Optimal. Leaf size=393 \[ \frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {843, 718, 424, 419} \[ \frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 843
Rubi steps
\begin {align*} \int \frac {d+e x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx &=\frac {e \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{g}+\frac {(-e f+d g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{g}\\ &=\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g 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} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\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 g \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (-e f+d g) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\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+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, 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 g \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g 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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 5.88, size = 814, normalized size = 2.07 \[ -\frac {(f+g x)^{3/2} \left (-\frac {4 e \sqrt {\frac {c f^2+g (a g-b f)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} (a+x (b+c x)) g^2}{(f+g x)^2}+\frac {i \sqrt {2} e \left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \sqrt {\frac {-2 a g^2+2 c f x g+b (f-g x) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {\frac {2 a g^2-2 c f x g+b (g x-f) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{\sqrt {f+g x}}-\frac {i \sqrt {2} \left (2 c d g+e \left (\sqrt {\left (b^2-4 a c\right ) g^2}-b g\right )\right ) \sqrt {\frac {-2 a g^2+2 c f x g+b (f-g x) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {\frac {2 a g^2-2 c f x g+b (g x-f) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{\sqrt {f+g x}}\right )}{2 c g^2 \sqrt {\frac {c f^2+g (a g-b f)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \sqrt {g x + f}}{c g x^{3} + {\left (c f + b g\right )} x^{2} + a f + {\left (b f + a g\right )} x}, x\right ) \]
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 \[ \int \frac {e x + d}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1014, normalized size = 2.58 \[ \frac {\left (-2 a e \,g^{2} \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )+2 a e \,g^{2} \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )-b d \,g^{2} \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )+2 b e f g \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )-b e f g \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )+2 c d f g \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )-2 c e \,f^{2} \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )-\sqrt {-4 a c +b^{2}}\, d \,g^{2} \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )+\sqrt {-4 a c +b^{2}}\, e f g \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}, \sqrt {-\frac {b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\right )\right ) \sqrt {\frac {\left (2 c x +b +\sqrt {-4 a c +b^{2}}\right ) g}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}\, \sqrt {\frac {\left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) g}{-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g}}\, \sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{b g -2 c f +\sqrt {-4 a c +b^{2}}\, g}}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{\left (c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +a f \right ) c \,g^{2}} \]
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 \[ \int \frac {e x + d}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {d+e\,x}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {d + e x}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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