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

Optimal. Leaf size=635 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (3 a e g^2 (e f-10 d g)+c f \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{315 c g^3}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{315 c e g^3}+\frac{2 e \sqrt{a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{63 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 e} \]

[Out]

(-2*(6*a*e^2*g^2*(e*f - 10*d*g) - c*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*g^
2 - 35*d^3*g^3))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(315*c*e*g^3) + (2*(d + e*x)^3*S
qrt[f + g*x]*Sqrt[a + c*x^2])/(9*e) + (4*(7*a*e^2*g^2 - c*(8*e^2*f^2 - 24*d*e*f*
g + 21*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^3) + (2*e*(e*f - 3*d*
g)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^3) + (4*Sqrt[-a]*(21*a^2*e^2*g^4 + 3*a
*c*g^2*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g +
21*d^2*g^2))*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)])/(315*c^(3/2)*g^4
*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-
a]*(c*f^2 + a*g^2)*(3*a*e*g^2*(e*f - 10*d*g) + c*f*(8*e^2*f^2 - 24*d*e*f*g + 21*
d^2*g^2))*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]*Sq
rt[c]*f - a*g)])/(315*c^(3/2)*g^4*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 2.94972, antiderivative size = 635, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (3 a e g^2 (e f-10 d g)+c f \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{315 c g^3}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{315 c e g^3}+\frac{2 e \sqrt{a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{63 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 e} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(6*a*e^2*g^2*(e*f - 10*d*g) - c*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*g^
2 - 35*d^3*g^3))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(315*c*e*g^3) + (2*(d + e*x)^3*S
qrt[f + g*x]*Sqrt[a + c*x^2])/(9*e) + (4*(7*a*e^2*g^2 - c*(8*e^2*f^2 - 24*d*e*f*
g + 21*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^3) + (2*e*(e*f - 3*d*
g)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^3) + (4*Sqrt[-a]*(21*a^2*e^2*g^4 + 3*a
*c*g^2*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g +
21*d^2*g^2))*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)])/(315*c^(3/2)*g^4
*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-
a]*(c*f^2 + a*g^2)*(3*a*e*g^2*(e*f - 10*d*g) + c*f*(8*e^2*f^2 - 24*d*e*f*g + 21*
d^2*g^2))*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]*Sq
rt[c]*f - a*g)])/(315*c^(3/2)*g^4*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(g*x+f)**(1/2)*(c*x**2+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 12.7148, size = 910, normalized size = 1.43 \[ \frac{2 \left (-2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (21 a^2 e^2 g^4-3 a c \left (-3 e^2 f^2+16 d e g f+21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) \left (\frac{a g^2}{(f+g x)^2}+c \left (\frac{f}{f+g x}-1\right )^2\right )+\frac{2 i \sqrt{c} \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (21 a^2 e^2 g^4-3 a c \left (-3 e^2 f^2+16 d e g f+21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{f+g x}}+\frac{2 \sqrt{a} \sqrt{c} g \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (-21 i a^{3/2} e^2 g^3+3 a \sqrt{c} e (e f-10 d g) g^2+3 i \sqrt{a} c \left (-2 e^2 f^2+6 d e g f+21 d^2 g^2\right ) g+c^{3/2} f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{f+g x}}\right ) (f+g x)^{3/2}}{315 c^2 g^5 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{c (f+g x)^2 \left (\frac{f}{f+g x}-1\right )^2}{g^2}+a}}+\sqrt{c x^2+a} \left (\frac{2 e^2 x^3}{9}+\frac{2 e (e f+18 d g) x^2}{63 g}+\frac{2 \left (-6 c e^2 f^2+18 c d e g f+63 c d^2 g^2+14 a e^2 g^2\right ) x}{315 c g^2}+\frac{2 \left (8 c e^2 f^3-24 c d e g f^2+21 c d^2 g^2 f+8 a e^2 g^2 f+60 a d e g^3\right )}{315 c g^3}\right ) \sqrt{f+g x} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[f + g*x]*Sqrt[a + c*x^2]*((2*(8*c*e^2*f^3 - 24*c*d*e*f^2*g + 21*c*d^2*f*g^2
+ 8*a*e^2*f*g^2 + 60*a*d*e*g^3))/(315*c*g^3) + (2*(-6*c*e^2*f^2 + 18*c*d*e*f*g
+ 63*c*d^2*g^2 + 14*a*e^2*g^2)*x)/(315*c*g^2) + (2*e*(e*f + 18*d*g)*x^2)/(63*g)
+ (2*e^2*x^3)/9) + (2*(f + g*x)^(3/2)*(-2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(21*a
^2*e^2*g^4 + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f
^2 + 16*d*e*f*g + 21*d^2*g^2))*((a*g^2)/(f + g*x)^2 + c*(-1 + f/(f + g*x))^2) +
((2*I)*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(21*a^2*e^2*g^4 + c^2*f^2*(8*e^2*f^2 -
24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f^2 + 16*d*e*f*g + 21*d^2*g^2))*Sqr
t[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I
*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqr
t[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt
[f + g*x] + (2*Sqrt[a]*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((-21*I)*a^(3/2)*e^2*
g^3 + 3*a*Sqrt[c]*e*g^2*(e*f - 10*d*g) + c^(3/2)*f*(8*e^2*f^2 - 24*d*e*f*g + 21*
d^2*g^2) + (3*I)*Sqrt[a]*c*g*(-2*e^2*f^2 + 6*d*e*f*g + 21*d^2*g^2))*Sqrt[1 - f/(
f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*
g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqr
t[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]
))/(315*c^2*g^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + (c*(f + g*x)^2*(-1 + f
/(f + g*x))^2)/g^2])

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Maple [B]  time = 0.041, size = 4351, normalized size = 6.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x)

[Out]

-2/315*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(6*x^3*c^3*d*e*f^2*g^4-7*x^2*a*c^2*e^2*f^2*
g^4+24*x^2*c^3*d*e*f^3*g^3-60*x*a^2*c*d*e*g^6-22*x*a^2*c*e^2*f*g^5-84*x*a*c^2*d^
2*f*g^5-2*x*a*c^2*e^2*f^3*g^3-108*x^4*c^3*d*e*f*g^5-150*x^3*a*c^2*d*e*g^6-62*x^3
*a*c^2*e^2*f*g^5-60*a^2*c*d*e*f*g^5+24*a*c^2*d*e*f^3*g^3-35*x^6*c^3*e^2*g^6-63*x
^4*c^3*d^2*g^6-108*(-(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)^(1/2)*a*c*d*e*f^2*g^4-2*x^3*c^3*e^2*f^3*g^3-60*(-(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))*a
^2*c*e^2*f^2*g^4+84*(-(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))*a*c^2*d^2*f^2*g^4-34*(-(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))*a*c^2*e^2*f^4*g^2+48*(-(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*e*f
^5*g-60*(-(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)^(1/2)*a^2*d*e*g^6+6*(-(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)^(1/2)*a^2*e^2*f*g^5+42*(-(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)
^(1/2)*c^2*d^2*f^3*g^3+16*(-(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)^(1/2)*c^2*e^2*f^5*g+54*(-(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^2*c*e^2*f^2*g^4-
126*(-(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^2*d^2*f^2*g^4+12*(-(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^2*e^2*f^4*g^2-14*x^2*a^2*c*e^2*g^6-63*x^2*a*c
^2*d^2*g^6-21*x^2*c^3*d^2*f^2*g^4-8*x^2*c^3*e^2*f^4*g^2-90*x^5*c^3*d*e*g^6-40*x^
5*c^3*e^2*f*g^5-49*x^4*a*c^2*e^2*g^6+x^4*c^3*e^2*f^2*g^4-84*x^3*c^3*d^2*f*g^5+6*
x*a*c^2*d*e*f^2*g^4+42*(-(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^3*e^2*g^6-42*(-(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))*a^3*e^2*g^6-16*(-(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^2*f^6-168*x^
2*a*c^2*d*e*f*g^5+126*(-(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))*a^2*c*d^2*g^6-42*(-(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^2*f^4*g^2+96*(-(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))*a^2*c*d*e*f*g
^5+144*(-(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))*a*c^2*d*e*f^3*g^3+42*(-(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)^(1/2)*a*c*d^2*f*g^5+22*(-(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)^(1/2)*
a*c*e^2*f^3*g^3-48*(-(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)^(1/2)*c^2*d*e*f^4*g^2-36*(-(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^2*c*d*e*f*g^5-36*(-(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^2*d*e*f^3*g^3-8*a^2*c*e^2*f^2*g^4-21*a*c^2*d^2*f^2*g^4-8*a*c^2*e^2*f^4*g^2-12
6*(-(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^2*c*d^2*g^6)/c^2/(c*g*x^3+c*f*x^2+a*g*x+a*f)/g^5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(g*x+f)**(1/2)*(c*x**2+a)**(1/2),x)

[Out]

Integral(sqrt(a + c*x**2)*(d + e*x)**2*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)^2*sqrt(g*x + f),x, algorithm="giac")

[Out]

Exception raised: RuntimeError