3.658 \(\int \frac{1}{x^5 \left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=185 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^4} (2 b c-a d)}{4 a^2 c \left (a+b x^4\right ) (b c-a d)}-\frac{\sqrt{c+d x^4}}{4 a c x^4 \left (a+b x^4\right )} \]

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^4])/(4*a^2*c*(b*c - a*d)*(a + b*x^4)) - Sqrt[c +
d*x^4]/(4*a*c*x^4*(a + b*x^4)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]
)/(4*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/S
qrt[b*c - a*d]])/(4*a^3*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.648915, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^4} (2 b c-a d)}{4 a^2 c \left (a+b x^4\right ) (b c-a d)}-\frac{\sqrt{c+d x^4}}{4 a c x^4 \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^4])/(4*a^2*c*(b*c - a*d)*(a + b*x^4)) - Sqrt[c +
d*x^4]/(4*a*c*x^4*(a + b*x^4)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]
)/(4*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/S
qrt[b*c - a*d]])/(4*a^3*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 75.4899, size = 158, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{4}}}{4 a c x^{4} \left (a + b x^{4}\right )} - \frac{b \sqrt{c + d x^{4}} \left (a d - 2 b c\right )}{4 a^{2} c \left (a + b x^{4}\right ) \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{4 a^{3} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\left (a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{4 a^{3} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**4)/(4*a*c*x**4*(a + b*x**4)) - b*sqrt(c + d*x**4)*(a*d - 2*b*c)/(
4*a**2*c*(a + b*x**4)*(a*d - b*c)) + b**(3/2)*(5*a*d - 4*b*c)*atan(sqrt(b)*sqrt(
c + d*x**4)/sqrt(a*d - b*c))/(4*a**3*(a*d - b*c)**(3/2)) + (a*d + 4*b*c)*atanh(s
qrt(c + d*x**4)/sqrt(c))/(4*a**3*c**(3/2))

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Mathematica [C]  time = 1.21083, size = 489, normalized size = 2.64 \[ \frac{\frac{5 b d x^4 \left (-a^2 d \left (3 c+2 d x^4\right )+3 a b \left (c^2+c d x^4-d^2 x^8\right )+2 b^2 c x^4 \left (c+3 d x^4\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+3 \left (c+d x^4\right ) \left (a^2 d+a b \left (d x^4-c\right )-2 b^2 c x^4\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}{c (b c-a d) \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}+\frac{6 a b d x^8 (a d-2 b c) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{(a d-b c) \left (x^4 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}}{12 a^2 x^4 \left (a+b x^4\right ) \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [B]  time = 0.02, size = 938, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/a^2/c/x^4*(d*x^4+c)^(1/2)+1/4/a^2*d/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^4+c)^(1/
2))/x^2)-1/8/a^3*b*(-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^
2*b*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)))+1/8/a
^3*b*(-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^2*b*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)))+b/a^3/c^(1/2)*ln((2
*c+2*c^(1/2)*(d*x^4+c)^(1/2))/x^2)-1/2/a^3*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)))-1/2/a^3*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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^5), x)

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Fricas [A]  time = 0.358729, 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^5),x, algorithm="fricas")

[Out]

[1/8*(((4*b^3*c^2 - 5*a*b^2*c*d)*x^8 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^4)*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)*s
qrt(b/(b*c - a*d)))/(b*x^4 + a)) - 2*((2*a*b^2*c - a^2*b*d)*x^4 + a^2*b*c - a^3*
d)*sqrt(d*x^4 + c)*sqrt(c) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^8 + (4*a*b
^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^4)*log(((d*x^4 + 2*c)*sqrt(c) + 2*sqrt(d*x^4 +
c)*c)/x^4))/(((a^3*b^2*c^2 - a^4*b*c*d)*x^8 + (a^4*b*c^2 - a^5*c*d)*x^4)*sqrt(c
)), -1/8*(2*((4*b^3*c^2 - 5*a*b^2*c*d)*x^8 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^4)*sq
rt(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*((2*a*b^2*c - a^2*b*d)*x^4 + a^2*b*c - a^3*d)*sqrt(d*x^4 + c)*sqrt(
c) - ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^8 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a
^3*d^2)*x^4)*log(((d*x^4 + 2*c)*sqrt(c) + 2*sqrt(d*x^4 + c)*c)/x^4))/(((a^3*b^2*
c^2 - a^4*b*c*d)*x^8 + (a^4*b*c^2 - a^5*c*d)*x^4)*sqrt(c)), 1/8*(((4*b^3*c^2 - 5
*a*b^2*c*d)*x^8 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^4)*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*((2*a*b^2*c - a^2*b*d)*x^4 + a^2*b*c - a^3*d)*sqrt(d*x^4 + c)*s
qrt(-c) - 2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^8 + (4*a*b^2*c^2 - 3*a^2*b*
c*d - a^3*d^2)*x^4)*arctan(c/(sqrt(d*x^4 + c)*sqrt(-c))))/(((a^3*b^2*c^2 - a^4*b
*c*d)*x^8 + (a^4*b*c^2 - a^5*c*d)*x^4)*sqrt(-c)), -1/4*(((4*b^3*c^2 - 5*a*b^2*c*
d)*x^8 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^4)*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*a*b^2*c - a^2*b*d)*x
^4 + a^2*b*c - a^3*d)*sqrt(d*x^4 + c)*sqrt(-c) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2
*b*d^2)*x^8 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^4)*arctan(c/(sqrt(d*x^4 +
c)*sqrt(-c))))/(((a^3*b^2*c^2 - a^4*b*c*d)*x^8 + (a^4*b*c^2 - a^5*c*d)*x^4)*sqrt
(-c))]

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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**5/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222698, size = 362, normalized size = 1.96 \[ \frac{1}{4} \, d^{3}{\left (\frac{{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{4} + c\right )}^{\frac{3}{2}} b^{2} c - 2 \, \sqrt{d x^{4} + c} b^{2} c^{2} -{\left (d x^{4} + c\right )}^{\frac{3}{2}} a b d + 2 \, \sqrt{d x^{4} + c} a b c d - \sqrt{d x^{4} + c} a^{2} d^{2}}{{\left (a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}{\left ({\left (d x^{4} + c\right )}^{2} b - 2 \,{\left (d x^{4} + c\right )} b c + b c^{2} +{\left (d x^{4} + c\right )} a d - a c d\right )}} - \frac{{\left (4 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^5),x, algorithm="giac")

[Out]

1/4*d^3*((4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/((
a^3*b*c*d^3 - a^4*d^4)*sqrt(-b^2*c + a*b*d)) - (2*(d*x^4 + c)^(3/2)*b^2*c - 2*sq
rt(d*x^4 + c)*b^2*c^2 - (d*x^4 + c)^(3/2)*a*b*d + 2*sqrt(d*x^4 + c)*a*b*c*d - sq
rt(d*x^4 + c)*a^2*d^2)/((a^2*b*c^2*d^2 - a^3*c*d^3)*((d*x^4 + c)^2*b - 2*(d*x^4
+ c)*b*c + b*c^2 + (d*x^4 + c)*a*d - a*c*d)) - (4*b*c + a*d)*arctan(sqrt(d*x^4 +
c)/sqrt(-c))/(a^3*sqrt(-c)*c*d^3))