3.42 \(\int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=501 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right )}{15 e f^3 \sqrt{e+f x^2}}-\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{15 e f^3}+\frac{d x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (6 b e-5 a f)}{5 e f^2}-\frac{x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}} \]

[Out]

-((5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d*e*f +
19*c^2*f^2))*x*Sqrt[c + d*x^2])/(15*e*f^3*Sqrt[e + f*x^2]) - ((b*e - a*f)*x*(c
+ d*x^2)^(5/2))/(e*f*Sqrt[e + f*x^2]) - (d*(b*e*(24*d*e - 23*c*f) - 5*a*f*(4*d*e
- 3*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*e*f^3) + (d*(6*b*e - 5*a*f)*x*
(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*e*f^2) + ((5*a*f*(8*d^2*e^2 - 13*c*d*e*f +
3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d*e*f + 19*c^2*f^2))*Sqrt[c + d*x^2]*Elli
pticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*Sqrt[e]*f^(7/2)*Sqrt[(e
*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(10*a*d*f*(2*d*e - 3*
c*f) - b*(24*d^2*e^2 - 41*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTa
n[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*f^(7/2)*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.69633, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right )}{15 e f^3 \sqrt{e+f x^2}}-\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{15 e f^3}+\frac{d x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (6 b e-5 a f)}{5 e f^2}-\frac{x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]

[Out]

-((5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d*e*f +
19*c^2*f^2))*x*Sqrt[c + d*x^2])/(15*e*f^3*Sqrt[e + f*x^2]) - ((b*e - a*f)*x*(c
+ d*x^2)^(5/2))/(e*f*Sqrt[e + f*x^2]) - (d*(b*e*(24*d*e - 23*c*f) - 5*a*f*(4*d*e
- 3*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*e*f^3) + (d*(6*b*e - 5*a*f)*x*
(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*e*f^2) + ((5*a*f*(8*d^2*e^2 - 13*c*d*e*f +
3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d*e*f + 19*c^2*f^2))*Sqrt[c + d*x^2]*Elli
pticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*Sqrt[e]*f^(7/2)*Sqrt[(e
*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(10*a*d*f*(2*d*e - 3*
c*f) - b*(24*d^2*e^2 - 41*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTa
n[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*f^(7/2)*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((b*x**2+a)*(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.05784, size = 369, normalized size = 0.74 \[ \frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (5 a d f (9 c f-8 d e)+b \left (15 c^2 f^2-64 c d e f+48 d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )-5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (5 a f \left (3 c^2 f^2-6 c d e f+d^2 e \left (4 e+f x^2\right )\right )+b e \left (-15 c^2 f^2+c d f \left (41 e+11 f x^2\right )-3 d^2 \left (8 e^2+2 e f x^2-f^2 x^4\right )\right )\right )}{15 e f^4 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(5*a*f*(-6*c*d*e*f + 3*c^2*f^2 + d^2*e*(4*e + f*x^2))
+ b*e*(-15*c^2*f^2 + c*d*f*(41*e + 11*f*x^2) - 3*d^2*(8*e^2 + 2*e*f*x^2 - f^2*x
^4))) - I*d*e*(-5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + 2*b*e*(24*d^2*e^2 -
44*c*d*e*f + 19*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*A
rcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(5*a*d*f*(-8*d*e + 9*c*f)
+ b*(48*d^2*e^2 - 64*c*d*e*f + 15*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)])/(15*Sqrt[d/c]*e*f^4*Sqrt[c
+ d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.069, size = 1169, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)

[Out]

1/15*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(-88*((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))*b*c*d^2*e^3*f-48*((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))*b*d^3*e^4+48*((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))*b*
d^3*e^4+3*(-d/c)^(1/2)*x^7*b*d^3*e*f^3+5*(-d/c)^(1/2)*x^5*a*d^3*e*f^3-6*(-d/c)^(
1/2)*x^5*b*d^3*e^2*f^2+15*(-d/c)^(1/2)*x^3*a*c^2*d*f^4+20*(-d/c)^(1/2)*x^3*a*d^3
*e^2*f^2-24*(-d/c)^(1/2)*x^3*b*d^3*e^3*f-15*(-d/c)^(1/2)*x*b*c^3*e*f^3-30*(-d/c)
^(1/2)*x*a*c^2*d*e*f^3+20*(-d/c)^(1/2)*x*a*c*d^2*e^2*f^2+40*((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*d^3*e^3*f+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))*b
*c^3*e*f^3+41*(-d/c)^(1/2)*x*b*c^2*d*e^2*f^2-24*(-d/c)^(1/2)*x*b*c*d^2*e^3*f-40*
((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*d^3*e^3*f+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*c^2*d*e*f^3+38*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^2*d*e^2*f^2-15*((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*c^2*d*e*f^3+65*((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*
c*d^2*e^2*f^2-85*((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*c*d^2*e^2*f^2-79*((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))*b*c^2*d*e^2*f^2+112*((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))*b*c*d^2*e^3*f+14*(
-d/c)^(1/2)*x^5*b*c*d^2*e*f^3-25*(-d/c)^(1/2)*x^3*a*c*d^2*e*f^3-4*(-d/c)^(1/2)*x
^3*b*c^2*d*e*f^3+35*(-d/c)^(1/2)*x^3*b*c*d^2*e^2*f^2+15*x*a*c^3*f^4*(-d/c)^(1/2)
)/f^4/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/(-d/c)^(1/2)/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b d^{2} x^{6} +{\left (2 \, b c d + a d^{2}\right )} x^{4} + a c^{2} +{\left (b c^{2} + 2 \, a c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e)^(3/2),x, algorithm="fricas")

[Out]

integral((b*d^2*x^6 + (2*b*c*d + a*d^2)*x^4 + a*c^2 + (b*c^2 + 2*a*c*d)*x^2)*sqr
t(d*x^2 + c)/(f*x^2 + e)^(3/2), 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((b*x**2+a)*(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e)^(3/2), x)