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

Optimal. Leaf size=284 \[ \frac{\sqrt{e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a c f+a d e+2 b c e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.63039, antiderivative size = 284, 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 \[ \frac{\sqrt{e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a c f+a d e+2 b c e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x*Sqrt[e + f*x^2])/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*a*d*(
d*e - 2*c*f) + b*c*(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)*Sqrt[d]*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(
c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt[f]*(2*b*c*e + a*d*e - 3*a*c*f)*
Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*
(d*e - c*f)^2*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 = 90.132, size = 257, normalized size = 0.9 \[ - \frac{x \sqrt{e + f x^{2}} \left (a d - b c\right )}{3 c \left (c + d x^{2}\right )^{\frac{3}{2}} \left (c f - d e\right )} + \frac{\sqrt{e} \sqrt{f} \sqrt{c + d x^{2}} \left (3 a c f - a d e - 2 b c e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 c^{2} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (c f - d e\right )^{2}} + \frac{\sqrt{e + f x^{2}} \left (- 4 a c d f + 2 a d^{2} e + b c^{2} f + b c d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 c^{\frac{3}{2}} \sqrt{d} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (c f - d e\right )^{2}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 2.13819, size = 302, normalized size = 1.06 \[ \frac{x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (a d \left (-5 c^2 f+c d \left (3 e-4 f x^2\right )+2 d^2 e x^2\right )+b c \left (2 c^2 f+c d f x^2+d^2 e x^2\right )\right )+i \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (-3 a c f+2 a d e+b c e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 c^2 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d/c]*x*(e + f*x^2)*(b*c*(2*c^2*f + d^2*e*x^2 + c*d*f*x^2) + a*d*(-5*c^2*f
+ 2*d^2*e*x^2 + c*d*(3*e - 4*f*x^2))) + I*e*(2*a*d*(d*e - 2*c*f) + b*c*(d*e + 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*(-(d*e) + c*f)*(b*c*e + 2*a*d*e - 3*a*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^2*Sqrt[d/c]*(d*e - c*f)^2*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.049, size = 1352, normalized size = 4.8 \[ \text{result too large to display} \]

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)^(1/2),x)

[Out]

1/3*(-4*x^5*a*c*d^2*f^2*(-d/c)^(1/2)+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)-5*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)+4*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)-x^3*a*c*d^2*e*f
*(-d/c)^(1/2)+x^3*b*c^2*d*e*f*(-d/c)^(1/2)-5*x*a*c^2*d*e*f*(-d/c)^(1/2)+2*Ellipt
icF(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*Elliptic
E(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)-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)+3*EllipticF(x*(-d/c)^(1/2),(c*
f/d/e)^(1/2))*a*c^3*f^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)+3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^2*d*f^2*((d*x^2
+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*x^5*a*d^3*e*f*(-d/c)^(1/2)+x^5*b*c^2*d*f^2*(-
d/c)^(1/2)-5*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)+2*x*b*c^3*e*f*(-d/c)^(1/2)-5*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)+4*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)-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)/(c*f-d*e)^2/c^2/(-d/
c)^(1/2)/(d*x^2+c)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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