3.723 \(\int \frac{x^{19}}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=123 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{8 b^2 d^{3/2}}+\frac{x^4 \sqrt{c+d x^8}}{8 b d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.417163, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{8 b^2 d^{3/2}}+\frac{x^4 \sqrt{c+d x^8}}{8 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^19/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.626, size = 107, normalized size = 0.87 \[ \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{4 b^{2} \sqrt{a d - b c}} + \frac{x^{4} \sqrt{c + d x^{8}}}{8 b d} - \frac{\left (2 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x^{4}}{\sqrt{c + d x^{8}}} \right )}}{8 b^{2} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.220045, size = 118, normalized size = 0.96 \[ \frac{\frac{2 a^{3/2} \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^8}+d x^4\right )}{d^{3/2}}+\frac{b x^4 \sqrt{c+d x^8}}{d}}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{\frac{{x}^{19}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^19/(b*x^8+a)/(d*x^8+c)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.346774, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[1/16*(2*sqrt(d*x^8 + c)*b*sqrt(d)*x^4 + a*d^(3/2)*sqrt(-a/(b*c - a*d))*log(((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 +
4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d*x^8
+ c)*sqrt(-a/(b*c - a*d)))/(b^2*x^16 + 2*a*b*x^8 + a^2)) + (b*c + 2*a*d)*log(2*s
qrt(d*x^8 + c)*d*x^4 - (2*d*x^8 + c)*sqrt(d)))/(b^2*d^(3/2)), 1/16*(2*sqrt(d*x^8
+ c)*b*sqrt(-d)*x^4 + a*sqrt(-d)*d*sqrt(-a/(b*c - a*d))*log(((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 + 4*((b^2*c^2 - 3
*a*b*c*d + 2*a^2*d^2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d*x^8 + c)*sqrt(-a/(b
*c - a*d)))/(b^2*x^16 + 2*a*b*x^8 + a^2)) - 2*(b*c + 2*a*d)*arctan(sqrt(-d)*x^4/
sqrt(d*x^8 + c)))/(b^2*sqrt(-d)*d), 1/16*(2*sqrt(d*x^8 + c)*b*sqrt(d)*x^4 + 2*a*
d^(3/2)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d*x^8 + c
)*(b*c - a*d)*x^4*sqrt(a/(b*c - a*d)))) + (b*c + 2*a*d)*log(2*sqrt(d*x^8 + c)*d*
x^4 - (2*d*x^8 + c)*sqrt(d)))/(b^2*d^(3/2)), 1/8*(sqrt(d*x^8 + c)*b*sqrt(-d)*x^4
+ a*sqrt(-d)*d*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d
*x^8 + c)*(b*c - a*d)*x^4*sqrt(a/(b*c - a*d)))) - (b*c + 2*a*d)*arctan(sqrt(-d)*
x^4/sqrt(d*x^8 + c)))/(b^2*sqrt(-d)*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**19/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.308071, size = 140, normalized size = 1.14 \[ \frac{\sqrt{d x^{8} + c} x^{4}}{8 \, b d} - \frac{a^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \, \sqrt{a b c - a^{2} d} b^{2}} + \frac{{\left (b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{d + \frac{c}{x^{8}}}}{\sqrt{-d}}\right )}{8 \, b^{2} \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

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