3.639 \(\int \frac{1}{x \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.208555, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.4942, size = 71, normalized size = 0.84 \[ - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a \sqrt{a d - b c}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*atan(sqrt(b)*sqrt(c + d*x**4)/sqrt(a*d - b*c))/(2*a*sqrt(a*d - b*c)) -
atanh(sqrt(c + d*x**4)/sqrt(c))/(2*a*sqrt(c))

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Mathematica [C]  time = 0.0764951, size = 162, normalized size = 1.91 \[ \frac{5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )}{6 \left (a+b x^4\right ) \sqrt{c+d x^4} \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*b*d*x^4*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^4)), -(a/(b*x^4))])/(6*(a + b*x^4
)*Sqrt[c + d*x^4]*(-5*b*d*x^4*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^4)), -(a/(b*x^
4))] + 2*a*d*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^4)), -(a/(b*x^4))] + b*c*Appell
F1[5/2, 3/2, 1, 7/2, -(c/(d*x^4)), -(a/(b*x^4))]))

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Maple [B]  time = 0.021, size = 347, normalized size = 4.1 \[ -{\frac{1}{2\,a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{4}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{1}{4\,a}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{1}{4\,a}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x), x)

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Fricas [A]  time = 0.26572, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{4} + a}\right ) + \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right )}{4 \, a \sqrt{c}}, \frac{2 \, \sqrt{c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{4} + c} b}\right ) + \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right )}{4 \, a \sqrt{c}}, \frac{\sqrt{-c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{4} + a}\right ) + 2 \, \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right )}{4 \, a \sqrt{-c}}, \frac{\sqrt{-c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{4} + c} b}\right ) + \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right )}{2 \, a \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x),x, algorithm="fricas")

[Out]

[1/4*(sqrt(c)*sqrt(b/(b*c - a*d))*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)
*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^4 + a)) + log(((d*x^4 + 2*c)*sqrt(c) - 2*
sqrt(d*x^4 + c)*c)/x^4))/(a*sqrt(c)), 1/4*(2*sqrt(c)*sqrt(-b/(b*c - a*d))*arctan
(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^4 + c)*b)) + log(((d*x^4 + 2*c)*sqr
t(c) - 2*sqrt(d*x^4 + c)*c)/x^4))/(a*sqrt(c)), 1/4*(sqrt(-c)*sqrt(b/(b*c - a*d))
*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))
/(b*x^4 + a)) + 2*arctan(c/(sqrt(d*x^4 + c)*sqrt(-c))))/(a*sqrt(-c)), 1/2*(sqrt(
-c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^4 +
c)*b)) + arctan(c/(sqrt(d*x^4 + c)*sqrt(-c))))/(a*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.21277, size = 107, normalized size = 1.26 \[ -\frac{1}{2} \, d{\left (\frac{b \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a d} - \frac{\arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x),x, algorithm="giac")

[Out]

-1/2*d*(b*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a
*d) - arctan(sqrt(d*x^4 + c)/sqrt(-c))/(a*sqrt(-c)*d))