Optimal. Leaf size=955 \[ \frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b 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 )}{3 a e \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 {2 \sqrt {2} \sqrt {b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {4 \sqrt {2} \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 )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\sqrt {2} c \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 {a} \left (c+b x+a x^2\right )} \]
[Out]
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Rubi [A]
time = 2.13, antiderivative size = 955, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1463, 932,
6874, 732, 430, 948, 175, 552, 551, 857, 435} \begin {gather*} \frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b e) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\text {ArcSin}\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 ) x}{3 a e \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 {2 \sqrt {2} \sqrt {b^2-4 a c} d (a d+b e) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \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 (\text {ArcSin}\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 ) x}{3 a e \sqrt {d+e x} \left (a x^2+b x+c\right )}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \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 (\text {ArcSin}\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 ) x}{3 a \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {\sqrt {2} c \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \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};\text {ArcSin}\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 ) x}{\sqrt {a} \left (a x^2+b x+c\right )}+\frac {2}{3} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 175
Rule 430
Rule 435
Rule 551
Rule 552
Rule 732
Rule 857
Rule 932
Rule 948
Rule 1463
Rule 6874
Rubi steps
\begin {align*} \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, 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} \, dx}{\sqrt {c+b x+a x^2}}\\ &=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}-\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {-3 c d-2 (b d+c e) x-(a d+b e) x^2}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{3 \sqrt {c+b x+a x^2}}\\ &=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}-\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \left (-\frac {2 (b d+c e)}{\sqrt {d+e x} \sqrt {c+b x+a x^2}}-\frac {3 c d}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}}-\frac {(a d+b e) x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}}\right ) \, dx}{3 \sqrt {c+b x+a x^2}}\\ &=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\left (c d \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}{\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 {x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{3 \sqrt {c+b x+a x^2}}+\frac {\left (2 (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}{3 \sqrt {c+b x+a x^2}}\\ &=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\left (c d \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}{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 {\sqrt {d+e x}}{\sqrt {c+b x+a x^2}} \, dx}{3 e \sqrt {c+b x+a x^2}}+\frac {\left (d (-a d-b 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}{3 e \sqrt {c+b x+a x^2}}+\frac {\left (4 \sqrt {2} \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 ) \text {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 )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {4 \sqrt {2} \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 )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left (2 c d \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 ) \text {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 (\sqrt {2} \sqrt {b^2-4 a c} (-a d-b 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 ) \text {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 )}{3 a e \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 (2 \sqrt {2} \sqrt {b^2-4 a c} d (-a d-b 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 ) \text {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 )}{3 a e \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b 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 )}{3 a e \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 {2 \sqrt {2} \sqrt {b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {4 \sqrt {2} \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 )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left (2 c d \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 ) \text {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 {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b 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 )}{3 a e \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 {2 \sqrt {2} \sqrt {b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {4 \sqrt {2} \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 )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left (2 c d \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 ) \text {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 {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b 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 )}{3 a e \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 {2 \sqrt {2} \sqrt {b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {4 \sqrt {2} \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 )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\sqrt {2} c \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 {a} \left (c+b x+a x^2\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 27.55, size = 1258, normalized size = 1.32 \begin {gather*} \frac {2}{3} x \sqrt {d+e x} \sqrt {a+\frac {c+b x}{x^2}}+\frac {x (d+e x)^{3/2} \sqrt {a+\frac {c+b x}{x^2}} \left (\frac {4 e^2 (a d+b e) \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}{(d+e x)^2}-\frac {i \sqrt {2} (a d+b e) \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {-2 c e^2+2 a d e x+b e (d-e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2-2 a d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+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 )}{\sqrt {d+e x}}+\frac {i \sqrt {2} \left (b e \left (-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+a \left (3 b d e-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \sqrt {\frac {-2 c e^2+2 a d e x+b e (d-e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2-2 a d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+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 )}{\sqrt {d+e x}}+\frac {6 i \sqrt {2} a c e^2 \sqrt {\frac {-2 c e^2+2 a d e x+b e (d-e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2-2 a d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \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 (-b d+c e)\right )};i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+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 )}{\sqrt {d+e x}}\right )}{6 a e^2 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3022\) vs.
\(2(836)=1672\).
time = 0.23, size = 3023, normalized size = 3.17
method | result | size |
default | \(\text {Expression too large to display}\) | \(3023\) |
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 \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, 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 \sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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