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

Optimal. Leaf size=401 \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{d} \sqrt{e+f x^2} (b c (5 d e-4 c f)-a d (2 d e-c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{d e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (b c-a d) (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d x \sqrt{e+f x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)} \]

[Out]

-(d*x*Sqrt[e + f*x^2])/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (Sqrt[d]*(b*c*(5*d*
e - 4*c*f) - a*d*(2*d*e - c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqr
t[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*(b*c - a*d)^2*(d*e - c*f)*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (d*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*Ellipt
icF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)*(d*e - c*f
)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*Sqrt[c +
d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)
])/(a*c*(b*c - a*d)^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x
^2])

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Rubi [A]  time = 1.08284, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{d} \sqrt{e+f x^2} (b c (5 d e-4 c f)-a d (2 d e-c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{d e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (b c-a d) (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d x \sqrt{e+f x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x*Sqrt[e + f*x^2])/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (Sqrt[d]*(b*c*(5*d*
e - 4*c*f) - a*d*(2*d*e - c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqr
t[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*(b*c - a*d)^2*(d*e - c*f)*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (d*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*Ellipt
icF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)*(d*e - c*f
)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*Sqrt[c +
d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)
])/(a*c*(b*c - a*d)^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x
^2])

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Rubi in Sympy [A]  time = 127.482, size = 333, normalized size = 0.83 \[ \frac{d x \sqrt{e + f x^{2}}}{3 c \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{\sqrt{d} f \sqrt{e + f x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 \sqrt{c} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (c f - d e\right )} + \frac{\sqrt{d} \sqrt{e + f x^{2}} \left (c f \left (a d - 4 b c\right ) - d e \left (2 a d - 5 b c\right )\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 c^{\frac{3}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2} \left (c f - d e\right )} + \frac{b^{2} e^{\frac{3}{2}} \sqrt{c + d x^{2}} \Pi \left (1 - \frac{b e}{a f}; \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x*sqrt(e + f*x**2)/(3*c*(c + d*x**2)**(3/2)*(a*d - b*c)) + sqrt(d)*f*sqrt(e +
f*x**2)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(3*sqrt(c)*sqrt(c*(e
+ f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(a*d - b*c)*(c*f - d*e)) + sqrt(d)
*sqrt(e + f*x**2)*(c*f*(a*d - 4*b*c) - d*e*(2*a*d - 5*b*c))*elliptic_e(atan(sqrt
(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(3*c**(3/2)*sqrt(c*(e + f*x**2)/(e*(c + d*x**2))
)*sqrt(c + d*x**2)*(a*d - b*c)**2*(c*f - d*e)) + b**2*e**(3/2)*sqrt(c + d*x**2)*
elliptic_pi(1 - b*e/(a*f), atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(a*c*sqrt(f)*
sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)*(a*d - b*c)**2)

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Mathematica [C]  time = 6.41633, size = 427, normalized size = 1.06 \[ \frac{a c x \left (\frac{d}{c}\right )^{3/2} \left (e+f x^2\right ) \left (a d \left (2 c^2 f-3 c d e+c d f x^2-2 d^2 e x^2\right )+b c \left (-5 c^2 f+6 c d e-4 c d f x^2+5 d^2 e x^2\right )\right )-3 i b c^2 \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (c f-d e) \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i a \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (2 a d^2 e+b c (3 c f-5 d e)\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i a d e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a d (2 d e-c f)+b c (4 c f-5 d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a c^2 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (b c-a d)^2 (c f-d e)} \]

Antiderivative was successfully verified.

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

[Out]

(a*c*(d/c)^(3/2)*x*(e + f*x^2)*(b*c*(6*c*d*e - 5*c^2*f + 5*d^2*e*x^2 - 4*c*d*f*x
^2) + a*d*(-3*c*d*e + 2*c^2*f - 2*d^2*e*x^2 + c*d*f*x^2)) - I*a*d*e*(a*d*(2*d*e
- c*f) + b*c*(-5*d*e + 4*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/
e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-(d*e) + c*f)*(2*a*d^2*
e + b*c*(-5*d*e + 3*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*El
lipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b*c^2*(b*e - a*f)*(-(d*e) +
c*f)*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d)
, I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*(-(d*e)
+ c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.055, size = 2068, normalized size = 5.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(-2*x^3*a^2*d^4*e^2*(-d/c)^(1/2)+x^5*a^2*c*d^3*f^2*(-d/c)^(1/2)-8*EllipticF(
x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c^2*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c^2*d^2*e*f*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(
1/2)/(-d/c)^(1/2))*x^2*a*b*c^2*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+5
*x^3*a*b*c*d^3*e^2*(-d/c)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*
a^2*c*d^3*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticF(x*(-d/c)^(1/2)
,(c*f/d/e)^(1/2))*x^2*a*b*c^3*d*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+5*El
lipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c*d^3*e^2*((d*x^2+c)/c)^(1/2)*((
f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*c*d^3*e*f*((
d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*x^2*a*b*c*d^3*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticPi(x*(-d/c
)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*a*b*c^3*d*f^2*((d*x^2+c)/c)^(1/2)
*((f*x^2+e)/e)^(1/2)+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/
2))*x^2*b^2*c^3*d*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticF(x*(-d/
c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^3*d*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+
4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^3*d*e*f*((d*x^2+c)/c)^(1/2)*((
f*x^2+e)/e)^(1/2)-5*x*a*b*c^3*d*e*f*(-d/c)^(1/2)-2*x^5*a^2*d^4*e*f*(-d/c)^(1/2)+
2*x^3*a^2*c^2*d^2*f^2*(-d/c)^(1/2)-3*x*a^2*c*d^3*e^2*(-d/c)^(1/2)+2*x*a^2*c^2*d^
2*e*f*(-d/c)^(1/2)+6*x*a*b*c^2*d^2*e^2*(-d/c)^(1/2)-2*EllipticF(x*(-d/c)^(1/2),(
c*f/d/e)^(1/2))*x^2*a^2*d^4*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*Ellipt
icE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*d^4*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)-2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c*d^3*e^2*((d*x^2+c)
/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^
4*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticE(x*(-d/c)^(1/2),(c*f/d/
e)^(1/2))*a^2*c*d^3*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticPi(x*(
-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b*c^4*f^2*((d*x^2+c)/c)^(1/2)*(
(f*x^2+e)/e)^(1/2)+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2)
)*b^2*c^4*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticPi(x*(-d/c)^(1/2
),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c^3*d*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e
)/e)^(1/2)-4*x^5*a*b*c^2*d^2*f^2*(-d/c)^(1/2)-2*x^3*a^2*c*d^3*e*f*(-d/c)^(1/2)-5
*x^3*a*b*c^3*d*f^2*(-d/c)^(1/2)-3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)
/(-d/c)^(1/2))*x^2*b^2*c^2*d^2*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*Ell
ipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c^2*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x
^2+e)/e)^(1/2)+5*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^2*d^2*e^2*((d*x
^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2
*c^2*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*EllipticE(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*a*b*c^2*d^2*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+5*x^5*a
*b*c*d^3*e*f*(-d/c)^(1/2)+2*x^3*a*b*c^2*d^2*e*f*(-d/c)^(1/2)+3*EllipticPi(x*(-d/
c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b*c^3*d*e*f*((d*x^2+c)/c)^(1/2)*((
f*x^2+e)/e)^(1/2))/(f*x^2+e)^(1/2)/(a*d-b*c)^2/c^2/(c*f-d*e)/(-d/c)^(1/2)/a/(d*x
^2+c)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(f*x^2 + e)/((b*x^2 + a)*(d*x^2 + c)^(5/2)), 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(sqrt(f*x^2 + e)/((b*x^2 + a)*(d*x^2 + c)^(5/2)),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((f*x**2+e)**(1/2)/(b*x**2+a)/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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