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

Optimal. Leaf size=544 \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (7 a d f (3 c f+d e)-b \left (6 c^2 f^2-6 c d e f+4 d^2 e^2\right )\right )}{105 d f^2}-\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (d e-9 c f)-b \left (-3 c^2 f^2-9 c d e f+4 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right )}{105 d^2 f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d} \]

[Out]

-((7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f +
9*c^2*d*e*f^2 - 6*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d^2*f^2*Sqrt[e + f*x^2]) +
((7*a*d*f*(d*e + 3*c*f) - b*(4*d^2*e^2 - 6*c*d*e*f + 6*c^2*f^2))*x*Sqrt[c + d*x^
2]*Sqrt[e + f*x^2])/(105*d*f^2) + ((b*d*e - 2*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/
2)*Sqrt[e + f*x^2])/(35*d*f) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*d) + (
Sqrt[e]*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e
^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/
Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2)
)]*Sqrt[e + f*x^2]) - (e^(3/2)*(7*a*d*f*(d*e - 9*c*f) - b*(4*d^2*e^2 - 9*c*d*e*f
- 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/
(c*f)])/(105*d*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.89573, antiderivative size = 544, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (7 a d f (3 c f+d e)-b \left (6 c^2 f^2-6 c d e f+4 d^2 e^2\right )\right )}{105 d f^2}-\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (d e-9 c f)-b \left (-3 c^2 f^2-9 c d e f+4 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right )}{105 d^2 f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d} \]

Antiderivative was successfully verified.

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

[Out]

-((7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f +
9*c^2*d*e*f^2 - 6*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d^2*f^2*Sqrt[e + f*x^2]) +
((7*a*d*f*(d*e + 3*c*f) - b*(4*d^2*e^2 - 6*c*d*e*f + 6*c^2*f^2))*x*Sqrt[c + d*x^
2]*Sqrt[e + f*x^2])/(105*d*f^2) + ((b*d*e - 2*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/
2)*Sqrt[e + f*x^2])/(35*d*f) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*d) + (
Sqrt[e]*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e
^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/
Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2)
)]*Sqrt[e + f*x^2]) - (e^(3/2)*(7*a*d*f*(d*e - 9*c*f) - b*(4*d^2*e^2 - 9*c*d*e*f
- 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/
(c*f)])/(105*d*f^(5/2)*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)**(3/2)*(f*x**2+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.06531, size = 373, normalized size = 0.69 \[ \frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (b \left (3 c^2 f^2-15 c d e f+8 d^2 e^2\right )-14 a d f (d e-3 c f)\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (6 c f+d \left (e+3 f x^2\right )\right )+b \left (3 c^2 f^2+3 c d f \left (3 e+8 f x^2\right )+d^2 \left (-4 e^2+3 e f x^2+15 f^2 x^4\right )\right )\right )+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )+b \left (6 c^3 f^3-9 c^2 d e f^2+19 c d^2 e^2 f-8 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 d f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(6*c*f + d*(e + 3*f*x^2)) + b*(3
*c^2*f^2 + 3*c*d*f*(3*e + 8*f*x^2) + d^2*(-4*e^2 + 3*e*f*x^2 + 15*f^2*x^4))) + I
*e*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + b*(-8*d^3*e^3 + 19*c*d^2*e^2*f
- 9*c^2*d*e*f^2 + 6*c^3*f^3))*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)*(-14*a*d*f*(d*e - 3*c
*f) + b*(8*d^2*e^2 - 15*c*d*e*f + 3*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)])/(105*d*Sqrt[d/c]*f^3*Sqrt[
c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.06, size = 1332, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(d*x^2+c)^(3/2)*(f*x^2+e)^(1/2),x)

[Out]

1/105*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(-19*((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-8*((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+8*((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+39*(-d/c)^(1/2)*x^7*b*c*d^2*f^4+18*(-d/c)^(1/2)*x^7*b*d^3*e*f^3+63*(-d/c)
^(1/2)*x^5*a*c*d^2*f^4+28*(-d/c)^(1/2)*x^5*a*d^3*e*f^3+27*(-d/c)^(1/2)*x^5*b*c^2
*d*f^4-(-d/c)^(1/2)*x^5*b*d^3*e^2*f^2+42*(-d/c)^(1/2)*x^3*a*c^2*d*f^4+7*(-d/c)^(
1/2)*x^3*a*d^3*e^2*f^2-4*(-d/c)^(1/2)*x^3*b*d^3*e^3*f+3*(-d/c)^(1/2)*x*b*c^3*e*f
^3+3*(-d/c)^(1/2)*x^3*b*c^3*f^4+15*(-d/c)^(1/2)*x^9*b*d^3*f^4+42*(-d/c)^(1/2)*x*
a*c^2*d*e*f^3+7*(-d/c)^(1/2)*x*a*c*d^2*e^2*f^2+14*((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+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^3*e*f^3+
9*(-d/c)^(1/2)*x*b*c^2*d*e^2*f^2-4*(-d/c)^(1/2)*x*b*c*d^2*e^3*f-14*((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
-6*((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^3+21*(-d/c)^(1/2)*x^7*a*d^3*f^4+42*((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+9*((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*f^2+21*((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+49*((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^2-56*((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^2-18*((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*f^2+23*((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+51*(-d/c)^(1/2)*x^5*b*c*d^2*e*f^3+70*(-d/c)^(1/2)*x^3
*a*c*d^2*e*f^3+36*(-d/c)^(1/2)*x^3*b*c^2*d*e*f^3+8*(-d/c)^(1/2)*x^3*b*c*d^2*e^2*
f^2)/d/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/f^3/(-d/c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)*(d*x**2+c)**(3/2)*(f*x**2+e)**(1/2),x)

[Out]

Integral((a + b*x**2)*(c + d*x**2)**(3/2)*sqrt(e + f*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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