3.623 \(\int \frac{\sqrt{c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{c+d x^4}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.253304, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{c+d x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.6608, size = 63, normalized size = 0.83 \[ - \frac{\sqrt{c + d x^{4}}}{2 a x^{2}} + \frac{\sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**4+c)**(1/2)/x**3/(b*x**4+a),x)

[Out]

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

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Mathematica [A]  time = 1.21772, size = 136, normalized size = 1.79 \[ \frac{\sqrt{c+d x^4} \left (-\frac{x^4 (b c-a d) \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{c \sqrt{\frac{b x^4}{a}+1} \sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}-a\right )}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]

[Out]

(Sqrt[c + d*x^4]*(-a - ((b*c - a*d)*x^4*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b
*x^4)/a]])/(c*Sqrt[(b/a - d/c)*x^4]*Sqrt[1 + (b*x^4)/a]*Sqrt[(a*(c + d*x^4))/(c*
(a + b*x^4))])))/(2*a^2*x^2)

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Maple [B]  time = 0.021, size = 1075, normalized size = 14.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x)

[Out]

-1/2/a/c/x^2*(d*x^4+c)^(3/2)+1/2/a*d/c*x^2*(d*x^4+c)^(1/2)+1/2/a*d^(1/2)*ln(x^2*
d^(1/2)+(d*x^4+c)^(1/2))-1/4*b/a/(-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/a*d^(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/(-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)))*d+1/4*b/a/(-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*b/a/(-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/a*d^(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/(-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)))*
d-1/4*b/a/(-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. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3), x)

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Fricas [A]  time = 0.233045, size = 1, normalized size = 0.01 \[ \left [\frac{x^{2} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) - 4 \, \sqrt{d x^{4} + c}}{8 \, a x^{2}}, \frac{x^{2} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} a x^{2} \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \, \sqrt{d x^{4} + c}}{4 \, a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3),x, algorithm="fricas")

[Out]

[1/8*(x^2*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)) - 4*sqrt(d*x^4 +
c))/(a*x^2), 1/4*(x^2*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))) - 2*sqrt(d*x^4 + c))/(a*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{x^{3} \left (a + b x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**4+c)**(1/2)/x**3/(b*x**4+a),x)

[Out]

Integral(sqrt(c + d*x**4)/(x**3*(a + b*x**4)), x)

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GIAC/XCAS [A]  time = 0.218513, size = 89, normalized size = 1.17 \[ \frac{{\left (b c - a d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{2 \, \sqrt{a b c - a^{2} d} a} - \frac{\sqrt{d + \frac{c}{x^{4}}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3),x, algorithm="giac")

[Out]

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