3.974 \(\int \frac{1}{x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=149 \[ \frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 a^2 c^2 x^2}-\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} c^{5/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{4 a c x^4} \]

[Out]

-(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(4*a*c*x^4) + (3*(b*c + a*d)*Sqrt[a + b*x^2]*
Sqrt[c + d*x^2])/(8*a^2*c^2*x^2) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTanh[
(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.430652, antiderivative size = 149, 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 \[ \frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 a^2 c^2 x^2}-\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} c^{5/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

-(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(4*a*c*x^4) + (3*(b*c + a*d)*Sqrt[a + b*x^2]*
Sqrt[c + d*x^2])/(8*a^2*c^2*x^2) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTanh[
(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.0844, size = 131, normalized size = 0.88 \[ - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{4 a c x^{4}} + \frac{3 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d + b c\right )}{8 a^{2} c^{2} x^{2}} + \frac{\left (a b c d - \frac{3 \left (a d + b c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**2)*sqrt(c + d*x**2)/(4*a*c*x**4) + 3*sqrt(a + b*x**2)*sqrt(c + d*
x**2)*(a*d + b*c)/(8*a**2*c**2*x**2) + (a*b*c*d - 3*(a*d + b*c)**2/4)*atanh(sqrt
(c)*sqrt(a + b*x**2)/(sqrt(a)*sqrt(c + d*x**2)))/(2*a**(5/2)*c**(5/2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.379321, size = 224, normalized size = 1.5 \[ \frac{\frac{2 b d x^6 \left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}+\left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 a c+3 a d x^2+3 b c x^2\right )}{8 a^2 c^2 x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

((a + b*x^2)*(c + d*x^2)*(-2*a*c + 3*b*c*x^2 + 3*a*d*x^2) + (2*b*d*(3*b^2*c^2 +
2*a*b*c*d + 3*a^2*d^2)*x^6*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^2)), -(c/(d*x^2))])
/(-4*b*d*x^2*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^2)), -(c/(d*x^2))] + b*c*AppellF1
[2, 1/2, 3/2, 3, -(a/(b*x^2)), -(c/(d*x^2))] + a*d*AppellF1[2, 3/2, 1/2, 3, -(a/
(b*x^2)), -(c/(d*x^2))]))/(8*a^2*c^2*x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.034, size = 355, normalized size = 2.4 \[ -{\frac{1}{16\,{a}^{2}{c}^{2}{x}^{4}} \left ( 3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{a}^{2}{d}^{2}+2\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}abcd+3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{b}^{2}{c}^{2}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}da{x}^{2}\sqrt{ac}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}bc{x}^{2}\sqrt{ac}+4\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}ca\sqrt{ac} \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16/a^2/c^2*(3*ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)
^(1/2)+2*a*c)/x^2)*x^4*a^2*d^2+2*ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/2)*(b*d*x^4+a*d*
x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)*x^4*a*b*c*d+3*ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/
2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)*x^4*b^2*c^2-6*(b*d*x^4+a*d*x^
2+b*c*x^2+a*c)^(1/2)*d*a*x^2*(a*c)^(1/2)-6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*b
*c*x^2*(a*c)^(1/2)+4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*c*a*(a*c)^(1/2))*(d*x^2
+c)^(1/2)*(b*x^2+a)^(1/2)/(a*c)^(1/2)/x^4/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.323605, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} -{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{a c}}{x^{4}}\right ) + 4 \,{\left (3 \,{\left (b c + a d\right )} x^{2} - 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{a c}}{32 \, \sqrt{a c} a^{2} c^{2} x^{4}}, -\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} a c}\right ) - 2 \,{\left (3 \,{\left (b c + a d\right )} x^{2} - 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-a c}}{16 \, \sqrt{-a c} a^{2} c^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^5),x, algorithm="fricas")

[Out]

[1/32*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*x^4*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a
^2*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c) - ((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*
x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2)*sqrt(a*c))/x^4) + 4*(3*(b*c + a*d)*
x^2 - 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a*c))/(sqrt(a*c)*a^2*c^2*x^4),
-1/16*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*x^4*arctan(1/2*((b*c + a*d)*x^2 + 2*
a*c)*sqrt(-a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a*c)) - 2*(3*(b*c + a*d)*x^2 -
2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a*c))/(sqrt(-a*c)*a^2*c^2*x^4)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**5*sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError