∖int a+bx2 _______________________________________________∖left(c+dx2 ∖right)7∕2∘ ____ e+fx2 ∖,dx

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3.41 \(\int \frac{a+b x^2}{\left (c+d x^2\right )^{7/2} \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=401 \[ \frac{x \sqrt{e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (d e-c f)^2}+\frac{\sqrt{e+f x^2} \left (a d \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b c \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} \left (a \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )+b c e (d e-9 c f)\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c \left (c+d x^2\right )^{5/2} (d e-c f)} \]

[Out]

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

d*e - 2*c*f) + b*c*(d*e + 3*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(d*e - c*f)^2*(c +

d*x^2)^(3/2)) + ((b*c*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + a*d*(8*d^2*e^2 - 23*

c*d*e*f + 23*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1

- (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(d*e - c*f)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f

*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt[f]*(b*c*e*(d*e - 9*c*f) + a*(4*d^2*e^2

- 11*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]

], 1 - (d*e)/(c*f)])/(15*c^3*(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 = 1.18249, antiderivative size = 401, 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 \[ \frac{x \sqrt{e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (d e-c f)^2}+\frac{\sqrt{e+f x^2} \left (a d \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b c \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} \left (a \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )+b c e (d e-9 c f)\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c \left (c+d x^2\right )^{5/2} (d e-c f)} \]

Antiderivative was successfully veriﬁed.

[In] Int[(a + b*x^2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]

[Out]

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

d*e - 2*c*f) + b*c*(d*e + 3*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(d*e - c*f)^2*(c +

d*x^2)^(3/2)) + ((b*c*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + a*d*(8*d^2*e^2 - 23*

c*d*e*f + 23*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1

- (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(d*e - c*f)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f

*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt[f]*(b*c*e*(d*e - 9*c*f) + a*(4*d^2*e^2

- 11*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]

], 1 - (d*e)/(c*f)])/(15*c^3*(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)**(7/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C] time = 2.28568, size = 393, normalized size = 0.98 \[ \frac{-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a d \left (-23 c^2 f^2+23 c d e f-8 d^2 e^2\right )+b c \left (3 c^2 f^2+7 c d e f-2 d^2 e^2\right )\right )+3 c^2 (b c-a d) (d e-c f)^2+c \left (c+d x^2\right ) (c f-d e) (4 a d (d e-2 c f)+b c (3 c f+d e))\right )-i \sqrt{\frac{d x^2}{c}+1} \left (c+d x^2\right )^2 \sqrt{\frac{f x^2}{e}+1} \left ((d e-c f) \left (a \left (15 c^2 f^2-19 c d e f+8 d^2 e^2\right )+2 b c e (d e-3 c f)\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+e \left (a d \left (-23 c^2 f^2+23 c d e f-8 d^2 e^2\right )+b c \left (3 c^2 f^2+7 c d e f-2 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 c^3 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (d e-c f)^3} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(a + b*x^2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]

[Out]

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

4*a*d*(d*e - 2*c*f) + b*c*(d*e + 3*c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 23*c*d

*e*f - 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2))*(c + d*x^2)^2)) -

I*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(e*(a*d*(-8*d^2*e^2 + 2

3*c*d*e*f - 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2))*EllipticE[I*

ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*e - c*f)*(2*b*c*e*(d*e - 3*c*f) + a*(8*d

^2*e^2 - 19*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)

]))/(15*c^3*Sqrt[d/c]*(d*e - c*f)^3*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])

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Maple [B] time = 0.059, size = 3039, normalized size = 7.6 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((b*x^2+a)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)

[Out]

1/15*(9*x^3*b*c^5*f^3*(-d/c)^(1/2)-2*x^5*b*c*d^4*e^3*(-d/c)^(1/2)-34*x^3*a*c^4*d

*f^3*(-d/c)^(1/2)-20*x^3*a*c*d^4*e^3*(-d/c)^(1/2)-5*x^3*b*c^2*d^3*e^3*(-d/c)^(1/

2)-15*x*a*c^2*d^3*e^3*(-d/c)^(1/2)+9*x*b*c^5*e*f^2*(-d/c)^(1/2)+15*EllipticF(x*(

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

3*x^7*a*c^2*d^3*f^3*(-d/c)^(1/2)+23*x^7*a*c*d^4*e*f^2*(-d/c)^(1/2)+7*x^7*b*c^2*d

^3*e*f^2*(-d/c)^(1/2)-2*x^7*b*c*d^4*e^2*f*(-d/c)^(1/2)+35*x^5*a*c^2*d^3*e*f^2*(-

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

2*x^5*b*c^2*d^3*e^2*f*(-d/c)^(1/2)-13*x^3*a*c^3*d^2*e*f^2*(-d/c)^(1/2)+43*x^3*a*

c^2*d^3*e^2*f*(-d/c)^(1/2)+8*x^3*b*c^4*d*e*f^2*(-d/c)^(1/2)-2*EllipticF(x*(-d/c)

^(1/2),(c*f/d/e)^(1/2))*x^4*b*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+

12*x^3*b*c^3*d^2*e^2*f*(-d/c)^(1/2)-34*x*a*c^4*d*e*f^2*(-d/c)^(1/2)+41*x*a*c^3*d

^2*e^2*f*(-d/c)^(1/2)-x*b*c^4*d*e^2*f*(-d/c)^(1/2)-8*EllipticF(x*(-d/c)^(1/2),(c

*f/d/e)^(1/2))*x^4*a*d^5*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticE

(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*d^5*e^3*((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))*a*c^2*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))*b*c^5*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))*b*c^3*d^2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticE(x*(-d/c)^

(1/2),(c*f/d/e)^(1/2))*a*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*E

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

)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*x^7*a*d^5*e^2*f*(-d/c)^(1/2)+3*x^7*b*c^3*d^2*f^

3*(-d/c)^(1/2)-54*x^5*a*c^3*d^2*f^3*(-d/c)^(1/2)+9*x^5*b*c^4*d*f^3*(-d/c)^(1/2)+

15*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c^3*d^2*f^3*((d*x^2+c)/c)^(1/

2)*((f*x^2+e)/e)^(1/2)+30*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^4*d*

f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+23*EllipticE(x*(-d/c)^(1/2),(c*f/d/e

)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-23*Elliptic

E(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*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^4*b*c^3*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^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-68*EllipticF(x*(-d/c

)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(

1/2)+46*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^3*d^2*e*f^2*((d*x^2+c)

/c)^(1/2)*((f*x^2+e)/e)^(1/2)-46*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a

*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticE(x*(-d/c)^(1/2

),(c*f/d/e)^(1/2))*x^2*b*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-14*

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

)*((f*x^2+e)/e)^(1/2)+54*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^2*d^3

*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-12*EllipticF(x*(-d/c)^(1/2),(c*f/

d/e)^(1/2))*x^2*b*c^4*d*e*f^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))*x^4*b*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)

/e)^(1/2)-16*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((d*x^2+c

)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b

*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+16*EllipticE(x*(-d/c)^(1/2)

,(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*Elli

pticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*

x^2+e)/e)^(1/2)-34*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^4*d*e*f^2*((d*x

^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+27*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*

a*c^3*d^2*e^2*f*((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))*b*c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+23*Ell

ipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2

+e)/e)^(1/2)-23*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*d^2*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))*b*

c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+16*EllipticF(x*(-d/c)^(1/2),

(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-34*

EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/2

)*((f*x^2+e)/e)^(1/2)+27*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*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^4*b*c^3*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^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2

+e)/e)^(1/2)-8*x^5*a*d^5*e^3*(-d/c)^(1/2))/(f*x^2+e)^(1/2)/(c*f-d*e)^3/c^3/(-d/c

)^(1/2)/(d*x^2+c)^(5/2)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac{7}{2}} \sqrt{f x^{2} + e}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)), x)

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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x^{2} + a}{{\left (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)/((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*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)**(7/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac{7}{2}} \sqrt{f x^{2} + e}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)), x)

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