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

Optimal. Leaf size=608 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} \left (15 a^2 d^2 f^2-5 a b d f (7 c f+d e)+b^2 \left (23 c^2 f^2+12 c d e f-2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d e^{3/2} \sqrt{c+d x^2} \left (15 a^2 d^2 f-40 a b c d f+b^2 c (34 c f-d e)\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b^3 c f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^3 \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt{e+f x^2}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{d x \sqrt{c+d x^2} \left (\frac{3 c^2 f}{d}+7 c e-\frac{2 d e^2}{f}\right )}{15 b \sqrt{e+f x^2}}+\frac{d^2 x \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac{2 d x \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-3 c f)}{15 b f} \]

[Out]

(d*(7*c*e - (2*d*e^2)/f + (3*c^2*f)/d)*x*Sqrt[c + d*x^2])/(15*b*Sqrt[e + f*x^2])
+ ((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*Sqrt[e + f
*x^2]) + (d*(b*c - a*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b^2) - (2*d*(d*e -
3*c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b*f) + (d^2*x*Sqrt[c + d*x^2]*(e
+ f*x^2)^(3/2))/(5*b*f) - (Sqrt[e]*(15*a^2*d^2*f^2 - 5*a*b*d*f*(d*e + 7*c*f) + b
^2*(-2*d^2*e^2 + 12*c*d*e*f + 23*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqr
t[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b^3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e +
f*x^2))]*Sqrt[e + f*x^2]) + (d*e^(3/2)*(-40*a*b*c*d*f + 15*a^2*d^2*f + b^2*c*(-
(d*e) + 34*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e
)/(c*f)])/(15*b^3*c*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2
]) + ((b*c - a*d)^3*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^3*c*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 = 2.2298, antiderivative size = 776, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281 \[ \frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) (b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^3 \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \left (\frac{3 c^2 f}{d}+7 c e-\frac{2 d e^2}{f}\right )}{15 b \sqrt{e+f x^2}}+\frac{d^2 x \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac{d e^{3/2} \sqrt{c+d x^2} (d e-9 c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{2 d x \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-3 c f)}{15 b f} \]

Antiderivative was successfully verified.

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

[Out]

(d*(7*c*e - (2*d*e^2)/f + (3*c^2*f)/d)*x*Sqrt[c + d*x^2])/(15*b*Sqrt[e + f*x^2])
+ ((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*Sqrt[e + f
*x^2]) + (d*(b*c - a*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b^2) - (2*d*(d*e -
3*c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b*f) + (d^2*x*Sqrt[c + d*x^2]*(e
+ f*x^2)^(3/2))/(5*b*f) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqrt[
c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*Sqrt[
f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(2*d^2*e^2
- 7*c*d*e*f - 3*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]],
1 - (d*e)/(c*f)])/(15*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f
*x^2]) + (d*(5*b*c - 3*a*d)*(b*c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan
[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(
c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*e^(3/2)*(d*e - 9*c*f)*Sqrt[c + d*x^2]*Elli
pticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b*f^(3/2)*Sqrt[(e*(c +
d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^3*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^3*c*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 4.74015, size = 456, normalized size = 0.75 \[ \frac{-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (15 a^2 d^2 f^2-5 a b d f (7 c f+d e)+b^2 \left (23 c^2 f^2+12 c d e f-2 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (-15 a^3 d^3 f^3+45 a^2 b c d^2 f^3+5 a b^2 d f \left (-9 c^2 f^2-c d e f+d^2 e^2\right )+b^3 \left (15 c^3 f^3+11 c^2 d e f^2-13 c d^2 e^2 f+2 d^3 e^3\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f \left (a b^2 d x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-5 a d f+11 b c f+b d \left (e+3 f x^2\right )\right )-15 i f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 (b e-a f) \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 a b^4 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*a*b*d*e*(15*a^2*d^2*f^2 - 5*a*b*d*f*(d*e + 7*c*f) + b^2*(-2*d^2*e^2 + 12*c
*d*e*f + 23*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSin
h[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(45*a^2*b*c*d^2*f^3 - 15*a^3*d^3*f^3 + 5*a*b^
2*d*f*(d^2*e^2 - c*d*e*f - 9*c^2*f^2) + b^3*(2*d^3*e^3 - 13*c*d^2*e^2*f + 11*c^2
*d*e*f^2 + 15*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcS
inh[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)*
(11*b*c*f - 5*a*d*f + b*d*(e + 3*f*x^2)) - (15*I)*(b*c - a*d)^3*f*(b*e - a*f)*Sq
rt[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)]))/(15*a*b^4*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.056, size = 1891, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(15*((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^4*d^3*f^3-45*((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*b*c*d^2*f^3+45*
((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^2*c^2*d*f^3-5*((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^2*d^3*e^2*f-15*((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^3*b*d^3*e*f^2+5*((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^2*d^3*e
^2*f+5*((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^2*c*d^2*e*f^2+45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elliptic
Pi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^3*b*c*d^2*f^3+15*((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*b*d^3*e*f^2-45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elliptic
Pi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*b^2*c^2*d*f^3-15*(-d/c)
^(1/2)*x^3*a*b^3*c*d^2*e*f^2+5*(-d/c)^(1/2)*x*a^2*b^2*c*d^2*e*f^2-11*(-d/c)^(1/2
)*x*a*b^3*c^2*d*e*f^2-(-d/c)^(1/2)*x*a*b^3*c*d^2*e^2*f-15*((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
^4*d^3*f^3+5*(-d/c)^(1/2)*x^5*a^2*b^2*d^3*f^3-3*(-d/c)^(1/2)*x^7*a*b^3*d^3*f^3-1
4*(-d/c)^(1/2)*x^5*a*b^3*c*d^2*f^3-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*El
lipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^4*c^3*e*f^2-15*((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^3*c^3*f^3-2*((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^3*d^3*e^3+2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elliptic
E(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*d^3*e^3+15*((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^3*c^
3*f^3-4*(-d/c)^(1/2)*x^5*a*b^3*d^3*e*f^2+5*(-d/c)^(1/2)*x^3*a^2*b^2*c*d^2*f^3+5*
(-d/c)^(1/2)*x^3*a^2*b^2*d^3*e*f^2-11*(-d/c)^(1/2)*x^3*a*b^3*c^2*d*f^3-(-d/c)^(1
/2)*x^3*a*b^3*d^3*e^2*f-23*((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^3*c^2*d*e*f^2-12*((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^3*c*d^2*e^2*f-45*((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^2*c*d^2*e*f^2+45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ell
ipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b^3*c^2*d*e*f^2-11*(
(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^3*c^2*d*e*f^2+13*((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^3*c*d^2*e^2*f+35*((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^2*c*d^2*e*f^2)/(d*f*x^4+c*f
*x^2+d*e*x^2+c*e)/b^4/f^2/(-d/c)^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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