3.643 \(\int \frac{(f+g x)^{5/2}}{(d+e x) \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=600 \[ \frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g^2+c \left (-3 d^2 g^2+6 d e f g-2 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{a \sqrt{c} x}{(-a)^{3/2}}+1}}{\sqrt{2}}\right )|\frac{2 a g}{a g-\sqrt{-a} \sqrt{c} f}\right )}{3 c^{3/2} e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 (e f-d g)^2 \sqrt{\frac{g \left (\sqrt{-a}-\sqrt{c} x\right )}{\sqrt{-a} g+\sqrt{c} f}} \sqrt{-\frac{g \left (\sqrt{-a}+\sqrt{c} x\right )}{\sqrt{c} f-\sqrt{-a} g}} \Pi \left (\frac{e \left (f+\frac{\sqrt{-a} g}{\sqrt{c}}\right )}{e f-d g};\sin ^{-1}\left (\sqrt{\frac{c}{c f+\sqrt{-a} \sqrt{c} g}} \sqrt{f+g x}\right )|\frac{\sqrt{c} f+\sqrt{-a} g}{\sqrt{c} f-\sqrt{-a} g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{\frac{c}{\sqrt{-a} \sqrt{c} g+c f}}}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} (7 e f-3 d g) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{a \sqrt{c} x}{(-a)^{3/2}}+1}}{\sqrt{2}}\right )|\frac{2 a g}{a g-\sqrt{-a} \sqrt{c} f}\right )}{3 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 g^2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 c e} \]

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) - (2*Sqrt[-a]*g*(7*e*f - 3*d*g)*Sq
rt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 + (a*Sqrt[c]*x)/(-a)^(3/
2)]/Sqrt[2]], (2*a*g)/(-(Sqrt[-a]*Sqrt[c]*f) + a*g)])/(3*Sqrt[c]*e^2*Sqrt[(Sqrt[
c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*g*(a*e^2*
g^2 + c*(-2*e^2*f^2 + 6*d*e*f*g - 3*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 + (a*Sqrt[c]*x)/(-a
)^(3/2)]/Sqrt[2]], (2*a*g)/(-(Sqrt[-a]*Sqrt[c]*f) + a*g)])/(3*c^(3/2)*e^3*Sqrt[f
+ g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)^2*Sqrt[(g*(Sqrt[-a] - Sqrt[c]*x))/(Sqr
t[c]*f + Sqrt[-a]*g)]*Sqrt[-((g*(Sqrt[-a] + Sqrt[c]*x))/(Sqrt[c]*f - Sqrt[-a]*g)
)]*EllipticPi[(e*(f + (Sqrt[-a]*g)/Sqrt[c]))/(e*f - d*g), ArcSin[Sqrt[c/(c*f + S
qrt[-a]*Sqrt[c]*g)]*Sqrt[f + g*x]], (Sqrt[c]*f + Sqrt[-a]*g)/(Sqrt[c]*f - Sqrt[-
a]*g)])/(e^3*Sqrt[c/(c*f + Sqrt[-a]*Sqrt[c]*g)]*Sqrt[a + c*x^2])

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Rubi [A]  time = 2.89956, antiderivative size = 808, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right ) (e f-d g)^3}{e^3 \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right ) (e f-d g)^2}{\sqrt{c} e^3 \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right ) (e f-d g)}{\sqrt{c} e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{8 \sqrt{-a} f g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} e \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{2 \sqrt{-a} g \left (c f^2+a g^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 c^{3/2} e \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 g^2 \sqrt{f+g x} \sqrt{c x^2+a}}{3 c e} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)^(5/2)/((d + e*x)*Sqrt[a + c*x^2]),x]

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) - (8*Sqrt[-a]*f*g*Sqrt[f + g*x]*Sq
rt[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)])/(3*Sqrt[c]*e*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]
*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*g*(e*f - d*g)*Sqrt[f + g*x]*Sqr
t[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)])/(Sqrt[c]*e^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*
f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*g*(e*f - d*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)])/(Sqrt[c]*
e^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*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)])/(3*c^(
3/2)*e*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)^3*Sqrt[(Sqrt[c]*(f + g*x)
)/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticPi[(2*e)/((Sqrt[c]*d)/Sq
rt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (2*Sqrt[-a]*g)/(Sqr
t[c]*f + Sqrt[-a]*g)])/(e^3*((Sqrt[c]*d)/Sqrt[-a] + e)*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((g*x+f)**(5/2)/(e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.7764, size = 1440, normalized size = 2.4 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+a} g^2}{3 c e}+\frac{2 (f+g x)^{3/2} \left (\frac{7 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3}{(f+g x)^2}+\frac{3 i c e^2 \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} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};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 ) f^2}{\sqrt{f+g x}}-\frac{14 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{f+g x}-\frac{3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{(f+g x)^2}-\frac{6 i c d e g \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} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};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 ) f}{\sqrt{f+g x}}+7 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+\frac{6 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{f+g x}+\frac{7 a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{(f+g x)^2}+\frac{\sqrt{c} e \left (\sqrt{a} g-i \sqrt{c} f\right ) (7 e f-3 d g) \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{i e \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (i \sqrt{a} e g+\sqrt{c} (6 e f-3 d g)\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}}+\frac{3 i c d^2 g^2 \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} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};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}}-3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}-\frac{3 a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{(f+g x)^2}\right )}{3 c e^3 \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}} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)^(5/2)/((d + e*x)*Sqrt[a + c*x^2]),x]

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) + (2*(f + g*x)^(3/2)*(7*c*e^2*f*Sq
rt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (7
*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*c*d*e*f^2*g*Sqrt[-
f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (7*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)
/Sqrt[c]])/(f + g*x)^2 - (3*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x
)^2 - (14*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (6*c*d*e*f*g*S
qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a
]*g)*(7*e*f - 3*d*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*S
qrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqr
t[-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 + g*x] + (I*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(I*Sqrt[a]*e*
g + Sqrt[c]*(6*e*f - 3*d*g))*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*Arc
Sinh[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 + g*x] + ((3*I)*c*e^2*f^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)/(Sqr
t[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)),
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 + g*x] - ((6*I)*c*d*e*f*g*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))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a
]*g)), 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 + g*x] + ((3*I)*c*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*Sqr
t[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*
Sqrt[a]*g)), 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 + g*x]))/(3*c*e^3*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 = 0.057, size = 3164, normalized size = 5.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(5/2)/(e*x+d)/(c*x^2+a)^(1/2),x)

[Out]

-2/3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(3*(-(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*e*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^2*f*g^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*g^3-3*(-(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*g^2+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))*c^2*d*e*f^2*g-9*((-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
llipticF((-(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))*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*c^2*e^2*f^3+3*(-(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*g^3-6*((-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))*(-(g*x+f)*c/(g
*(-a*c)^(1/2)-c*f))^(1/2)*(-a*c)^(1/2)*c*d*e*f*g^2+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^2*g
-3*(-(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*g^3+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)
*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*g^2-3*(-(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^2*g+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)*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^3+3*(-(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)*EllipticPi((-(g*x+f)*c/(
g*(-a*c)^(1/2)-c*f))^(1/2),(g*(-a*c)^(1/2)-c*f)*e/c/(d*g-e*f),(-(g*(-a*c)^(1/2)-
c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*c^2*d^2*f*g^2-6*((-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)*EllipticP
i((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(g*(-a*c)^(1/2)-c*f)*e/c/(d*g-e*f),(-(
g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f
))^(1/2)*c^2*d*e*f^2*g+3*((-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)*EllipticPi((-(g*x+f)*c/(g*(-a*c)^(
1/2)-c*f))^(1/2),(g*(-a*c)^(1/2)-c*f)*e/c/(d*g-e*f),(-(g*(-a*c)^(1/2)-c*f)/(g*(-
a*c)^(1/2)+c*f))^(1/2))*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2)*c^2*e^2*f^3-3*(-
(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)*EllipticPi((-(g*x+f)*
c/(g*(-a*c)^(1/2)-c*f))^(1/2),(g*(-a*c)^(1/2)-c*f)*e/c/(d*g-e*f),(-(g*(-a*c)^(1/
2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*(-a*c)^(1/2)*c*d^2*g^3+6*((-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)*EllipticPi((-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(g*(-a*c)^(1/2)-c*f)*e/c/(
d*g-e*f),(-(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*(-(g*x+f)*c/(g*(-a*
c)^(1/2)-c*f))^(1/2)*(-a*c)^(1/2)*c*d*e*f*g^2-3*((-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)*EllipticPi(
(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))^(1/2),(g*(-a*c)^(1/2)-c*f)*e/c/(d*g-e*f),(-(g*
(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f))^(1/2))*(-(g*x+f)*c/(g*(-a*c)^(1/2)-c*f))
^(1/2)*(-a*c)^(1/2)*c*e^2*f^2*g-x^3*c^2*e^2*g^3-x^2*c^2*e^2*f*g^2-x*a*c*e^2*g^3-
a*c*e^2*f*g^2)/e^3/c^2/(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. \[ \int \frac{{\left (g x + f\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + a}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + a)*(e*x + d)),x, algorithm="maxima")

[Out]

integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + a)*(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + a)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**(5/2)/(e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x + f\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + a}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + a)*(e*x + d)),x, algorithm="giac")

[Out]

integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + a)*(e*x + d)), x)