Optimal. Leaf size=393 \[ \frac{\sqrt{2} e \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{c g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 g \sqrt{b^2-4 a c}}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}\right )}{c g \sqrt{f+g x} \sqrt{a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.209592, antiderivative size = 393, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 843
Rule 718
Rule 424
Rule 419
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*}
Mathematica [C] time = 5.69684, size = 814, normalized size = 2.07 \[ -\frac{(f+g x)^{3/2} \left (-\frac{4 e \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}} (a+x (b+c x)) g^2}{(f+g x)^2}+\frac{i \sqrt{2} e \left (2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) \sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt{\frac{2 a g^2-2 c f x g+b (g x-f) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{\sqrt{f+g x}}-\frac{i \sqrt{2} \left (2 c d g+e \left (\sqrt{\left (b^2-4 a c\right ) g^2}-b g\right )\right ) \sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt{\frac{2 a g^2-2 c f x g+b (g x-f) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\left (-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right ),-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{\sqrt{f+g x}}\right )}{2 c g^2 \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}} \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.317, size = 1014, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x + d}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\sqrt{f + g x} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]