3.645 \(\int \frac{(d+e x)^3}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=457 \[ \frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g^2 (7 e f-15 d g)-c \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{8 e^2 \sqrt{a+c x^2} \sqrt{f+g x} (e f-3 d g)}{15 c g^2}+\frac{2 e^2 \sqrt{a+c x^2} (d+e x) \sqrt{f+g x}}{5 c g} \]

[Out]

(-8*e^2*(e*f - 3*d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(15*c*g^2) + (2*e^2*(d + e*
x)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(5*c*g) + (2*Sqrt[-a]*e*(9*a*e^2*g^2 - c*(8*e^
2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Ar
cSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g
)])/(15*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]*(a*e^2*g^2*(7*e*f - 15*d*g) - c*(8*e^3*f^3 - 30*d*e^2*f^2*
g + 45*d^2*e*f*g^2 - 15*d^3*g^3))*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)])/(15*c^(3/2)*g^3*Sqrt[f + g*x]*Sqrt[a +
c*x^2])

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Rubi [A]  time = 1.40465, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g^2 (7 e f-15 d g)-c \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{8 e^2 \sqrt{a+c x^2} \sqrt{f+g x} (e f-3 d g)}{15 c g^2}+\frac{2 e^2 \sqrt{a+c x^2} (d+e x) \sqrt{f+g x}}{5 c g} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

(-8*e^2*(e*f - 3*d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(15*c*g^2) + (2*e^2*(d + e*
x)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(5*c*g) + (2*Sqrt[-a]*e*(9*a*e^2*g^2 - c*(8*e^
2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Ar
cSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g
)])/(15*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]*(a*e^2*g^2*(7*e*f - 15*d*g) - c*(8*e^3*f^3 - 30*d*e^2*f^2*
g + 45*d^2*e*f*g^2 - 15*d^3*g^3))*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)])/(15*c^(3/2)*g^3*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)**3/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 7.70356, size = 625, normalized size = 1.37 \[ \frac{2 \sqrt{f+g x} \left (\frac{\sqrt{c} g \sqrt{f+g x} \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (9 a^{3/2} e^3 g^2+\sqrt{a} c e \left (-45 d^2 g^2+30 d e f g-8 e^2 f^2\right )-i a \sqrt{c} e^2 g (15 d g+2 e f)+15 i c^{3/2} d^3 g^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+\frac{e g^2 \left (-9 a^2 e^2 g^2+a c \left (45 d^2 g^2-30 d e f g+e^2 \left (8 f^2-9 g^2 x^2\right )\right )+c^2 x^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right )}{f+g x}+i c e \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )-9 a e^2 g^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+c e^2 g^2 \left (a+c x^2\right ) (15 d g-4 e f+3 e g x)\right )}{15 c^2 g^4 \sqrt{a+c x^2}} \]

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]*(c*e^2*g^2*(-4*e*f + 15*d*g + 3*e*g*x)*(a + c*x^2) + (e*g^2*(-9
*a^2*e^2*g^2 + c^2*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2)*x^2 + a*c*(-30*d*e*f*g
+ 45*d^2*g^2 + e^2*(8*f^2 - 9*g^2*x^2))))/(f + g*x) + I*c*e*Sqrt[-f - (I*Sqrt[a]
*g)/Sqrt[c]]*(-9*a*e^2*g^2 + c*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*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*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f +
g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (Sqrt[c]*g*((15*I
)*c^(3/2)*d^3*g^2 + 9*a^(3/2)*e^3*g^2 - I*a*Sqrt[c]*e^2*g*(2*e*f + 15*d*g) + Sqr
t[a]*c*e*(-8*e^2*f^2 + 30*d*e*f*g - 45*d^2*g^2))*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]*El
lipticF[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[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(15*
c^2*g^4*Sqrt[a + c*x^2])

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Maple [B]  time = 0.061, size = 2949, normalized size = 6.5 \[ \text{output too large to display} \]

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]

-2/15*(-15*x*a*c*d*e^2*g^4+x*a*c*e^3*f*g^3+4*a*c*e^3*f^2*g^2-3*x^4*c^2*e^3*g^4-1
5*x^2*c^2*d*e^2*f*g^3-15*a*c*d*e^2*f*g^3+15*(-(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*d^3*g^4-8*(-(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*e^3*f^3*g-45*(-(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^2*e*g^4-15*(-(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*d*e^2*g^4+7*(-(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*e^3*f*g^3-6*(-(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*e^3*f^2*g^2+45*(-(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*d^2*e*g^4-(-(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*e^3*f^2*g^2
+45*(-(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^2*d^2*e*f^2*g^2-30*(-(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^2*d*e^2*f^3*g+x^3*c^2*e^3*f*g^3-3*x^2*a*c*e^3*g
^4+4*x^2*c^2*e^3*f^2*g^2-45*(-(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*d^2*e*f*g^3+30*(-(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*d
*e^2*f^2*g^2+45*(-(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)*Ell
ipticF((-(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*e^2*f*g^3-30*(-(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*d*e^2*f*g^3+9*(-(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*e^3*g^4-9*(-(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*e^3*g^4+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)*Ellipt
icE((-(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^3*f^4-15*(-(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^3*f*g^3-15*x^3*c^2*d*e^2*g^4)*(g*x+f)^(1
/2)*(c*x^2+a)^(1/2)/c^2/g^4/(c*g*x^3+c*f*x^2+a*g*x+a*f)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{\sqrt{c x^{2} + a} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(sqrt(c*x^2 + a)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

integrate((e*x + d)^3/(sqrt(c*x^2 + a)*sqrt(g*x + f)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{\sqrt{c x^{2} + a} \sqrt{g x + f}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(sqrt(c*x^2 + a)*sqrt(g*x + f)),x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)/(sqrt(c*x^2 + a)*sqrt(g*x + f
)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\sqrt{a + c x^{2}} \sqrt{f + g x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)

[Out]

Integral((d + e*x)**3/(sqrt(a + c*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((e*x + d)^3/(sqrt(c*x^2 + a)*sqrt(g*x + f)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError