Optimal. Leaf size=1287 \[ \frac {\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right ) (b d+c e)^2}{4 \sqrt {2} \sqrt {a} c d^2 \left (a x^2+b x+c\right )}+\frac {\sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) (b d+c e)}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac {\sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) (b d+c e)}{2 \sqrt {2} c \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} (b d+c e)}{4 c d}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {(a d+b e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (a x^2+b x+c\right )}-\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x}}{2 x} \]
[Out]
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Rubi [A] time = 5.30, antiderivative size = 1287, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1573, 916, 6742, 718, 419, 939, 934, 169, 538, 537, 843, 424} \[ \frac {\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right ) (b d+c e)^2}{4 \sqrt {2} \sqrt {a} c d^2 \left (a x^2+b x+c\right )}+\frac {\sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) (b d+c e)}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac {\sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) (b d+c e)}{2 \sqrt {2} c \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} (b d+c e)}{4 c d}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {(a d+b e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (a x^2+b x+c\right )}-\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x}}{2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 169
Rule 419
Rule 424
Rule 537
Rule 538
Rule 718
Rule 843
Rule 916
Rule 934
Rule 939
Rule 1573
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx &=\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\sqrt {d+e x} \sqrt {c+b x+a x^2}}{x^3} \, dx}{\sqrt {c+b x+a x^2}}\\ &=-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {b d+c e+2 (a d+b e) x+3 a e x^2}{x^2 \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{4 \sqrt {c+b x+a x^2}}\\ &=-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \left (\frac {3 a e}{\sqrt {d+e x} \sqrt {c+b x+a x^2}}+\frac {b d+c e}{x^2 \sqrt {d+e x} \sqrt {c+b x+a x^2}}+\frac {2 (a d+b e)}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}}\right ) \, dx}{4 \sqrt {c+b x+a x^2}}\\ &=-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\left (3 a e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{4 \sqrt {c+b x+a x^2}}+\frac {\left ((a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{2 \sqrt {c+b x+a x^2}}+\frac {\left ((b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{x^2 \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{4 \sqrt {c+b x+a x^2}}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\left ((a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {b-\sqrt {b^2-4 a c}+2 a x} \sqrt {b+\sqrt {b^2-4 a c}+2 a x}\right ) \int \frac {1}{x \sqrt {b-\sqrt {b^2-4 a c}+2 a x} \sqrt {b+\sqrt {b^2-4 a c}+2 a x} \sqrt {d+e x}} \, dx}{2 \left (c+b x+a x^2\right )}-\frac {\left ((b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {b d+c e-a e x^2}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{8 c d \sqrt {c+b x+a x^2}}+\frac {\left (3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 a d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left ((a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {b-\sqrt {b^2-4 a c}+2 a x} \sqrt {b+\sqrt {b^2-4 a c}+2 a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}+\frac {2 a x^2}{e}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}+\frac {2 a x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{c+b x+a x^2}-\frac {\left ((b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \left (\frac {b d+c e}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}}-\frac {a e x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}}\right ) \, dx}{8 c d \sqrt {c+b x+a x^2}}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {\left (a e (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{8 c d \sqrt {c+b x+a x^2}}-\frac {\left ((b d+c e)^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{8 c d \sqrt {c+b x+a x^2}}-\frac {\left ((a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {b+\sqrt {b^2-4 a c}+2 a x} \sqrt {1+\frac {2 a (d+e x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}+\frac {2 a x^2}{e}} \sqrt {1+\frac {2 a x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}} \, dx,x,\sqrt {d+e x}\right )}{c+b x+a x^2}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left ((b d+c e)^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {b-\sqrt {b^2-4 a c}+2 a x} \sqrt {b+\sqrt {b^2-4 a c}+2 a x}\right ) \int \frac {1}{x \sqrt {b-\sqrt {b^2-4 a c}+2 a x} \sqrt {b+\sqrt {b^2-4 a c}+2 a x} \sqrt {d+e x}} \, dx}{8 c d \left (c+b x+a x^2\right )}-\frac {\left (a (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{8 c \sqrt {c+b x+a x^2}}+\frac {\left (a (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c+b x+a x^2}} \, dx}{8 c d \sqrt {c+b x+a x^2}}-\frac {\left ((a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1+\frac {2 a (d+e x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}} \sqrt {1+\frac {2 a (d+e x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right ) \sqrt {1+\frac {2 a x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}} \sqrt {1+\frac {2 a x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}} \, dx,x,\sqrt {d+e x}\right )}{c+b x+a x^2}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {(a d+b e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (c+b x+a x^2\right )}+\frac {\left ((b d+c e)^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {b-\sqrt {b^2-4 a c}+2 a x} \sqrt {b+\sqrt {b^2-4 a c}+2 a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}+\frac {2 a x^2}{e}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}+\frac {2 a x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{4 c d \left (c+b x+a x^2\right )}+\frac {\left (\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 a d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \left (c+b x+a x^2\right )}-\frac {\left (\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 a d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{2 \sqrt {2} c \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {2} c \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {(a d+b e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (c+b x+a x^2\right )}+\frac {\left ((b d+c e)^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {b+\sqrt {b^2-4 a c}+2 a x} \sqrt {1+\frac {2 a (d+e x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}+\frac {2 a x^2}{e}} \sqrt {1+\frac {2 a x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}} \, dx,x,\sqrt {d+e x}\right )}{4 c d \left (c+b x+a x^2\right )}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {2} c \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {(a d+b e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (c+b x+a x^2\right )}+\frac {\left ((b d+c e)^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1+\frac {2 a (d+e x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}} \sqrt {1+\frac {2 a (d+e x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right ) \sqrt {1+\frac {2 a x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}} \sqrt {1+\frac {2 a x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}\right ) e}}} \, dx,x,\sqrt {d+e x}\right )}{4 c d \left (c+b x+a x^2\right )}\\ &=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {3 \sqrt {b^2-4 a c} e \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {2} c \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {(a d+b e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (c+b x+a x^2\right )}+\frac {(b d+c e)^2 \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{4 \sqrt {2} \sqrt {a} c d^2 \left (c+b x+a x^2\right )}\\ \end {align*}
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Mathematica [C] time = 12.53, size = 811, normalized size = 0.63 \[ \frac {x \sqrt {a+\frac {c+b x}{x^2}} \left (-\frac {8 c d^3}{x^2}-\frac {8 c e d^2}{x}-\frac {4 (b d+c e) d^2}{x}-\frac {i (d+e x)^{3/2} \sqrt {1-\frac {2 \left (a d^2+e (c e-b d)\right )}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {4 \left (a d^2+e (c e-b d)\right )}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+2} \left (d (b d+c e) \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b e d+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\left (b^2 e d^2+b \left (d \sqrt {\left (b^2-4 a c\right ) e^2}-5 c e^2\right ) d+c e \left (4 a d^2+\sqrt {\left (b^2-4 a c\right ) e^2} d+2 c e^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b e d+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+2 e \left (b^2 d^2-2 b c e d+c \left (c e^2-4 a d^2\right )\right ) \Pi \left (\frac {d \left (2 a d-b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )}{2 \left (a d^2+e (c e-b d)\right )};i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b e d+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{e \sqrt {\frac {a d^2+e (c e-b d)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}\right )}{16 c d^2 \sqrt {d+e x}} \]
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 \[ \int \frac {\sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 4957, normalized size = 3.85 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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