^2}}+\frac {\left (3 c \sqrt {-b^2+4 a c} (2 c d-b e)^2 \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 )}{2 \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 (3 c \sqrt {-b^2+4 a c} (2 c d-b e)^2 \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 )}{2 \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 (a+b x+c x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\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 {3 \sqrt [4]{-b^2+4 a c} (2 c d-b e) \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 )}{4 \sqrt [4]{c} e^{5/2} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}+\frac {3 \sqrt [4]{-b^2+4 a c} (2 c d-b e) \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 )}{4 \sqrt [4]{c} e^{5/2} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}-\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}+\frac {3 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} \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} e^2 (b+2 c x)}-\frac {3 \left (b^2-4 a c\right ) (2 c d-b 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 )}{4 \sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c} e^3 \sqrt {c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}+\frac {3 \left (b^2-4 a c\right ) (2 c d-b 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 )}{4 \sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c} e^3 \sqrt {c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 185, normalized size = 0.15 \[ \frac {4 \sqrt {2} (a+x (b+c x))^{3/4} F_1\left (-\frac {1}{2};-\frac {3}{4},-\frac {3}{4};\frac {1}{2};\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{e (d+e x) \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.61, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}{\left (e x +d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}}{{\left (e x + d\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/4}}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{4}}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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