3.745 \(\int \frac{x^{11}}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=93 \[ \frac{c \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^4 \sqrt{c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]

[Out]

-(x^4*Sqrt[c + d*x^8])/(8*(b*c - a*d)*(a + b*x^8)) + (c*ArcTan[(Sqrt[b*c - a*d]*
x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(8*Sqrt[a]*(b*c - a*d)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.273649, antiderivative size = 93, 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 \[ \frac{c \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^4 \sqrt{c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(x^4*Sqrt[c + d*x^8])/(8*(b*c - a*d)*(a + b*x^8)) + (c*ArcTan[(Sqrt[b*c - a*d]*
x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(8*Sqrt[a]*(b*c - a*d)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.8303, size = 76, normalized size = 0.82 \[ \frac{x^{4} \sqrt{c + d x^{8}}}{8 \left (a + b x^{8}\right ) \left (a d - b c\right )} - \frac{c \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{8 \sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.172646, size = 90, normalized size = 0.97 \[ \frac{\frac{x^4 \sqrt{c+d x^8}}{a+b x^8}-\frac{c \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{\sqrt{a} \sqrt{b c-a d}}}{8 a d-8 b c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((x^4*Sqrt[c + d*x^8])/(a + b*x^8) - (c*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sq
rt[c + d*x^8])])/(Sqrt[a]*Sqrt[b*c - a*d]))/(-8*b*c + 8*a*d)

_______________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{{x}^{11}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(x^11/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

integrate(x^11/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.328178, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{d x^{8} + c} \sqrt{-a b c + a^{2} d} x^{4} +{\left (b c x^{8} + a c\right )} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{12} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \,{\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4} -{\left (b c x^{8} + a c\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4}}\right )}{16 \,{\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[-1/32*(4*sqrt(d*x^8 + c)*sqrt(-a*b*c + a^2*d)*x^4 + (b*c*x^8 + a*c)*log(-(4*((a
*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^12 - (a^2*b*c^2 - a^3*c*d)*x^4)*sqrt(d*x^8
+ c) - ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^8
+ a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8 + a^2)))/(((b^2*c - a*b*d
)*x^8 + a*b*c - a^2*d)*sqrt(-a*b*c + a^2*d)), -1/16*(2*sqrt(d*x^8 + c)*sqrt(a*b*
c - a^2*d)*x^4 - (b*c*x^8 + a*c)*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d*x^
8 + c)*sqrt(a*b*c - a^2*d)*x^4)))/(((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(a*
b*c - a^2*d))]

_______________________________________________________________________________________

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(x**11/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234679, size = 124, normalized size = 1.33 \[ -\frac{1}{8} \, c{\left (\frac{\arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d}{\left (b c - a d\right )}} + \frac{\sqrt{d + \frac{c}{x^{8}}}}{{\left (b c + a{\left (d + \frac{c}{x^{8}}\right )} - a d\right )}{\left (b c - a d\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

-1/8*c*(arctan(a*sqrt(d + c/x^8)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*(b*c
- a*d)) + sqrt(d + c/x^8)/((b*c + a*(d + c/x^8) - a*d)*(b*c - a*d)))