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

Optimal. Leaf size=91 \[ \frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 \sqrt{a} b}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.206905, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 \sqrt{a} b}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.265, size = 76, normalized size = 0.84 \[ \frac{\sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c + d x^{4}}} \right )}}{2 b} - \frac{\sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 \sqrt{a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0558978, size = 89, normalized size = 0.98 \[ \frac{\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{\sqrt{a}}+\sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^4}+d x^2\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 1000, normalized size = 11. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/(-a*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)+1/4*d^(1/2)/b*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/(-a*b)^(1/2)/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)))*a*d-1/4/(-a*b)^(1/2)/(-(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*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)+1/4*d^(1/2)/b*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/(-a*b)^(1/2)/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)))*a*d+1/4/(-a*b)^(1/2)/(-(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253652, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d} \log \left (-2 \, d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{d} x^{2} - c\right ) + \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \,{\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b}, \frac{4 \, \sqrt{-d} \arctan \left (\frac{d x^{2}}{\sqrt{d x^{4} + c} \sqrt{-d}}\right ) + \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \,{\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b}, -\frac{\sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} a x^{2} \sqrt{\frac{b c - a d}{a}}}\right ) - \sqrt{d} \log \left (-2 \, d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{d} x^{2} - c\right )}{4 \, b}, \frac{2 \, \sqrt{-d} \arctan \left (\frac{d x^{2}}{\sqrt{d x^{4} + c} \sqrt{-d}}\right ) - \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} a x^{2} \sqrt{\frac{b c - a d}{a}}}\right )}{4 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227485, size = 157, normalized size = 1.73 \[ -\frac{{\left (b c \sqrt{d} - a d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt{a b c d - a^{2} d^{2}} b} - \frac{\sqrt{d}{\rm ln}\left ({\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2}\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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