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

Optimal. Leaf size=274 \[ \frac{\sqrt{e+f x^2} (d e (2 a d+b c)-c f (a d+2 b c)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} d^{3/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 d \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}} \]

[Out]

-((b*c - a*d)*x*Sqrt[e + f*x^2])/(3*c*d*(c + d*x^2)^(3/2)) + ((d*(b*c + 2*a*d)*e
- c*(2*b*c + a*d)*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 -
(c*f)/(d*e)])/(3*c^(3/2)*d^(3/2)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2
))/(e*(c + d*x^2))]) + ((b*c - a*d)*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*d*(d*e - c*f)*Sqrt[(e*(c + d
*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.618998, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\sqrt{e+f x^2} (d e (2 a d+b c)-c f (a d+2 b c)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} d^{3/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 d \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x*Sqrt[e + f*x^2])/(3*c*d*(c + d*x^2)^(3/2)) + ((d*(b*c + 2*a*d)*e
- c*(2*b*c + a*d)*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 -
(c*f)/(d*e)])/(3*c^(3/2)*d^(3/2)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2
))/(e*(c + d*x^2))]) + ((b*c - a*d)*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*d*(d*e - c*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 = 66.2034, size = 231, normalized size = 0.84 \[ \frac{x \sqrt{e + f x^{2}} \left (a d - b c\right )}{3 c d \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{e^{\frac{3}{2}} \sqrt{f} \sqrt{c + d x^{2}} \left (a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 c^{2} d \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (c f - d e\right )} + \frac{\sqrt{e + f x^{2}} \left (c f \left (a d + 2 b c\right ) - d e \left (2 a d + b c\right )\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.73948, size = 297, normalized size = 1.08 \[ \frac{x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (a d \left (2 c^2 f-3 c d e+c d f x^2-2 d^2 e x^2\right )+b c \left (c^2 f+2 c d f x^2-d^2 e x^2\right )\right )-i e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d+b c) (c f-d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a d (c f-2 d e)+b c (2 c f-d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 c^3 \left (\frac{d}{c}\right )^{3/2} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (c f-d e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.067, size = 1236, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^5*a*c*d^2*f^2*(-d/c)^(1/2)+x^3*b*c^3*f^2*(-d/c)^(1/2)+EllipticE(x*(-d/c)^
(1/2),(c*f/d/e)^(1/2))*x^2*b*c*d^2*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*c^2*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*c^2*d*e*f*((d*x^2+c)/
c)^(1/2)*((f*x^2+e)/e)^(1/2)-x^5*b*c*d^2*e*f*(-d/c)^(1/2)-2*x^3*a*c*d^2*e*f*(-d/
c)^(1/2)+2*x^3*b*c^2*d*e*f*(-d/c)^(1/2)+2*x*a*c^2*d*e*f*(-d/c)^(1/2)-2*EllipticF
(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*d^3*e^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))*x^2*a*d^3*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*c*d^2*e^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
))*b*c^3*e*f*((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^2*d*e^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*c*d^2*e^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))*b*c^3*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))*b*c^2*d*e^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))*x^2*b*c
*d^2*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*x^3*a*d^3*e^2*(-d/c)^(1/2)-2*
x^5*a*d^3*e*f*(-d/c)^(1/2)+2*x^5*b*c^2*d*f^2*(-d/c)^(1/2)+2*x^3*a*c^2*d*f^2*(-d/
c)^(1/2)-x^3*b*c*d^2*e^2*(-d/c)^(1/2)-3*x*a*c*d^2*e^2*(-d/c)^(1/2)+x*b*c^3*e*f*(
-d/c)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^2*e*f*((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))*x^2*b*
c^2*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))*x^2*a*c*d^2*e*f*((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))*x^2*b*c^2*d*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2))/(f*x^2+e)^(1/2)/(-d/c)^(1/2)/(c*f-d*e)/c^2/d/(d*x^2+c)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)*sqrt(f*x^2 + e)/((d^2*x^4 + 2*c*d*x^2 + c^2)*sqrt(d*x^2 + c
)), 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)*(f*x**2+e)**(1/2)/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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