3.624 \(\int (d+e x) \sqrt{f+g x} \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=434 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (5 a e g^2+c f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 \sqrt{c} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{105 c g^2}+\frac{2 e \left (a+c x^2\right )^{3/2} \sqrt{f+g x}}{7 c} \]

[Out]

(-2*Sqrt[f + g*x]*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g) - 3*c*g*(e*f + 7*d*g)*x)*Sqrt
[a + c*x^2])/(105*c*g^2) + (2*e*Sqrt[f + g*x]*(a + c*x^2)^(3/2))/(7*c) - (4*Sqrt
[-a]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*Sqrt[f + g*x]*Sqrt[1 + (c*
x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt
[-a]*Sqrt[c]*f - a*g)])/(105*Sqrt[c]*g^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + S
qrt[-a]*g)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*(c*f^2 + a*g^2)*(5*a*e*g^2 + c*f*(4*e
*f - 7*d*g))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)
/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]
*Sqrt[c]*f - a*g)])/(105*c^(3/2)*g^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.2024, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (5 a e g^2+c f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 \sqrt{c} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{105 c g^2}+\frac{2 e \left (a+c x^2\right )^{3/2} \sqrt{f+g x}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]

[Out]

(-2*Sqrt[f + g*x]*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g) - 3*c*g*(e*f + 7*d*g)*x)*Sqrt
[a + c*x^2])/(105*c*g^2) + (2*e*Sqrt[f + g*x]*(a + c*x^2)^(3/2))/(7*c) - (4*Sqrt
[-a]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*Sqrt[f + g*x]*Sqrt[1 + (c*
x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt
[-a]*Sqrt[c]*f - a*g)])/(105*Sqrt[c]*g^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + S
qrt[-a]*g)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*(c*f^2 + a*g^2)*(5*a*e*g^2 + c*f*(4*e
*f - 7*d*g))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)
/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]
*Sqrt[c]*f - a*g)])/(105*c^(3/2)*g^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 174.839, size = 437, normalized size = 1.01 \[ \frac{2 e \left (a + c x^{2}\right )^{\frac{3}{2}} \sqrt{f + g x}}{7 c} - \frac{8 \sqrt{a + c x^{2}} \sqrt{f + g x} \left (\frac{5 a e g^{2}}{4} - \frac{7 c d f g}{4} + c e f^{2} - \frac{3 c g x \left (7 d g + e f\right )}{4}\right )}{105 c g^{2}} - \frac{4 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{f + g x} \left (21 a d g^{3} + 8 a e f g^{2} - 7 c d f^{2} g + 4 c e f^{3}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{105 \sqrt{c} g^{3} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{4 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a g^{2} + c f^{2}\right ) \left (5 a e g^{2} - 7 c d f g + 4 c e f^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{105 c^{\frac{3}{2}} g^{3} \sqrt{a + c x^{2}} \sqrt{f + g x}} \]

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]

2*e*(a + c*x**2)**(3/2)*sqrt(f + g*x)/(7*c) - 8*sqrt(a + c*x**2)*sqrt(f + g*x)*(
5*a*e*g**2/4 - 7*c*d*f*g/4 + c*e*f**2 - 3*c*g*x*(7*d*g + e*f)/4)/(105*c*g**2) -
4*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(f + g*x)*(21*a*d*g**3 + 8*a*e*f*g**2 - 7*c*d*
f**2*g + 4*c*e*f**3)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*g
/(a*g - sqrt(c)*f*sqrt(-a)))/(105*sqrt(c)*g**3*sqrt(sqrt(c)*sqrt(-a)*(-f - g*x)/
(a*g - sqrt(c)*f*sqrt(-a)))*sqrt(a + c*x**2)) + 4*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)
*(-f - g*x)/(a*g - sqrt(c)*f*sqrt(-a)))*sqrt(1 + c*x**2/a)*(a*g**2 + c*f**2)*(5*
a*e*g**2 - 7*c*d*f*g + 4*c*e*f**2)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a))
+ 1/2)), 2*a*g/(a*g - sqrt(c)*f*sqrt(-a)))/(105*c**(3/2)*g**3*sqrt(a + c*x**2)*s
qrt(f + g*x))

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Mathematica [C]  time = 7.98219, size = 610, normalized size = 1.41 \[ \frac{\sqrt{f+g x} \left (\frac{2 \left (a+c x^2\right ) \left (10 a e g^2+7 c d g (f+3 g x)+c e \left (-4 f^2+3 f g x+15 g^2 x^2\right )\right )}{c g^2}+\frac{4 \left (g^2 \left (a+c x^2\right ) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right )+i \sqrt{c} (f+g x)^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (c f^2 (7 d g-4 e f)-a g^2 (21 d g+8 e f)\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+\sqrt{a} g (f+g x)^{3/2} \left (-\sqrt{a} g+i \sqrt{c} f\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (3 \sqrt{a} \sqrt{c} g (7 d g+e f)+5 i a e g^2+i c f (4 e f-7 d g)\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{c g^4 (f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right )}{105 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]

[Out]

(Sqrt[f + g*x]*((2*(a + c*x^2)*(10*a*e*g^2 + 7*c*d*g*(f + 3*g*x) + c*e*(-4*f^2 +
3*f*g*x + 15*g^2*x^2)))/(c*g^2) + (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(c*f
^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*(a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*f
+ I*Sqrt[a]*g)*(c*f^2*(-4*e*f + 7*d*g) - a*g^2*(8*e*f + 21*d*g))*Sqrt[(g*((I*Sqr
t[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*
(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*
x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]*g*(I*Sqrt[c]
*f - Sqrt[a]*g)*((5*I)*a*e*g^2 + I*c*f*(4*e*f - 7*d*g) + 3*Sqrt[a]*Sqrt[c]*g*(e*
f + 7*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/
Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt
[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]
*g)]))/(c*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*Sqrt[a + c*x^2]
)

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Maple [B]  time = 0.03, size = 2549, normalized size = 5.9 \[ \text{result too large to display} \]

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]

-2/105*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(-28*x*a*c^2*d*f*g^4+x*a*c^2*e*f^2*g^3-28*x
^2*a*c^2*e*f*g^4-18*(-a*c)^(1/2)*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+
(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2
)-c*f))^(1/2)*EllipticF((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2
)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*a*c*e*f^2*g^3+14*(-a*c)^(1/2)*(-(g*x+f)*c/(g
*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c
*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticF((-(g*x+f)*c/(g*(-a*c)^(
1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*a*c*d*f*g^4
-15*x^5*c^3*e*g^5-21*x^4*c^3*d*g^5-6*a*c^2*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/
2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(
-a*c)^(1/2)-c*f))^(1/2)*EllipticF((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(
-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*e*f^3*g^2+16*a^2*c*(-(g*x+f)*c/(g*
(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*
x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticE((-(g*x+f)*c/(g*(-a*c)^(1
/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*e*f*g^4+28*a
*c^2*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1
/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticE((-(g
*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))
^(1/2))*d*f^2*g^3+24*a*c^2*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)
^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f)
)^(1/2)*EllipticE((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)
/(g*(-a*c)^(1/2)+c*f))^(1/2))*e*f^3*g^2+14*(-a*c)^(1/2)*(-(g*x+f)*c/(g*(-a*c)^(1
/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^
(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticF((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))
^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*c^2*d*f^3*g^2-8*(-a*c
)^(1/2)*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)
^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticF((
-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*
f))^(1/2))*c^2*e*f^4*g-6*a^2*c*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-
a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-
c*f))^(1/2)*EllipticF((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-
c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*e*f*g^4-42*a*c^2*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c
*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2)
)*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticF((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2
),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*d*f^2*g^3+x^3*c^3*e*f^2*g^
3-42*a^2*c*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a
*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*Elliptic
F((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)
+c*f))^(1/2))*d*g^5+8*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2
))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/
2)*EllipticE((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(
-a*c)^(1/2)+c*f))^(1/2))*c^3*e*f^5-21*x^2*a*c^2*d*g^5-7*x^2*c^3*d*f^2*g^3+4*x^2*
c^3*e*f^3*g^2-10*x*a^2*c*e*g^5-18*x^4*c^3*e*f*g^4-25*x^3*a*c^2*e*g^5-28*x^3*c^3*
d*f*g^4-10*a^2*c*e*f*g^4-7*a*c^2*d*f^2*g^3+4*a*c^2*e*f^3*g^2+42*a^2*c*(-(g*x+f)*
c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)
*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*EllipticE((-(g*x+f)*c/(g*(-a*
c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*d*g^5-1
0*(-a*c)^(1/2)*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(g
*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-c*f))^(1/2)*Elli
pticF((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(
1/2)+c*f))^(1/2))*a^2*e*g^5-14*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*((-c*x+(-
a*c)^(1/2))*g/(g*(-a*c)^(1/2)+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/(g*(-a*c)^(1/2)-
c*f))^(1/2)*EllipticE((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(-(g*(-a*c)^(1/2)-
c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*c^3*d*f^4*g)/(c*g*x^3+c*f*x^2+a*g*x+a*f)/g^4/c
^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + a}{\left (e x + d\right )} \sqrt{g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{2} + a}{\left (e x + d\right )} \sqrt{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + c x^{2}} \left (d + e x\right ) \sqrt{f + g x}\, dx \]

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]

Integral(sqrt(a + c*x**2)*(d + e*x)*sqrt(f + g*x), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f),x, algorithm="giac")

[Out]

Exception raised: RuntimeError