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

Optimal. Leaf size=551 \[ \frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a f \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )-b e \left (45 c^2 f^2-61 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (28 a d f (d e-2 c f)-b \left (15 c^2 f^2-43 c d e f+24 d^2 e^2\right )\right )}{105 f^3}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (-15 c^3 f^3+103 c^2 d e f^2-128 c d^2 e^2 f+48 d^3 e^3\right )\right )}{105 d f^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (-15 c^3 f^3+103 c^2 d e f^2-128 c d^2 e^2 f+48 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f} \]

[Out]

((7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 - 128*c*d^2*e^2*
f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d*f^3*Sqrt[e + f*x^2]
) - ((28*a*d*f*(d*e - 2*c*f) - b*(24*d^2*e^2 - 43*c*d*e*f + 15*c^2*f^2))*x*Sqrt[
c + d*x^2]*Sqrt[e + f*x^2])/(105*f^3) - ((6*b*d*e - 5*b*c*f - 7*a*d*f)*x*(c + d*
x^2)^(3/2)*Sqrt[e + f*x^2])/(35*f^2) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(
7*f) - (Sqrt[e]*(7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 -
128*c*d^2*e^2*f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(7/2)*Sqrt[(e*(c + d*x^2))/(
c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(7*a*f*(4*d^2*e^2 - 11*c*d*e*f + 15*
c^2*f^2) - b*e*(24*d^2*e^2 - 61*c*d*e*f + 45*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*f^(7/2)*Sqrt[(e*(c + d*x^2)
)/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.79634, antiderivative size = 551, 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 \[ \frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a f \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )-b e \left (45 c^2 f^2-61 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (28 a d f (d e-2 c f)-b \left (15 c^2 f^2-43 c d e f+24 d^2 e^2\right )\right )}{105 f^3}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (-15 c^3 f^3+103 c^2 d e f^2-128 c d^2 e^2 f+48 d^3 e^3\right )\right )}{105 d f^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (-15 c^3 f^3+103 c^2 d e f^2-128 c d^2 e^2 f+48 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f} \]

Antiderivative was successfully verified.

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

[Out]

((7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 - 128*c*d^2*e^2*
f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d*f^3*Sqrt[e + f*x^2]
) - ((28*a*d*f*(d*e - 2*c*f) - b*(24*d^2*e^2 - 43*c*d*e*f + 15*c^2*f^2))*x*Sqrt[
c + d*x^2]*Sqrt[e + f*x^2])/(105*f^3) - ((6*b*d*e - 5*b*c*f - 7*a*d*f)*x*(c + d*
x^2)^(3/2)*Sqrt[e + f*x^2])/(35*f^2) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(
7*f) - (Sqrt[e]*(7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 -
128*c*d^2*e^2*f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(7/2)*Sqrt[(e*(c + d*x^2))/(
c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(7*a*f*(4*d^2*e^2 - 11*c*d*e*f + 15*
c^2*f^2) - b*e*(24*d^2*e^2 - 61*c*d*e*f + 45*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*f^(7/2)*Sqrt[(e*(c + d*x^2)
)/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

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)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 2.19283, size = 386, normalized size = 0.7 \[ \frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (4 b e \left (15 c^2 f^2-26 c d e f+12 d^2 e^2\right )-7 a f \left (15 c^2 f^2-19 c d e f+8 d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (11 c f-4 d e+3 d f x^2\right )+b \left (45 c^2 f^2+c d f \left (45 f x^2-61 e\right )+3 d^2 \left (8 e^2-6 e f x^2+5 f^2 x^4\right )\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b \left (15 c^3 f^3-103 c^2 d e f^2+128 c d^2 e^2 f-48 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 f^4 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(-4*d*e + 11*c*f + 3*d*f*x^2) +
b*(45*c^2*f^2 + c*d*f*(-61*e + 45*f*x^2) + 3*d^2*(8*e^2 - 6*e*f*x^2 + 5*f^2*x^4)
)) - I*e*(7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) + b*(-48*d^3*e^3 + 128*c
*d^2*e^2*f - 103*c^2*d*e*f^2 + 15*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*(-(d*e) + c*f)*(4*b*e*(12
*d^2*e^2 - 26*c*d*e*f + 15*c^2*f^2) - 7*a*f*(8*d^2*e^2 - 19*c*d*e*f + 15*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*Sqrt[d/c]*f^4*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 1386, 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)^(5/2)/(f*x^2+e)^(1/2),x)

[Out]

1/105*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(128*((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+48*((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^4-48*((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+60*(-d/c)^(1/2)*x^7*b*c*d^2*f^4-3*(-d/c)^(1/2)*x^7*b*d^3*e*f^3+98*(-d/c
)^(1/2)*x^5*a*c*d^2*f^4-7*(-d/c)^(1/2)*x^5*a*d^3*e*f^3+90*(-d/c)^(1/2)*x^5*b*c^2
*d*f^4+6*(-d/c)^(1/2)*x^5*b*d^3*e^2*f^2+77*(-d/c)^(1/2)*x^3*a*c^2*d*f^4-28*(-d/c
)^(1/2)*x^3*a*d^3*e^2*f^2+24*(-d/c)^(1/2)*x^3*b*d^3*e^3*f+45*(-d/c)^(1/2)*x*b*c^
3*e*f^3+45*(-d/c)^(1/2)*x^3*b*c^3*f^4+15*(-d/c)^(1/2)*x^9*b*d^3*f^4+105*((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^3*
f^4+77*(-d/c)^(1/2)*x*a*c^2*d*e*f^3-28*(-d/c)^(1/2)*x*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*d^3
*e^3*f-60*((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^3-61*(-d/c)^(1/2)*x*b*c^2*d*e^2*f^2+24*(-d/c)^(1/2)*x*b*c*
d^2*e^3*f+56*((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+15*((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^4-238*((
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^3-103*((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+161*((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^3-161*((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+18
9*((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+164*((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^2-152*((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-19*(-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+29*(-d/c)^(1/2)*x^3*b*c^2*d*e*
f^3-55*(-d/c)^(1/2)*x^3*b*c*d^2*e^2*f^2)/f^4/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/(-d/c
)^(1/2)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)*(d*x**2+c)**(5/2)/(f*x**2+e)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

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)