Optimal. Leaf size=188 \[ \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}} \]
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Rubi [A] time = 0.07, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {718, 424} \[ \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}} \]
Antiderivative was successfully verified.
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Rule 424
Rule 718
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c 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 c 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 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{c \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}\\ &=\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.77, size = 365, normalized size = 1.94 \[ \frac {i \left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right ) \sqrt {\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt {b^2-4 a c}+b\right )-2 c d}} \sqrt {1-\frac {2 c (d+e x)}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}} \left (E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d}} \sqrt {d+e x}\right )|\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )-F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d}} \sqrt {d+e x}\right )|\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )\right )}{\sqrt {2} c e \sqrt {a+x (b+c x)} \sqrt {\frac {c}{e \left (\sqrt {b^2-4 a c}+b\right )-2 c d}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 747, normalized size = 3.97 \[ \frac {\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}\, \sqrt {\frac {\left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\, \sqrt {\frac {\left (2 c x +b +\sqrt {-4 a c +b^{2}}\right ) e}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}\, \left (-b e \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}, \sqrt {-\frac {b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\right )+b e \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}, \sqrt {-\frac {b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\right )+2 c d \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}, \sqrt {-\frac {b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\right )-2 c d \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}, \sqrt {-\frac {b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\right )+\sqrt {-4 a c +b^{2}}\, e \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}, \sqrt {-\frac {b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\right )-\sqrt {-4 a c +b^{2}}\, e \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}}, \sqrt {-\frac {b e -2 c d +\sqrt {-4 a c +b^{2}}\, e}{-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e}}\right )\right )}{2 \left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) c^{2} e} \]
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 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {\sqrt {d + e x}}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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