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

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

[Out]

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

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Rubi [A]  time = 0.223052, 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 \[ \frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.5395, size = 70, normalized size = 0.82 \[ - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a} + \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.287604, size = 162, normalized size = 1.91 \[ -\frac{3 b d x^4 \sqrt{c+d x^4} F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )}{2 \left (a+b x^4\right ) \left (3 b d x^4 F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-2 a d F_1\left (\frac{3}{2};-\frac{1}{2},2;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*b*d*x^4*Sqrt[c + d*x^4]*AppellF1[1/2, -1/2, 1, 3/2, -(c/(d*x^4)), -(a/(b*x^4
))])/(2*(a + b*x^4)*(3*b*d*x^4*AppellF1[1/2, -1/2, 1, 3/2, -(c/(d*x^4)), -(a/(b*
x^4))] - 2*a*d*AppellF1[3/2, -1/2, 2, 5/2, -(c/(d*x^4)), -(a/(b*x^4))] + b*c*App
ellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^4)), -(a/(b*x^4))]))

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Maple [B]  time = 0.022, size = 1037, normalized size = 12.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.235215, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right ) + \sqrt{c} \log \left (\frac{d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{c} + 2 \, c}{x^{4}}\right )}{4 \, a}, \frac{2 \, \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) + \sqrt{c} \log \left (\frac{d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{c} + 2 \, c}{x^{4}}\right )}{4 \, a}, -\frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right ) - \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right )}{4 \, a}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right ) - \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223032, size = 117, normalized size = 1.38 \[ -\frac{1}{2} \, d{\left (\frac{{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a d} - \frac{c \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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