Optimal. Leaf size=288 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.542987, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 93.1342, size = 270, normalized size = 0.94 \[ - \frac{2 e \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{f + g x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{c} g \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{2 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (d g - e f\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{c} g \sqrt{a + c x^{2}} \sqrt{f + g x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 2.66158, size = 439, normalized size = 1.52 \[ -\frac{2 \left (\sqrt{c} g (f+g x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-e g^2 \left (a+c x^2\right ) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}+i \sqrt{c} e (f+g x)^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{c g^2 \sqrt{a+c x^2} \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.048, size = 520, normalized size = 1.8 \[ 2\,{\frac{\sqrt{gx+f}\sqrt{c{x}^{2}+a}}{{g}^{2}c \left ( cg{x}^{3}+cf{x}^{2}+agx+fa \right ) } \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) ae{g}^{2}+{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) cdfg-{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) \sqrt{-ac}d{g}^{2}+{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) \sqrt{-ac}efg-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) ae{g}^{2}-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) ce{f}^{2} \right ) \sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e x + d}{\sqrt{c x^{2} + a} \sqrt{g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(sqrt(c*x^2 + a)*sqrt(g*x + f)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e x + d}{\sqrt{c x^{2} + a} \sqrt{g x + f}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(sqrt(c*x^2 + a)*sqrt(g*x + f)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\sqrt{a + c x^{2}} \sqrt{f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(sqrt(c*x^2 + a)*sqrt(g*x + f)),x, algorithm="giac")
[Out]