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

Optimal. Leaf size=375 \[ \frac{\sqrt{f} \sqrt{c+d x^2} \left (a \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )+b c e (7 c f+d e)\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e} \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x (2 a d (d e-3 c f)+b c (3 c f+d e))}{3 c^2 \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)^2}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-9 c f)+b c (3 c f+5 d e)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (d e-c f)} \]

[Out]

-((b*c - a*d)*x)/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) + ((2*a*d*(
d*e - 3*c*f) + b*c*(d*e + 3*c*f))*x)/(3*c^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e
+ f*x^2]) + (Sqrt[f]*(b*c*e*(d*e + 7*c*f) + a*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^
2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*
c^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
- (Sqrt[e]*Sqrt[f]*(a*d*(d*e - 9*c*f) + b*c*(5*d*e + 3*c*f))*Sqrt[c + d*x^2]*El
lipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(d*e - c*f)^3*Sqrt
[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.12029, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\sqrt{f} \sqrt{c+d x^2} \left (a \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )+b c e (7 c f+d e)\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e} \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x (2 a d (d e-3 c f)+b c (3 c f+d e))}{3 c^2 \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)^2}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-9 c f)+b c (3 c f+5 d e)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x)/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) + ((2*a*d*(
d*e - 3*c*f) + b*c*(d*e + 3*c*f))*x)/(3*c^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e
+ f*x^2]) + (Sqrt[f]*(b*c*e*(d*e + 7*c*f) + a*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^
2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*
c^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
- (Sqrt[e]*Sqrt[f]*(a*d*(d*e - 9*c*f) + b*c*(5*d*e + 3*c*f))*Sqrt[c + d*x^2]*El
lipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(d*e - c*f)^3*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 = 3.82665, size = 428, normalized size = 1.14 \[ \frac{-i d e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (a \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )+b c e (7 c f+d e)\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+x \sqrt{\frac{d}{c}} \left (a \left (3 c^4 f^3+6 c^3 d f^3 x^2+c^2 d^2 f \left (8 e^2+8 e f x^2+3 f^2 x^4\right )+c d^3 e \left (-3 e^2+4 e f x^2+7 f^2 x^4\right )-2 d^4 e^2 x^2 \left (e+f x^2\right )\right )-b c e \left (3 c^3 f^2+c^2 d f \left (5 e+11 f x^2\right )+c d^2 f x^2 \left (4 e+7 f x^2\right )+d^3 e x^2 \left (e+f x^2\right )\right )\right )-i e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (2 a d (d e-3 c f)+b c (3 c f+d e)) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 c^2 e \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (c f-d e)^3} \]

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]*x*(-(b*c*e*(3*c^3*f^2 + d^3*e*x^2*(e + f*x^2) + c*d^2*f*x^2*(4*e + 7*
f*x^2) + c^2*d*f*(5*e + 11*f*x^2))) + a*(3*c^4*f^3 + 6*c^3*d*f^3*x^2 - 2*d^4*e^2
*x^2*(e + f*x^2) + c^2*d^2*f*(8*e^2 + 8*e*f*x^2 + 3*f^2*x^4) + c*d^3*e*(-3*e^2 +
4*e*f*x^2 + 7*f^2*x^4))) - I*d*e*(b*c*e*(d*e + 7*c*f) + a*(2*d^2*e^2 - 7*c*d*e*
f - 3*c^2*f^2))*(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*e*(-(d*e) + c*f)*(2*a*d*(d*e - 3*c*f) + b
*c*(d*e + 3*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]*e*(-(d*e) + c*f)^3*(c + d
*x^2)^(3/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.056, size = 1742, normalized size = 4.7 \[ \text{result too large to display} \]

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]

-1/3*(-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*d^4*e^3*((d*x^2+c)/c)^(
1/2)*((f*x^2+e)/e)^(1/2)-7*x^5*a*c*d^3*e*f^2*(-d/c)^(1/2)+7*x^5*b*c^2*d^2*e*f^2*
(-d/c)^(1/2)+x^5*b*c*d^3*e^2*f*(-d/c)^(1/2)-3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)
^(1/2))*x^2*b*c^3*d*e*f^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))*x^2*b*c^2*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)
/e)^(1/2)+3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^2*d^2*e*f^2*((d*x^
2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+7*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^
2*a*c*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*EllipticE(x*(-d/c)^(1/
2),(c*f/d/e)^(1/2))*x^2*b*c^2*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+
6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^2*d^2*e*f^2*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)-8*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^3*
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))*x^2*b*c*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*x^5*a*c^2*d^2*
f^3*(-d/c)^(1/2)+2*x^5*a*d^4*e^2*f*(-d/c)^(1/2)-6*x^3*a*c^3*d*f^3*(-d/c)^(1/2)+x
^3*b*c*d^3*e^3*(-d/c)^(1/2)+3*x*a*c*d^3*e^3*(-d/c)^(1/2)+3*x*b*c^4*e*f^2*(-d/c)^
(1/2)-3*x*a*c^4*f^3*(-d/c)^(1/2)-8*x^3*a*c^2*d^2*e*f^2*(-d/c)^(1/2)-4*x^3*a*c*d^
3*e^2*f*(-d/c)^(1/2)+11*x^3*b*c^3*d*e*f^2*(-d/c)^(1/2)+4*x^3*b*c^2*d^2*e^2*f*(-d
/c)^(1/2)-8*x*a*c^2*d^2*e^2*f*(-d/c)^(1/2)+5*x*b*c^3*d*e^2*f*(-d/c)^(1/2)+2*Elli
pticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*d^4*e^3*((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^3*e^3*((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))*b*c^4*e*
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*c^2*d^2*e^3*((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^3*e^3*((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^2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)+2*x^3*a*d^4*e^3*(-d/c)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2
))*x^2*b*c*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+6*EllipticF(x*(-d/c)^
(1/2),(c*f/d/e)^(1/2))*a*c^3*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*E
llipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d^2*e^2*f*((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))*b*c^3*d*e^2*f*((d*x
^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a
*c^3*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+7*EllipticE(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*a*c^2*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*Ellip
ticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*d*e^2*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)^3/c^2/(-d/c)^(1/2)/e/(d*x^2+c)^(3/2)

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

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. \[{\rm integral}\left (\frac{b x^{2} + a}{{\left (d^{2} f x^{6} +{\left (d^{2} e + 2 \, c d f\right )} x^{4} + c^{2} e +{\left (2 \, c d e + c^{2} f\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

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)