3.636 \(\int \frac{(d+e x)^3 \sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx\)
Optimal. Leaf size=531 \[ -\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{105 c^2 g^2}+\frac{2 e^2 \sqrt{a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{35 c g^2}+\frac{2 e \sqrt{a+c x^2} (d+e x)^2 \sqrt{f+g x}}{7 c} \]
[Out]
(-2*e*(25*a*e^2*g^2 + c*(7*e^2*f^2 + 12*d*e*f*g - 90*d^2*g^2))*Sqrt[f + g*x]*Sqr
t[a + c*x^2])/(105*c^2*g^2) + (2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7
*c) + (2*e^2*(e*f + 11*d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(35*c*g^2) + (2*Sqr
t[-a]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*
f*g^2 + 105*d^3*g^3))*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*c^(
3/2)*g^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (
2*Sqrt[-a]*e*(c*f^2 + a*g^2)*(25*a*e^2*g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2
*g^2))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*El
lipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[
c]*f - a*g)])/(105*c^(5/2)*g^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])
_______________________________________________________________________________________
Rubi [A] time = 2.19036, antiderivative size = 527, normalized size of antiderivative =
0.99, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214
\[ -\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x} \left (-e^2 \left (\frac{25 a}{c}+\frac{7 f^2}{g^2}\right )+90 d^2-\frac{12 d e f}{g}\right )}{105 c}+\frac{2 e^2 \sqrt{a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{35 c g^2}+\frac{2 e \sqrt{a+c x^2} (d+e x)^2 \sqrt{f+g x}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]
[Out]
(2*e*(90*d^2 - e^2*((25*a)/c + (7*f^2)/g^2) - (12*d*e*f)/g)*Sqrt[f + g*x]*Sqrt[a
+ c*x^2])/(105*c) + (2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*c) + (2*
e^2*(e*f + 11*d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(35*c*g^2) + (2*Sqrt[-a]*(a*
e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 1
05*d^3*g^3))*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*c^(3/2)*g^3*
Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a
]*e*(c*f^2 + a*g^2)*(25*a*e^2*g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2))*Sq
rt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[A
rcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*
g)])/(105*c^(5/2)*g^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])
_______________________________________________________________________________________
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)**3*(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [C] time = 10.9903, size = 747, normalized size = 1.41 \[ \frac{2 \sqrt{f+g x} \left (\frac{g \sqrt{f+g x} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (25 a^{3/2} e^3 g^2+\sqrt{a} c e \left (-105 d^2 g^2+42 d e f g-8 e^2 f^2\right )+3 i a \sqrt{c} e^2 g (2 e f-63 d g)+105 i c^{3/2} d^3 g^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+\frac{g^2 \left (-a^2 e^2 g^2 (189 d g+19 e f)+a c \left (105 d^3 g^3+105 d^2 e f g^2-21 d e^2 g \left (2 f^2+9 g^2 x^2\right )+e^3 \left (8 f^3-19 f g^2 x^2\right )\right )+c^2 x^2 \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )}{f+g x}+g^2 \left (-\left (a+c x^2\right )\right ) \left (25 a e^3 g^2+c e \left (-105 d^2 g^2-21 d e g (f+3 g x)+e^2 \left (4 f^2-3 f g x-15 g^2 x^2\right )\right )\right )+i c \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )-a e^2 g^2 (189 d g+19 e f)\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{105 c^2 g^4 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]
[Out]
(2*Sqrt[f + g*x]*(-(g^2*(a + c*x^2)*(25*a*e^3*g^2 + c*e*(-105*d^2*g^2 - 21*d*e*g
*(f + 3*g*x) + e^2*(4*f^2 - 3*f*g*x - 15*g^2*x^2)))) + (g^2*(-(a^2*e^2*g^2*(19*e
*f + 189*d*g)) + c^2*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3
)*x^2 + a*c*(105*d^2*e*f*g^2 + 105*d^3*g^3 - 21*d*e^2*g*(2*f^2 + 9*g^2*x^2) + e^
3*(8*f^3 - 19*f*g^2*x^2))))/(f + g*x) + I*c*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-(
a*e^2*g^2*(19*e*f + 189*d*g)) + c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2
+ 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]
*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sq
rt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[
a]*g)] + (g*(Sqrt[c]*f + I*Sqrt[a]*g)*((105*I)*c^(3/2)*d^3*g^2 + 25*a^(3/2)*e^3*
g^2 + (3*I)*a*Sqrt[c]*e^2*g*(2*e*f - 63*d*g) + Sqrt[a]*c*e*(-8*e^2*f^2 + 42*d*e*
f*g - 105*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqr
t[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*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*S
qrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(105*c^2*g^4*Sqrt[a + c*x^2])
_______________________________________________________________________________________
Maple [B] time = 0.059, size = 3922, normalized size = 7.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(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)*(-105*x^2*c^3*d^2*e*f*g^4-21*x^2*c^3*d*e^2*
f^2*g^3-105*x*a*c^2*d^2*e*g^5+x*a*c^2*e^3*f^2*g^3-84*x^3*c^3*d*e^2*f*g^4-63*x^2*
a*c^2*d*e^2*g^5+7*x^2*a*c^2*e^3*f*g^4+25*a^2*c*e^3*f*g^4+4*a*c^2*e^3*f^3*g^2-15*
x^5*c^3*e^3*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)*E
llipticE((-(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^3*f^5-84*x*a*c^2*d*e^2*f*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*c)^(1/2)*a*c*d
*e^2*f*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)*Ellip
ticF((-(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^2*f^3*g^2-105*(-(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^3*g^5+25*(-(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^3*g^5-105*(-(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^3*d^3*f^2*g^3+105*(-(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^3*g^5+105*(-(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^3*f
^2*g^3+189*(-(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))*a^2*c*d*e^2*g^5-6*(-(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*e^3*f*g^4-6*(-(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^2*e^3*f^3*g^2-8*(-(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)*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^3*f^4*g-189*(-(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^2*g^5-19*(-(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^3*f*g^4-11*(-(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^3*f^3*g^2+105*(-(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*e*f^3*g^2-42*(-(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*e^2*f^4*g-18*x^4*c^3*e^3*f*g
^4+10*x^3*a*c^2*e^3*g^5-105*x^3*c^3*d^2*e*g^5+x^3*c^3*e^3*f^2*g^3+4*x^2*c^3*e^3*
f^3*g^2+25*x*a^2*c*e^3*g^5+105*(-(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*e*f*g^4-231*(-(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^2*f^2*g^3+1
89*(-(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^2*f^2*g^3-105*(-(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*e*g^5+17*(-(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^3*f^2*g^3-105*(-(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*e*f^2*g^3-63*x^4*c^3*d*e^2*g^5-21*a*c
^2*d*e^2*f^2*g^3-105*a*c^2*d^2*e*f*g^4)/(c*g*x^3+c*f*x^2+a*g*x+a*f)/c^3/g^4
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3} \sqrt{g x + f}}{\sqrt{c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + a),x, algorithm="maxima")
[Out]
integrate((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + a), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{g x + f}}{\sqrt{c x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + a),x, algorithm="fricas")
[Out]
integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(g*x + f)/sqrt(c*x^2 + a)
, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3} \sqrt{f + g x}}{\sqrt{a + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
[Out]
Integral((d + e*x)**3*sqrt(f + g*x)/sqrt(a + c*x**2), 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((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + a),x, algorithm="giac")
[Out]
Exception raised: RuntimeError