3.657 \(\int \frac{1}{x \left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=132 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.384089, antiderivative size = 132, 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 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.7179, size = 114, normalized size = 0.86 \[ - \frac{b \sqrt{c + d x^{4}}}{4 a \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{\sqrt{b} \left (\frac{3 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a^{2} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.442581, size = 396, normalized size = 3. \[ \frac{b \left (\frac{6 c d x^4 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{x^4 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}+\frac{5 d x^4 \left (2 a d+b \left (c+3 d x^4\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-3 \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}{a \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}\right )}{12 \left (a+b x^4\right ) \sqrt{c+d x^4} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(b*((6*c*d*x^4*AppellF1[1, 1/2, 1, 2, -((d*x^4)/c), -((b*x^4)/a)])/(-4*a*c*Appel
lF1[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*d*x^4*(2*a*d + b*(c + 3*d*x^4))*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*
x^4)), -(a/(b*x^4))] - 3*(c + d*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))])
)/(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*(-(b*c) + a*d)*(a + b*x^4)*Sqrt[c +
d*x^4])

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 880, normalized size = 6.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/a^2/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^4+c)^(1/2))/x^2)+1/8/a^2*(-a*b)^(1/2)/(a
*d-b*c)/(x^2-1/b*(-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)+1/8/a*d/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*l
n((-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)))-1/8/a^2*(-a*b)^(1/2)/(a*d-b*c)/(x^2+1/b*(-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)+1/8/a*d/(a*d-b*c)/(-(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^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
)))+1/4/a^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)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.286573, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215774, size = 207, normalized size = 1.57 \[ -\frac{1}{4} \, d^{2}{\left (\frac{{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x^{4} + c} b}{{\left (a b c d - a^{2} d^{2}\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x),x, algorithm="giac")

[Out]

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