3.624 \(\int \frac{\sqrt{c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx\)

Optimal. Leaf size=115 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a^2}-\frac{\sqrt{c+d x^4}}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.384249, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a^2}-\frac{\sqrt{c+d x^4}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.369, size = 100, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{4}}}{4 a x^{4}} - \frac{\sqrt{b} \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a^{2}} - \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a^{2} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**4+c)**(1/2)/x**5/(b*x**4+a),x)

[Out]

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

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Mathematica [C]  time = 0.956401, size = 407, normalized size = 3.54 \[ \frac{\frac{6 b c d x^8 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{x^4 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}+\frac{5 b d x^4 \left (3 a c+4 a d x^4+b c x^4+3 b d x^8\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-3 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}{a \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}}{12 x^4 \left (a+b x^4\right ) \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]

[Out]

((6*b*c*d*x^8*AppellF1[1, 1/2, 1, 2, -((d*x^4)/c), -((b*x^4)/a)])/(-4*a*c*Appell
F1[1, 1/2, 1, 2, -((d*x^4)/c), -((b*x^4)/a)] + x^4*(2*b*c*AppellF1[2, 1/2, 2, 3,
-((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[2, 3/2, 1, 3, -((d*x^4)/c), -((b*x^4
)/a)])) + (5*b*d*x^4*(3*a*c + b*c*x^4 + 4*a*d*x^4 + 3*b*d*x^8)*AppellF1[3/2, 1/2
, 1, 5/2, -(c/(d*x^4)), -(a/(b*x^4))] - 3*(a + b*x^4)*(c + d*x^4)*(2*a*d*AppellF
1[5/2, 1/2, 2, 7/2, -(c/(d*x^4)), -(a/(b*x^4))] + b*c*AppellF1[5/2, 3/2, 1, 7/2,
-(c/(d*x^4)), -(a/(b*x^4))]))/(a*(-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*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^4)), -(a/(b*x^4))])))/(12*x^4*(a +
b*x^4)*Sqrt[c + d*x^4])

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Maple [B]  time = 0.023, size = 1107, normalized size = 9.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5), x)

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Fricas [A]  time = 0.243514, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} c - a b d} \sqrt{c} x^{4} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{b x^{4} + a}\right ) -{\left (2 \, b c - a d\right )} x^{4} \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right ) - 2 \, \sqrt{d x^{4} + c} a \sqrt{c}}{8 \, a^{2} \sqrt{c} x^{4}}, \frac{4 \, \sqrt{-b^{2} c + a b d} \sqrt{c} x^{4} \arctan \left (\frac{\sqrt{-b^{2} c + a b d}}{\sqrt{d x^{4} + c} b}\right ) -{\left (2 \, b c - a d\right )} x^{4} \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right ) - 2 \, \sqrt{d x^{4} + c} a \sqrt{c}}{8 \, a^{2} \sqrt{c} x^{4}}, -\frac{{\left (2 \, b c - a d\right )} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right ) - \sqrt{b^{2} c - a b d} \sqrt{-c} x^{4} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{b x^{4} + a}\right ) + \sqrt{d x^{4} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} x^{4}}, -\frac{{\left (2 \, b c - a d\right )} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right ) - 2 \, \sqrt{-b^{2} c + a b d} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-b^{2} c + a b d}}{\sqrt{d x^{4} + c} b}\right ) + \sqrt{d x^{4} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{x^{5} \left (a + b x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**4+c)**(1/2)/x**5/(b*x**4+a),x)

[Out]

Integral(sqrt(c + d*x**4)/(x**5*(a + b*x**4)), x)

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GIAC/XCAS [A]  time = 0.215831, size = 163, normalized size = 1.42 \[ \frac{1}{4} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c - a b d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{4} + c}}{a d^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5),x, algorithm="giac")

[Out]

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