3.976 \(\int \frac{x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=261 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x \sqrt{a+b x^2} (a d+b c)}{3 b^2 d \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 b d} \]

[Out]

(-2*(b*c + a*d)*x*Sqrt[a + b*x^2])/(3*b^2*d*Sqrt[c + d*x^2]) + (x*Sqrt[a + b*x^2
]*Sqrt[c + d*x^2])/(3*b*d) + (2*Sqrt[c]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[Ar
cTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))
/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c +
d*x^2))]*Sqrt[c + d*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.502498, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x \sqrt{a+b x^2} (a d+b c)}{3 b^2 d \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

(-2*(b*c + a*d)*x*Sqrt[a + b*x^2])/(3*b^2*d*Sqrt[c + d*x^2]) + (x*Sqrt[a + b*x^2
]*Sqrt[c + d*x^2])/(3*b*d) + (2*Sqrt[c]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[Ar
cTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))
/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c +
d*x^2))]*Sqrt[c + d*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 63.58, size = 226, normalized size = 0.87 \[ - \frac{a^{\frac{3}{2}} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 b^{\frac{3}{2}} d \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{a} \sqrt{c + d x^{2}} \left (a d + b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 b^{\frac{3}{2}} d^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{3 b d} - \frac{2 x \sqrt{c + d x^{2}} \left (a d + b c\right )}{3 b d^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.474805, size = 201, normalized size = 0.77 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 b d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) + (2*I)*c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]
*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(2*b*c
+ a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x]
, (a*d)/(b*c)])/(3*b*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.033, size = 333, normalized size = 1.3 \[{\frac{1}{3\,{d}^{2}b \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( \sqrt{-{\frac{b}{a}}}{x}^{5}b{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}a{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}bcd+ac\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) d+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) b{c}^{2}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) acd-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) b{c}^{2}+\sqrt{-{\frac{b}{a}}}xacd \right ) \sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**4/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)