3.65 \(\int \frac{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx\)

Optimal. Leaf size=400 \[ \frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e)}{3 b^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b} \]

[Out]

((b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^2*Sqrt[e + f*x^2]) + (d*x*S
qrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b) - (Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqr
t[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^2*Sqr
t[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)
*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)]
)/(3*b^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*
c - a*d)^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*b^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^
2))]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.968117, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219 \[ \frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e)}{3 b^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

((b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^2*Sqrt[e + f*x^2]) + (d*x*S
qrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b) - (Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqr
t[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^2*Sqr
t[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)
*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)]
)/(3*b^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*
c - a*d)^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*b^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^
2))]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 130.427, size = 355, normalized size = 0.89 \[ \frac{d x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{3 b} + \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (3 a d f - 4 b c f - b d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 b^{2} \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} - \frac{x \sqrt{c + d x^{2}} \left (3 a d f - 4 b c f - b d e\right )}{3 b^{2} \sqrt{e + f x^{2}}} - \frac{d e^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (3 a d - 5 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 b^{2} c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} + \frac{e^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2} \Pi \left (1 - \frac{b e}{a f}; \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{a b^{2} c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 2.57959, size = 346, normalized size = 0.86 \[ \frac{-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (3 a^2 d^2 f^2-6 a b c d f^2+b^2 \left (3 c^2 f^2+c d e f-d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f \left (a b^2 d x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right )-3 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f) \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+4 b c f+b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a b^3 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*a*b*d*e*(b*d*e + 4*b*c*f - 3*a*d*f)*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*(-6*a*b*c*d*f^2 + 3*a^2*d
^2*f^2 + b^2*(-(d^2*e^2) + c*d*e*f + 3*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f
*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*d*Sqrt[d/c]*x
*(c + d*x^2)*(e + f*x^2) - (3*I)*(b*c - a*d)^2*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*S
qrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])
)/(3*a*b^3*Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 1059, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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