3.685 \(\int \frac{\sqrt{d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=392 \[ \frac{\sqrt{d+e x} \left (x \left (3 a e^2+4 c d^2\right )+a d e\right )}{6 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{x \sqrt{d+e x}}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}} \]

[Out]

(x*Sqrt[d + e*x])/(3*a*(a + c*x^2)^(3/2)) + (Sqrt[d + e*x]*(a*d*e + (4*c*d^2 + 3
*a*e^2)*x))/(6*a^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + ((4*c*d^2 + 3*a*e^2)*Sqrt[
d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqr
t[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*Sqrt[c]*(c*d^2 + a*e^
2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (2*d*Sq
rt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[A
rcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*
e)])/(3*(-a)^(3/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.01341, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sqrt{d+e x} \left (x \left (3 a e^2+4 c d^2\right )+a d e\right )}{6 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{x \sqrt{d+e x}}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[d + e*x])/(3*a*(a + c*x^2)^(3/2)) + (Sqrt[d + e*x]*(a*d*e + (4*c*d^2 + 3
*a*e^2)*x))/(6*a^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + ((4*c*d^2 + 3*a*e^2)*Sqrt[
d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqr
t[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*Sqrt[c]*(c*d^2 + a*e^
2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (2*d*Sq
rt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[A
rcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*
e)])/(3*(-a)^(3/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 157.367, size = 366, normalized size = 0.93 \[ - \frac{2 d \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 \sqrt{c} \left (- a\right )^{\frac{3}{2}} \sqrt{a + c x^{2}} \sqrt{d + e x}} + \frac{\sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (3 a e^{2} + 4 c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{6 \sqrt{c} \left (- a\right )^{\frac{3}{2}} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} + \frac{x \sqrt{d + e x}}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{d + e x} \left (\frac{a d e}{2} + x \left (\frac{3 a e^{2}}{2} + 2 c d^{2}\right )\right )}{3 a^{2} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*x**
2/a)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*
d*sqrt(-a)))/(3*sqrt(c)*(-a)**(3/2)*sqrt(a + c*x**2)*sqrt(d + e*x)) + sqrt(1 + c
*x**2/a)*sqrt(d + e*x)*(3*a*e**2 + 4*c*d**2)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*
sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(6*sqrt(c)*(-a)**(3/2)*sqrt
(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)*(a*e**
2 + c*d**2)) + x*sqrt(d + e*x)/(3*a*(a + c*x**2)**(3/2)) + sqrt(d + e*x)*(a*d*e/
2 + x*(3*a*e**2/2 + 2*c*d**2))/(3*a**2*sqrt(a + c*x**2)*(a*e**2 + c*d**2))

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Mathematica [C]  time = 4.67243, size = 619, normalized size = 1.58 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a^2 e (d+5 e x)+a c x \left (6 d^2+d e x+3 e^2 x^2\right )+4 c^2 d^2 x^3\right )}{a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x) \left (\frac{2 i \sqrt{c} \left (3 i a^{3/2} e^3+4 i \sqrt{a} c d^2 e+3 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}-\frac{2 e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (3 a^2 e^2+a c \left (4 d^2+3 e^2 x^2\right )+4 c^2 d^2 x^2\right )}{(d+e x)^2}+\frac{2 \sqrt{a} \sqrt{c} e \left (i \sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}\right )}{a^2 c e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right )}\right )}{12 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((2*(4*c^2*d^2*x^3 + a^2*e*(d + 5*e*x) + a*c*x*(6*d^2 + d*e*x + 3
*e^2*x^2)))/(a^2*(c*d^2 + a*e^2)*(a + c*x^2)) + ((d + e*x)*((-2*e^2*Sqrt[-d - (I
*Sqrt[a]*e)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)))/(d +
e*x)^2 + ((2*I)*Sqrt[c]*(4*c^(3/2)*d^3 + (4*I)*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*
e^2 + (3*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((
I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]
*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)
])/Sqrt[d + e*x] + (2*Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e
^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c]
- e*x)/(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d +
e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x]))/(a
^2*c*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2))))/(12*Sqrt[a + c*x^2])

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Maple [B]  time = 0.07, size = 2292, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)
/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-
a*c)^(1/2)*e-c*d))^(1/2)+3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-(
(-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a*c^2*d^2*e^2*(-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+4*EllipticF((-(e*x+d)*c/((-a*c)^
(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a*c*
d*e^3*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*
e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+
4*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*
c)^(1/2)*e+c*d))^(1/2))*x^2*c^2*d^3*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c
*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2)
)*e/((-a*c)^(1/2)*e-c*d))^(1/2)-3*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c*e^4*(-(e*x+d)*
c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)
*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-7*EllipticE((-(e*x+d)*c/((-a*
c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a
*c^2*d^2*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-4*Elli
pticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/
2)*e+c*d))^(1/2))*x^2*c^3*d^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a
*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c
*d))^(1/2)+3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e
-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/
2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e-c*d))^(1/2)+3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((
-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^2*e^2*(-(e*x+d)*c/((-a*c
)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(
-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+4*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)
*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*d*e^3*(-a
*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)
^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+4*Ellipti
cF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*
e+c*d))^(1/2))*a*c*d^3*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((
-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(
1/2)*e-c*d))^(1/2)-3*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)
^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c
*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2)
)*e/((-a*c)^(1/2)*e-c*d))^(1/2)-7*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^2*e^2*(-(e*x+d)*
c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)
*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-4*EllipticE((-(e*x+d)*c/((-a*
c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2
*d^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2
)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-3*x^4*a*c^2*e^
4-4*x^4*c^3*d^2*e^2-4*x^3*a*c^2*d*e^3-4*x^3*c^3*d^3*e-5*x^2*a^2*c*e^4-7*x^2*a*c^
2*d^2*e^2-6*x*a^2*c*d*e^3-6*x*a*c^2*d^3*e-a^2*c*d^2*e^2)/c/(e*x+d)^(1/2)/(a*e^2+
c*d^2)/a^2/e/(c*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x + d}}{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(e*x + d)/((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)), x)

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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((e*x+d)**(1/2)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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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(sqrt(e*x + d)/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError