3.975 \(\int \frac{x^6}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=342 \[ \frac{x \sqrt{a+b x^2} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{15 b^3 d^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^3 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{4 c^{3/2} \sqrt{a+b x^2} (a d+b c) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 x \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{15 b^2 d^2}+\frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 b d} \]

[Out]

((8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^3*d^2*Sqrt[c + d*x
^2]) - (4*(b*c + a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b^2*d^2) + (x^3*Sqr
t[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d) - (Sqrt[c]*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*
d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(1
5*b^3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (4*c^(3/2
)*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(
a*d)])/(15*b^2*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.854287, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x \sqrt{a+b x^2} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{15 b^3 d^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^3 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{4 c^{3/2} \sqrt{a+b x^2} (a d+b c) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 x \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{15 b^2 d^2}+\frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

((8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^3*d^2*Sqrt[c + d*x
^2]) - (4*(b*c + a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b^2*d^2) + (x^3*Sqr
t[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d) - (Sqrt[c]*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*
d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(1
5*b^3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (4*c^(3/2
)*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(
a*d)])/(15*b^2*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 97.8215, size = 313, normalized size = 0.92 \[ \frac{4 a^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d + b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 b^{\frac{5}{2}} d^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{\sqrt{a} \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} + 7 a b c d + 8 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 b^{\frac{5}{2}} d^{3} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{x^{3} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{5 b d} - \frac{4 x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d + b c\right )}{15 b^{2} d^{2}} + \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} + 7 a b c d + 8 b^{2} c^{2}\right )}{15 b^{2} d^{3} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

4*a**(3/2)*sqrt(c + d*x**2)*(a*d + b*c)*elliptic_f(atan(sqrt(b)*x/sqrt(a)), -a*d
/(b*c) + 1)/(15*b**(5/2)*d**2*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x
**2)) - sqrt(a)*sqrt(c + d*x**2)*(8*a**2*d**2 + 7*a*b*c*d + 8*b**2*c**2)*ellipti
c_e(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(15*b**(5/2)*d**3*sqrt(a*(c + d*x**
2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)) + x**3*sqrt(a + b*x**2)*sqrt(c + d*x**2)/
(5*b*d) - 4*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(a*d + b*c)/(15*b**2*d**2) + x*s
qrt(c + d*x**2)*(8*a**2*d**2 + 7*a*b*c*d + 8*b**2*c**2)/(15*b**2*d**3*sqrt(a + b
*x**2))

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Mathematica [C]  time = 0.827473, size = 249, normalized size = 0.73 \[ \frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \left (-\sqrt{\frac{b}{a}}\right ) \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d+4 b c-3 b d x^2\right )}{15 a^2 d^3 \left (\frac{b}{a}\right )^{5/2} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(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)*(4*b*c + 4*a*d - 3*b*d*x^2)) - I*c*(8*b
^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic
E[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(8*b^2*c^2 + 3*a*b*c*d + 4*a^2*d^2)
*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)
/(b*c)])/(15*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.036, size = 546, normalized size = 1.6 \[ -{\frac{1}{15\,{b}^{2}{d}^{3} \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( -3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{2}{d}^{3}+\sqrt{-{\frac{b}{a}}}{x}^{5}ab{d}^{3}+\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{2}c{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}{d}^{3}+5\,\sqrt{-{\frac{b}{a}}}{x}^{3}abc{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{2}{c}^{2}d+4\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d+8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}-7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}+4\,\sqrt{-{\frac{b}{a}}}x{a}^{2}c{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}xab{c}^{2}d \right ) \sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/15*(-3*(-b/a)^(1/2)*x^7*b^2*d^3+(-b/a)^(1/2)*x^5*a*b*d^3+(-b/a)^(1/2)*x^5*b^2
*c*d^2+4*(-b/a)^(1/2)*x^3*a^2*d^3+5*(-b/a)^(1/2)*x^3*a*b*c*d^2+4*(-b/a)^(1/2)*x^
3*b^2*c^2*d+4*((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))*a^2*c*d^2+3*((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))*a*b*c^2*d+8*((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^2*c^3-8*((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^2*c*d^2-7*((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*b*c^2*d-
8*((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^2*c^3+4*(-b/a)^(1/2)*x*a^2*c*d^2+4*(-b/a)^(1/2)*x*a*b*c^2*d)*(b*x^2+a)^(1/
2)*(d*x^2+c)^(1/2)/d^3/b^2/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

integral(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**6/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)