3.743 \(\int \frac{1}{x^9 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=185 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{8 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^8} (2 b c-a d)}{8 a^2 c \left (a+b x^8\right ) (b c-a d)}-\frac{\sqrt{c+d x^8}}{8 a c x^8 \left (a+b x^8\right )} \]

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*(a + b*x^8)) - Sqrt[c +
d*x^8]/(8*a*c*x^8*(a + b*x^8)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]]
)/(8*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/S
qrt[b*c - a*d]])/(8*a^3*(b*c - a*d)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.720181, 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 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{8 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^8} (2 b c-a d)}{8 a^2 c \left (a+b x^8\right ) (b c-a d)}-\frac{\sqrt{c+d x^8}}{8 a c x^8 \left (a+b x^8\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*(a + b*x^8)) - Sqrt[c +
d*x^8]/(8*a*c*x^8*(a + b*x^8)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]]
)/(8*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/S
qrt[b*c - a*d]])/(8*a^3*(b*c - a*d)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.0361, size = 158, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{8}}}{8 a c x^{8} \left (a + b x^{8}\right )} - \frac{b \sqrt{c + d x^{8}} \left (a d - 2 b c\right )}{8 a^{2} c \left (a + b x^{8}\right ) \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{8 a^{3} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\left (a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{8}}}{\sqrt{c}} \right )}}{8 a^{3} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

-sqrt(c + d*x**8)/(8*a*c*x**8*(a + b*x**8)) - b*sqrt(c + d*x**8)*(a*d - 2*b*c)/(
8*a**2*c*(a + b*x**8)*(a*d - b*c)) + b**(3/2)*(5*a*d - 4*b*c)*atan(sqrt(b)*sqrt(
c + d*x**8)/sqrt(a*d - b*c))/(8*a**3*(a*d - b*c)**(3/2)) + (a*d + 4*b*c)*atanh(s
qrt(c + d*x**8)/sqrt(c))/(8*a**3*c**(3/2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.2651, size = 489, normalized size = 2.64 \[ \frac{\frac{5 b d x^8 \left (-a^2 d \left (3 c+2 d x^8\right )+3 a b \left (c^2+c d x^8-d^2 x^{16}\right )+2 b^2 c x^8 \left (c+3 d x^8\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+3 \left (c+d x^8\right ) \left (a^2 d+a b \left (d x^8-c\right )-2 b^2 c x^8\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )\right )}{c (b c-a d) \left (-5 b d x^8 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )\right )}+\frac{6 a b d x^{16} (a d-2 b c) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{(a d-b c) \left (x^8 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}}{24 a^2 x^8 \left (a+b x^8\right ) \sqrt{c+d x^8}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^9*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((6*a*b*d*(-2*b*c + a*d)*x^16*AppellF1[1, 1/2, 1, 2, -((d*x^8)/c), -((b*x^8)/a)]
)/((-(b*c) + a*d)*(-4*a*c*AppellF1[1, 1/2, 1, 2, -((d*x^8)/c), -((b*x^8)/a)] + x
^8*(2*b*c*AppellF1[2, 1/2, 2, 3, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[2, 3
/2, 1, 3, -((d*x^8)/c), -((b*x^8)/a)]))) + (5*b*d*x^8*(-(a^2*d*(3*c + 2*d*x^8))
+ 2*b^2*c*x^8*(c + 3*d*x^8) + 3*a*b*(c^2 + c*d*x^8 - d^2*x^16))*AppellF1[3/2, 1/
2, 1, 5/2, -(c/(d*x^8)), -(a/(b*x^8))] + 3*(c + d*x^8)*(a^2*d - 2*b^2*c*x^8 + a*
b*(-c + d*x^8))*(2*a*d*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^8)), -(a/(b*x^8))] +
b*c*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^8)), -(a/(b*x^8))]))/(c*(b*c - a*d)*(-5*
b*d*x^8*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^8)), -(a/(b*x^8))] + 2*a*d*AppellF1[
5/2, 1/2, 2, 7/2, -(c/(d*x^8)), -(a/(b*x^8))] + b*c*AppellF1[5/2, 3/2, 1, 7/2, -
(c/(d*x^8)), -(a/(b*x^8))])))/(24*a^2*x^8*(a + b*x^8)*Sqrt[c + d*x^8])

_______________________________________________________________________________________

Maple [F]  time = 0.134, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^9/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^9),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^9), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.34291, 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^8 + a)^2*sqrt(d*x^8 + c)*x^9),x, algorithm="fricas")

[Out]

[1/16*(((4*b^3*c^2 - 5*a*b^2*c*d)*x^16 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^8)*sqrt(c
)*sqrt(b/(b*c - a*d))*log((b*d*x^8 + 2*b*c - a*d - 2*sqrt(d*x^8 + c)*(b*c - a*d)
*sqrt(b/(b*c - a*d)))/(b*x^8 + a)) - 2*((2*a*b^2*c - a^2*b*d)*x^8 + a^2*b*c - a^
3*d)*sqrt(d*x^8 + c)*sqrt(c) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^16 + (4*
a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^8)*log(((d*x^8 + 2*c)*sqrt(c) + 2*sqrt(d*x^
8 + c)*c)/x^8))/(((a^3*b^2*c^2 - a^4*b*c*d)*x^16 + (a^4*b*c^2 - a^5*c*d)*x^8)*sq
rt(c)), -1/16*(2*((4*b^3*c^2 - 5*a*b^2*c*d)*x^16 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x
^8)*sqrt(c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(
d*x^8 + c)*b)) + 2*((2*a*b^2*c - a^2*b*d)*x^8 + a^2*b*c - a^3*d)*sqrt(d*x^8 + c)
*sqrt(c) - ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^16 + (4*a*b^2*c^2 - 3*a^2*b*
c*d - a^3*d^2)*x^8)*log(((d*x^8 + 2*c)*sqrt(c) + 2*sqrt(d*x^8 + c)*c)/x^8))/(((a
^3*b^2*c^2 - a^4*b*c*d)*x^16 + (a^4*b*c^2 - a^5*c*d)*x^8)*sqrt(c)), 1/16*(((4*b^
3*c^2 - 5*a*b^2*c*d)*x^16 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^8)*sqrt(-c)*sqrt(b/(b*
c - a*d))*log((b*d*x^8 + 2*b*c - a*d - 2*sqrt(d*x^8 + c)*(b*c - a*d)*sqrt(b/(b*c
- a*d)))/(b*x^8 + a)) - 2*((2*a*b^2*c - a^2*b*d)*x^8 + a^2*b*c - a^3*d)*sqrt(d*
x^8 + c)*sqrt(-c) - 2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^16 + (4*a*b^2*c^2
- 3*a^2*b*c*d - a^3*d^2)*x^8)*arctan(c/(sqrt(d*x^8 + c)*sqrt(-c))))/(((a^3*b^2*
c^2 - a^4*b*c*d)*x^16 + (a^4*b*c^2 - a^5*c*d)*x^8)*sqrt(-c)), -1/8*(((4*b^3*c^2
- 5*a*b^2*c*d)*x^16 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^8)*sqrt(-c)*sqrt(-b/(b*c - a
*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^8 + c)*b)) + ((2*a*b^2*c
- a^2*b*d)*x^8 + a^2*b*c - a^3*d)*sqrt(d*x^8 + c)*sqrt(-c) + ((4*b^3*c^2 - 3*a*
b^2*c*d - a^2*b*d^2)*x^16 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^8)*arctan(c/
(sqrt(d*x^8 + c)*sqrt(-c))))/(((a^3*b^2*c^2 - a^4*b*c*d)*x^16 + (a^4*b*c^2 - a^5
*c*d)*x^8)*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**9/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21914, size = 362, normalized size = 1.96 \[ \frac{1}{8} \, d^{3}{\left (\frac{{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{8} + c\right )}^{\frac{3}{2}} b^{2} c - 2 \, \sqrt{d x^{8} + c} b^{2} c^{2} -{\left (d x^{8} + c\right )}^{\frac{3}{2}} a b d + 2 \, \sqrt{d x^{8} + c} a b c d - \sqrt{d x^{8} + c} a^{2} d^{2}}{{\left (a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}{\left ({\left (d x^{8} + c\right )}^{2} b - 2 \,{\left (d x^{8} + c\right )} b c + b c^{2} +{\left (d x^{8} + c\right )} a d - a c d\right )}} - \frac{{\left (4 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{8} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^9),x, algorithm="giac")

[Out]

1/8*d^3*((4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^8 + 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^8 + c)^(3/2)*b^2*c - 2*sq
rt(d*x^8 + c)*b^2*c^2 - (d*x^8 + c)^(3/2)*a*b*d + 2*sqrt(d*x^8 + c)*a*b*c*d - sq
rt(d*x^8 + c)*a^2*d^2)/((a^2*b*c^2*d^2 - a^3*c*d^3)*((d*x^8 + c)^2*b - 2*(d*x^8
+ c)*b*c + b*c^2 + (d*x^8 + c)*a*d - a*c*d)) - (4*b*c + a*d)*arctan(sqrt(d*x^8 +
c)/sqrt(-c))/(a^3*sqrt(-c)*c*d^3))