3.651 \(\int \frac{\sqrt{a+c x^2}}{(d+e x)^{7/2}} \, dx\)
Optimal. Leaf size=444 \[ -\frac{4 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^2 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{4 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 c \sqrt{a+c x^2} \left (c d^2-3 a e^2\right )}{15 e \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{4 c d \sqrt{a+c x^2}}{15 e (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac{2 \sqrt{a+c x^2}}{5 e (d+e x)^{5/2}} \]
[Out]
(-2*Sqrt[a + c*x^2])/(5*e*(d + e*x)^(5/2)) + (4*c*d*Sqrt[a + c*x^2])/(15*e*(c*d^
2 + a*e^2)*(d + e*x)^(3/2)) + (4*c*(c*d^2 - 3*a*e^2)*Sqrt[a + c*x^2])/(15*e*(c*d
^2 + a*e^2)^2*Sqrt[d + e*x]) + (4*Sqrt[-a]*c^(3/2)*(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]]/Sqrt[2]],
(-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^2*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(
d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*c^(3/2)*d*Sqr
t[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[Ar
cSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e
)])/(15*e^2*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])
_______________________________________________________________________________________
Rubi [A] time = 1.21041, antiderivative size = 444, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286
\[ -\frac{4 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^2 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{4 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 c \sqrt{a+c x^2} \left (c d^2-3 a e^2\right )}{15 e \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{4 c d \sqrt{a+c x^2}}{15 e (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac{2 \sqrt{a+c x^2}}{5 e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + c*x^2]/(d + e*x)^(7/2),x]
[Out]
(-2*Sqrt[a + c*x^2])/(5*e*(d + e*x)^(5/2)) + (4*c*d*Sqrt[a + c*x^2])/(15*e*(c*d^
2 + a*e^2)*(d + e*x)^(3/2)) + (4*c*(c*d^2 - 3*a*e^2)*Sqrt[a + c*x^2])/(15*e*(c*d
^2 + a*e^2)^2*Sqrt[d + e*x]) + (4*Sqrt[-a]*c^(3/2)*(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]]/Sqrt[2]],
(-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^2*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(
d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*c^(3/2)*d*Sqr
t[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[Ar
cSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e
)])/(15*e^2*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])
_______________________________________________________________________________________
Rubi in Sympy [A] time = 176.933, size = 413, normalized size = 0.93 \[ - \frac{4 c^{\frac{3}{2}} d \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 e^{2} \sqrt{a + c x^{2}} \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )} - \frac{4 c^{\frac{3}{2}} \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (3 a e^{2} - c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 e^{2} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{4 c d \sqrt{a + c x^{2}}}{15 e \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} - \frac{4 c \sqrt{a + c x^{2}} \left (3 a e^{2} - c d^{2}\right )}{15 e \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{2 \sqrt{a + c x^{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(1/2)/(e*x+d)**(7/2),x)
[Out]
-4*c**(3/2)*d*sqrt(-a)*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)))/(15*e**2*sqrt(a + c*x**2)*sqrt(d + e*x)*(a*e**2 +
c*d**2)) - 4*c**(3/2)*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(3*a*e**2 - 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)))/(15*e**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)**2) + 4*c*d*sqrt(a + c*x**2)/(15*e*(d + e*
x)**(3/2)*(a*e**2 + c*d**2)) - 4*c*sqrt(a + c*x**2)*(3*a*e**2 - c*d**2)/(15*e*sq
rt(d + e*x)*(a*e**2 + c*d**2)**2) - 2*sqrt(a + c*x**2)/(5*e*(d + e*x)**(5/2))
_______________________________________________________________________________________
Mathematica [C] time = 5.1269, size = 602, normalized size = 1.36 \[ \frac{2 \left (-e^2 \left (a+c x^2\right ) \left (3 a^2 e^4+2 a c e^2 \left (5 d^2+5 d e x+3 e^2 x^2\right )-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )\right )+\frac{2 c (d+e x)^2 \left (\sqrt{c} (d+e x)^{3/2} \left (3 a^{3/2} e^3-\sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2+i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (-3 a^2 e^2+a c \left (d^2-3 e^2 x^2\right )+c^2 d^2 x^2\right )+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (4 i \sqrt{a} \sqrt{c} d e-3 a e^2+c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{15 e^3 \sqrt{a+c x^2} (d+e x)^{5/2} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + c*x^2]/(d + e*x)^(7/2),x]
[Out]
(2*(-(e^2*(a + c*x^2)*(3*a^2*e^4 - c^2*d^2*(d^2 + 6*d*e*x + 2*e^2*x^2) + 2*a*c*e
^2*(5*d^2 + 5*d*e*x + 3*e^2*x^2))) + (2*c*(d + e*x)^2*(-(e^2*Sqrt[-d - (I*Sqrt[a
]*e)/Sqrt[c]]*(-3*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 - 3*e^2*x^2))) + Sqrt[c]*(I*c
^(3/2)*d^3 - Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*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))]*(d + e*x)^(3/2)*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[a]*Sqrt[c]
*e*(c*d^2 + (4*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))]*(d + e*x)^(3/2)*
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 - (I*Sqrt[a]*e)/Sqrt[c]]))/(
15*e^3*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2)*Sqrt[a + c*x^2])
_______________________________________________________________________________________
Maple [B] time = 0.078, size = 3411, normalized size = 7.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(1/2)/(e*x+d)^(7/2),x)
[Out]
-2/15*(12*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*a^2*c*d*e^5*(-(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)+8*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*a*c^2*d^3*e^3*(-(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)-12*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*a^2*c
*d*e^5*(-(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)-12*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*a*c^2*d^3*e^3*(-(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*c^2*d^4*e^2*(-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)+2*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*c*d^3*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)+3*a^3*e^
6+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))*x^2*a*c^2*d^2*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)-6*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^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)+2*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^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
)+10*x^3*a*c^2*d*e^5-6*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^6*(-(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)+10*a^2*c*d^2*e^4-c^2*d^4*a*e^2+4*Ellipti
cE((-(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^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)+2*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^5*(-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*a*c*d^2*e^4*
(-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)-2*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))*c^3*d^6*(-(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))*x*c^3*d^5*e*(-(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)-6*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*c^2*d^4*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)+6*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^6
*(-(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)-2*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*c^3*d^4*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)+2*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))*c^2*d^5*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)-6*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^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)+6*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^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)+6*x^4*a*c^2
*e^6-2*x^4*c^3*d^2*e^4-6*x^3*c^3*d^3*e^3+9*x^2*a^2*c*e^6-x^2*c^3*d^4*e^2+8*x^2*a
*c^2*d^2*e^4+10*x*a^2*c*d*e^5-6*x*a*c^2*d^3*e^3)/(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2
/e^3/(e*x+d)^(5/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
integrate(sqrt(c*x^2 + a)/(e*x + d)^(7/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + a}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
integral(sqrt(c*x^2 + a)/((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(e*x + d
)), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(1/2)/(e*x+d)**(7/2),x)
[Out]
Integral(sqrt(a + c*x**2)/(d + e*x)**(7/2), 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(sqrt(c*x^2 + a)/(e*x + d)^(7/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError