3.744 \(\int \frac{x^{19}}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 b^2 (b c-a d)^{3/2}}+\frac{a x^4 \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.472375, antiderivative size = 141, 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 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 b^2 (b c-a d)^{3/2}}+\frac{a x^4 \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.4692, size = 122, normalized size = 0.87 \[ - \frac{\sqrt{a} \left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{8 b^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{a x^{4} \sqrt{c + d x^{8}}}{8 b \left (a + b x^{8}\right ) \left (a d - b c\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x^{4}}{\sqrt{c + d x^{8}}} \right )}}{4 b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.340359, size = 135, normalized size = 0.96 \[ \frac{\frac{a b x^4 \sqrt{c+d x^8}}{\left (a+b x^8\right ) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{(b c-a d)^{3/2}}+\frac{2 \log \left (\sqrt{d} \sqrt{c+d x^8}+d x^4\right )}{\sqrt{d}}}{8 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{{x}^{19}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{19}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

integrate(x^19/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)

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Fricas [A]  time = 0.59376, 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)^2*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[1/32*(4*sqrt(d*x^8 + c)*a*b*sqrt(d)*x^4 + ((3*b^2*c - 2*a*b*d)*x^8 + 3*a*b*c -
2*a^2*d)*sqrt(d)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^1
6 - 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)) + 4*((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*log(-2*sqrt(d*x
^8 + c)*d*x^4 - (2*d*x^8 + c)*sqrt(d)))/(((b^4*c - a*b^3*d)*x^8 + a*b^3*c - a^2*
b^2*d)*sqrt(d)), 1/32*(4*sqrt(d*x^8 + c)*a*b*sqrt(-d)*x^4 + ((3*b^2*c - 2*a*b*d)
*x^8 + 3*a*b*c - 2*a^2*d)*sqrt(-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)) + 8*((b^2*c - a*b*d)*x^8 + a*b*c - a^2*
d)*arctan(sqrt(-d)*x^4/sqrt(d*x^8 + c)))/(((b^4*c - a*b^3*d)*x^8 + a*b^3*c - a^2
*b^2*d)*sqrt(-d)), 1/16*(2*sqrt(d*x^8 + c)*a*b*sqrt(d)*x^4 - ((3*b^2*c - 2*a*b*d
)*x^8 + 3*a*b*c - 2*a^2*d)*sqrt(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)))) + 2*((b^2*c -
a*b*d)*x^8 + a*b*c - a^2*d)*log(-2*sqrt(d*x^8 + c)*d*x^4 - (2*d*x^8 + c)*sqrt(d
)))/(((b^4*c - a*b^3*d)*x^8 + a*b^3*c - a^2*b^2*d)*sqrt(d)), 1/16*(2*sqrt(d*x^8
+ c)*a*b*sqrt(-d)*x^4 - ((3*b^2*c - 2*a*b*d)*x^8 + 3*a*b*c - 2*a^2*d)*sqrt(-d)*s
qrt(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)))) + 4*((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*arctan(
sqrt(-d)*x^4/sqrt(d*x^8 + c)))/(((b^4*c - a*b^3*d)*x^8 + a*b^3*c - a^2*b^2*d)*sq
rt(-d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238978, size = 205, normalized size = 1.45 \[ \frac{1}{8} \, c^{2}{\left (\frac{{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{{\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \sqrt{a b c - a^{2} d}} + \frac{a \sqrt{d + \frac{c}{x^{8}}}}{{\left (b^{2} c^{2} - a b c d\right )}{\left (b c + a{\left (d + \frac{c}{x^{8}}\right )} - a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d + \frac{c}{x^{8}}}}{\sqrt{-d}}\right )}{b^{2} c^{2} \sqrt{-d}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

1/8*c^2*((3*a*b*c - 2*a^2*d)*arctan(a*sqrt(d + c/x^8)/sqrt(a*b*c - a^2*d))/((b^3
*c^3 - a*b^2*c^2*d)*sqrt(a*b*c - a^2*d)) + a*sqrt(d + c/x^8)/((b^2*c^2 - a*b*c*d
)*(b*c + a*(d + c/x^8) - a*d)) - 2*arctan(sqrt(d + c/x^8)/sqrt(-d))/(b^2*c^2*sqr
t(-d)))