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

Optimal. Leaf size=381 \[ \frac{x \sqrt{c+d x^2} \left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right )}{15 d^2 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (-10 a d f+b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-2 b c f+b d e)}{15 d f}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 d} \]

[Out]

((5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*x*Sqrt[c + d*x^2])/(1
5*d^2*f*Sqrt[e + f*x^2]) + ((b*d*e - 2*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e
+ f*x^2])/(15*d*f) + (b*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*d) - (Sqrt[e]*(
5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*Sqrt[c + d*x^2]*Ellipti
cE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^2*f^(3/2)*Sqrt[(e*(c + d
*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (e^(3/2)*(b*d*e + b*c*f - 10*a*d*f)*S
qrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d*f^
(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.10216, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{c+d x^2} \left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right )}{15 d^2 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (-10 a d f+b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-2 b c f+b d e)}{15 d f}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

((5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*x*Sqrt[c + d*x^2])/(1
5*d^2*f*Sqrt[e + f*x^2]) + ((b*d*e - 2*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e
+ f*x^2])/(15*d*f) + (b*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*d) - (Sqrt[e]*(
5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*Sqrt[c + d*x^2]*Ellipti
cE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^2*f^(3/2)*Sqrt[(e*(c + d
*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (e^(3/2)*(b*d*e + b*c*f - 10*a*d*f)*S
qrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d*f^
(3/2)*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 = 115.453, size = 374, normalized size = 0.98 \[ \frac{b x \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{e + f x^{2}}}{5 d} + \frac{c^{\frac{3}{2}} \sqrt{e + f x^{2}} \left (10 a d f - b c f - b d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{15 d^{\frac{3}{2}} f \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}} \left (5 a d f - 2 b c f + b d e\right )}{15 d f} + \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (- 5 a c d f^{2} - 5 a d^{2} e f + 2 b c^{2} f^{2} - 2 b c d e f + 2 b d^{2} e^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{15 d^{2} f^{\frac{3}{2}} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} - \frac{x \sqrt{c + d x^{2}} \left (- 5 a c d f^{2} - 5 a d^{2} e f + 2 b c^{2} f^{2} - 2 b c d e f + 2 b d^{2} e^{2}\right )}{15 d^{2} f \sqrt{e + f x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*(c + d*x**2)**(3/2)*sqrt(e + f*x**2)/(5*d) + c**(3/2)*sqrt(e + f*x**2)*(10*a
*d*f - b*c*f - b*d*e)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(15*d*
*(3/2)*f*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)) + x*sqrt(c + d*
x**2)*sqrt(e + f*x**2)*(5*a*d*f - 2*b*c*f + b*d*e)/(15*d*f) + sqrt(e)*sqrt(c + d
*x**2)*(-5*a*c*d*f**2 - 5*a*d**2*e*f + 2*b*c**2*f**2 - 2*b*c*d*e*f + 2*b*d**2*e*
*2)*elliptic_e(atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(15*d**2*f**(3/2)*sqrt(e*
(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)) - x*sqrt(c + d*x**2)*(-5*a*c*d*
f**2 - 5*a*d**2*e*f + 2*b*c**2*f**2 - 2*b*c*d*e*f + 2*b*d**2*e**2)/(15*d**2*f*sq
rt(e + f*x**2))

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Mathematica [C]  time = 1.3225, size = 267, normalized size = 0.7 \[ \frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (2 b \left (c^2 f^2-c d e f+d^2 e^2\right )-5 a d f (c f+d e)\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (5 a d f+b c f+b d \left (e+3 f x^2\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (5 a d f+b c f-2 b d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{15 d f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(b*c*f + 5*a*d*f + b*d*(e + 3*f*x^2)) + I
*e*(-5*a*d*f*(d*e + c*f) + 2*b*(d^2*e^2 - c*d*e*f + c^2*f^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*e*(-(d*
e) + c*f)*(-2*b*d*e + b*c*f + 5*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*E
llipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*d*Sqrt[d/c]*f^2*Sqrt[c + d*x^
2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.023, size = 865, 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)^(1/2)*(f*x^2+e)^(1/2),x)

[Out]

1/15*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(3*(-d/c)^(1/2)*x^7*b*d^2*f^3+5*(-d/c)^(1/2
)*x^5*a*d^2*f^3+4*(-d/c)^(1/2)*x^5*b*c*d*f^3+4*(-d/c)^(1/2)*x^5*b*d^2*e*f^2+5*(-
d/c)^(1/2)*x^3*a*c*d*f^3+5*(-d/c)^(1/2)*x^3*a*d^2*e*f^2+(-d/c)^(1/2)*x^3*b*c^2*f
^3+5*(-d/c)^(1/2)*x^3*b*c*d*e*f^2+(-d/c)^(1/2)*x^3*b*d^2*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*c*d*e*f^2-5
*((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*d^2*e^2*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*e*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))*b*c*d*e^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))*b*d^2*e^3+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*c*d*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
*d^2*e^2*f-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^2*e*f^2+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*d*e^2*f-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*d^2*e^3+5*(-d/c)^(1/2)*x*a*c*d*
e*f^2+(-d/c)^(1/2)*x*b*c^2*e*f^2+(-d/c)^(1/2)*x*b*c*d*e^2*f)/(d*f*x^4+c*f*x^2+d*
e*x^2+c*e)/f^2/d/(-d/c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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