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

Optimal. Leaf size=469 \[ -\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]

[Out]

(-2*Sqrt[-a]*g*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)])/(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[(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^2*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)^
2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ellipti
cPi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt
[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(e^2*((Sqrt[c]*d)/Sqrt[-a] + e)*
Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 2.14008, antiderivative size = 469, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-a]*g*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)])/(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[(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^2*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)^
2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ellipti
cPi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt
[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(e^2*((Sqrt[c]*d)/Sqrt[-a] + e)*
Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 150.141, size = 478, normalized size = 1.02 \[ \frac{2 \sqrt{\frac{g \left (- \sqrt{c} x - \sqrt{- a}\right )}{\sqrt{c} f - g \sqrt{- a}}} \sqrt{\frac{g \left (- \sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} f + g \sqrt{- a}}} \left (d g - e f\right ) \Pi \left (- \frac{e \left (\sqrt{c} f + g \sqrt{- a}\right )}{\sqrt{c} \left (d g - e f\right )}; \operatorname{asin}{\left (\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{f + g x} \right )}\middle | \frac{\sqrt{c} f + g \sqrt{- a}}{\sqrt{c} f - g \sqrt{- a}}\right )}{e^{2} \sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{a + c x^{2}}} - \frac{2 g \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{f + g x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{c} e \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{2 g \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (d g - e f\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{c} e^{2} \sqrt{a + c x^{2}} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(g*(-sqrt(c)*x - sqrt(-a))/(sqrt(c)*f - g*sqrt(-a)))*sqrt(g*(-sqrt(c)*x +
sqrt(-a))/(sqrt(c)*f + g*sqrt(-a)))*(d*g - e*f)*elliptic_pi(-e*(sqrt(c)*f + g*sq
rt(-a))/(sqrt(c)*(d*g - e*f)), asin(sqrt(c/(sqrt(c)*g*sqrt(-a) + c*f))*sqrt(f +
g*x)), (sqrt(c)*f + g*sqrt(-a))/(sqrt(c)*f - g*sqrt(-a)))/(e**2*sqrt(c/(sqrt(c)*
g*sqrt(-a) + c*f))*sqrt(a + c*x**2)) - 2*g*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(f +
g*x)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*g/(a*g - sqrt(c)*
f*sqrt(-a)))/(sqrt(c)*e*sqrt(sqrt(c)*sqrt(-a)*(-f - g*x)/(a*g - sqrt(c)*f*sqrt(-
a)))*sqrt(a + c*x**2)) + 2*g*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-f - g*x)/(a*g - sq
rt(c)*f*sqrt(-a)))*sqrt(1 + c*x**2/a)*(d*g - e*f)*elliptic_f(asin(sqrt(-sqrt(c)*
x/(2*sqrt(-a)) + 1/2)), 2*a*g/(a*g - sqrt(c)*f*sqrt(-a)))/(sqrt(c)*e**2*sqrt(a +
c*x**2)*sqrt(f + g*x))

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Mathematica [C]  time = 1.68562, size = 927, normalized size = 1.98 \[ \frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}} \left (-\frac{\sqrt{a} \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right ) f^2}{i \sqrt{c} d+\sqrt{a} e}+\frac{2 i \sqrt{a} g \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right ) f}{\sqrt{c} e}+\frac{2 \sqrt{a} d g \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right ) f}{\sqrt{a} e^2+i \sqrt{c} d e}-\frac{i \sqrt{a} d g^2 \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right )}{\sqrt{c} e^2}+\frac{g \sqrt{\frac{g \left (i \sqrt{c} x+\sqrt{a}\right )}{\sqrt{a} g-i \sqrt{c} f}} \left (\sqrt{c} x+i \sqrt{a}\right ) \left (\left (\sqrt{c} f+i \sqrt{a} g\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-i \sqrt{a} g F\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{c e \sqrt{\frac{g \left (\sqrt{a}-i \sqrt{c} x\right )}{i \sqrt{c} f+\sqrt{a} g}}}-\frac{\sqrt{a} d^2 g^2 \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right )}{e^2 \left (i \sqrt{c} d+\sqrt{a} e\right )}\right )}{\sqrt{f+g x} \sqrt{c x^2+a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.052, size = 959, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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