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

Optimal. Leaf size=410 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) (e f-5 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x} (7 d g+e f)}{15 c g}+\frac{2 e \sqrt{a+c x^2} (d+e x) \sqrt{f+g x}}{5 c} \]

[Out]

(2*e*(e*f + 7*d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(15*c*g) + (2*e*(d + e*x)*Sqrt
[f + g*x]*Sqrt[a + c*x^2])/(5*c) + (2*Sqrt[-a]*(9*a*e^2*g^2 + c*(2*e^2*f^2 - 10*
d*e*f*g - 15*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)])/(15*c^(
3/2)*g^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (
4*Sqrt[-a]*e*(e*f - 5*d*g)*(c*f^2 + a*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]*Sqrt[c]*f - a*g)])/(15*c^(3/2)*g^2*Sqrt[f + g*x]*
Sqrt[a + c*x^2])

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Rubi [A]  time = 1.22907, antiderivative size = 410, 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 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) (e f-5 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x} (7 d g+e f)}{15 c g}+\frac{2 e \sqrt{a+c x^2} (d+e x) \sqrt{f+g x}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(e*f + 7*d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(15*c*g) + (2*e*(d + e*x)*Sqrt
[f + g*x]*Sqrt[a + c*x^2])/(5*c) + (2*Sqrt[-a]*(9*a*e^2*g^2 + c*(2*e^2*f^2 - 10*
d*e*f*g - 15*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)])/(15*c^(
3/2)*g^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (
4*Sqrt[-a]*e*(e*f - 5*d*g)*(c*f^2 + a*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]*Sqrt[c]*f - a*g)])/(15*c^(3/2)*g^2*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 = 7.58755, size = 596, normalized size = 1.45 \[ \frac{2 \sqrt{f+g x} \left (\frac{g^2 \left (-9 a^2 e^2 g^2+a c \left (15 d^2 g^2+10 d e f g+e^2 \left (-\left (2 f^2+9 g^2 x^2\right )\right )\right )+c^2 x^2 \left (15 d^2 g^2+10 d e f g-2 e^2 f^2\right )\right )}{f+g x}-i c \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+\frac{\sqrt{c} g \sqrt{f+g x} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (2 \sqrt{a} \sqrt{c} e (e f-5 d g)-9 i a e^2 g+15 i c d^2 g\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+c e g^2 \left (a+c x^2\right ) (10 d g+e (f+3 g x))\right )}{15 c^2 g^3 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*(c*e*g^2*(a + c*x^2)*(10*d*g + e*(f + 3*g*x)) + (g^2*(-9*a^2*e^
2*g^2 + c^2*(-2*e^2*f^2 + 10*d*e*f*g + 15*d^2*g^2)*x^2 + a*c*(10*d*e*f*g + 15*d^
2*g^2 - e^2*(2*f^2 + 9*g^2*x^2))))/(f + g*x) - I*c*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[
c]]*(9*a*e^2*g^2 + c*(2*e^2*f^2 - 10*d*e*f*g - 15*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]], (S
qrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (Sqrt[c]*g*(Sqrt[c]*f + I*S
qrt[a]*g)*((15*I)*c*d^2*g - (9*I)*a*e^2*g + 2*Sqrt[a]*Sqrt[c]*e*(e*f - 5*d*g))*S
qrt[(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]*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)])/Sqrt[-f -
(I*Sqrt[a]*g)/Sqrt[c]]))/(15*c^2*g^3*Sqrt[a + c*x^2])

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Maple [B]  time = 0.054, size = 2470, normalized size = 6. \[ \text{result too large to display} \]

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]

-2/15*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(-10*x^2*c^2*d*e*f*g^3-10*x*a*c*d*e*g^4-4*x*
a*c*e^2*f*g^3-10*a*c*d*e*f*g^3-3*x^4*c^2*e^2*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)*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^2*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^2*g
^4-2*(-(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))*c^2*e^2*f^4+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)^(1/2)*a*e^2*f*g^3+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)^(1/2)*c*e^2*f^3*g+
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*c*e^2*f^2*g^2-11*(-(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*e^2*f^2*g^2+10*(-(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*f^3*g-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^2*f^2
*g^2+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)*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*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)*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*f^2*g^2-10*(-(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*f^2*g^2+10
*(-(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*f*g^3-10*(-(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*g^4-10*x^3*c^2*d*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*d^2*
g^4-4*x^3*c^2*e^2*f*g^3-3*x^2*a*c*e^2*g^4-x^2*c^2*e^2*f^2*g^2-a*c*e^2*f^2*g^2)/c
^2/(c*g*x^3+c*f*x^2+a*g*x+a*f)/g^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RuntimeError