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

Optimal. Leaf size=621 \[ \frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}} \]

[Out]

(d*(b*c - a*d)*x*Sqrt[c + d*x^2])/(b^2*Sqrt[e + f*x^2]) - (2*d*(d*e - 2*c*f)*x*S
qrt[c + d*x^2])/(3*b*f*Sqrt[e + f*x^2]) + (d^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]
)/(3*b*f) - (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/
Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*S
qrt[e + f*x^2]) + (2*d*Sqrt[e]*(d*e - 2*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e +
f*x^2))]*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[Arc
Tan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c
*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[e]*(d*e - 3*c*f)*Sqrt[c + d*x^2]*Ellip
ticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*f^(3/2)*Sqrt[(e*(c + d*
x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)^2*Sqrt[e + f*x^2]
*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b
^2*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])

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Rubi [A]  time = 1.43776, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*c - a*d)*x*Sqrt[c + d*x^2])/(b^2*Sqrt[e + f*x^2]) - (2*d*(d*e - 2*c*f)*x*S
qrt[c + d*x^2])/(3*b*f*Sqrt[e + f*x^2]) + (d^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]
)/(3*b*f) - (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/
Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*S
qrt[e + f*x^2]) + (2*d*Sqrt[e]*(d*e - 2*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e +
f*x^2))]*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[Arc
Tan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c
*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[e]*(d*e - 3*c*f)*Sqrt[c + d*x^2]*Ellip
ticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*f^(3/2)*Sqrt[(e*(c + d*
x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)^2*Sqrt[e + f*x^2]
*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b
^2*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])

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Rubi in Sympy [A]  time = 153.277, size = 552, normalized size = 0.89 \[ \frac{c^{\frac{3}{2}} \sqrt{d} \sqrt{e + f x^{2}} \left (3 c f - d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 b e f \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{2 \sqrt{c} d^{\frac{3}{2}} \sqrt{e + f x^{2}} \left (2 c f - d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 b f^{2} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{d^{2} x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{3 b f} + \frac{2 d^{2} x \sqrt{e + f x^{2}} \left (2 c f - d e\right )}{3 b f^{2} \sqrt{c + d x^{2}}} - \frac{c^{\frac{3}{2}} \sqrt{d} \sqrt{e + f x^{2}} \left (a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{b^{2} e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{\sqrt{c} d^{\frac{3}{2}} \sqrt{e + f x^{2}} \left (a d - b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{b^{2} f \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{d^{2} x \sqrt{e + f x^{2}} \left (a d - b c\right )}{b^{2} f \sqrt{c + d x^{2}}} + \frac{c^{\frac{3}{2}} \sqrt{e + f x^{2}} \left (a d - b c\right )^{2} \Pi \left (1 - \frac{b c}{a d}; \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{a b^{2} \sqrt{d} e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**(3/2)*sqrt(d)*sqrt(e + f*x**2)*(3*c*f - d*e)*elliptic_f(atan(sqrt(d)*x/sqrt(c
)), -c*f/(d*e) + 1)/(3*b*e*f*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x*
*2)) - 2*sqrt(c)*d**(3/2)*sqrt(e + f*x**2)*(2*c*f - d*e)*elliptic_e(atan(sqrt(d)
*x/sqrt(c)), -c*f/(d*e) + 1)/(3*b*f**2*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqr
t(c + d*x**2)) + d**2*x*sqrt(c + d*x**2)*sqrt(e + f*x**2)/(3*b*f) + 2*d**2*x*sqr
t(e + f*x**2)*(2*c*f - d*e)/(3*b*f**2*sqrt(c + d*x**2)) - c**(3/2)*sqrt(d)*sqrt(
e + f*x**2)*(a*d - b*c)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(b**
2*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)) + sqrt(c)*d**(3/2)*s
qrt(e + f*x**2)*(a*d - b*c)*elliptic_e(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/
(b**2*f*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)) - d**2*x*sqrt(e
+ f*x**2)*(a*d - b*c)/(b**2*f*sqrt(c + d*x**2)) + c**(3/2)*sqrt(e + f*x**2)*(a*d
- b*c)**2*elliptic_pi(1 - b*c/(a*d), atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(
a*b**2*sqrt(d)*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2))

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Mathematica [C]  time = 2.2309, size = 350, normalized size = 0.56 \[ \frac{-i a d \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (3 a^2 d^2 f^2+3 a b d f (d e-3 c f)+b^2 \left (9 c^2 f^2-8 c d e f+2 d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f \left (a b^2 c d x \left (\frac{d}{c}\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )-3 i f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i a b d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+7 b c f-2 b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a b^3 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*a*b*d^2*e*(-2*b*d*e + 7*b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x
^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d*(3*a^2*d^2*f^2 + 3
*a*b*d*f*(d*e - 3*c*f) + b^2*(2*d^2*e^2 - 8*c*d*e*f + 9*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)] + f*(a*
b^2*c*d*(d/c)^(3/2)*x*(c + d*x^2)*(e + f*x^2) - (3*I)*(b*c - a*d)^3*f*Sqrt[1 + (
d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c
*f)/(d*e)]))/(3*a*b^3*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.036, size = 988, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((-d/c)^(1/2)*x^5*a*b^2*d^3*f^2+(-d/c)^(1/2)*x^3*a*b^2*c*d^2*f^2+(-d/c)^(1/2
)*x^3*a*b^2*d^3*e*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))*a^3*d^3*f^2-9*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*El
lipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b*c*d^2*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))*a^2*b*d^3*e*f+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*
b^2*c^2*d*f^2-8*((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))*a*b^2*c*d^2*e*f+2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*d^3*e^2-3*((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^2*b*d^3*e*f+7*((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*b^2*
c*d^2*e*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))*a*b^2*d^3*e^2-3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi
(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^3*d^3*f^2+9*((d*x^2+c)/c)^(
1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(
1/2))*a^2*b*c*d^2*f^2-9*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d
/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b^2*c^2*d*f^2+3*((d*x^2+c)/c)^(1/
2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/
2))*b^3*c^3*f^2+(-d/c)^(1/2)*x*a*b^2*c*d^2*e*f)*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/
a/(-d/c)^(1/2)/f^2/b^3/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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