3.659 \(\int \frac{x^{13}}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=191 \[ \frac{a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac{(4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{4 b^3 d^{3/2}}+\frac{x^2 \sqrt{c+d x^4} (b c-2 a d)}{4 b^2 d (b c-a d)}+\frac{a x^6 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]

[Out]

((b*c - 2*a*d)*x^2*Sqrt[c + d*x^4])/(4*b^2*d*(b*c - a*d)) + (a*x^6*Sqrt[c + d*x^
4])/(4*b*(b*c - a*d)*(a + b*x^4)) + (a^(3/2)*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c -
a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*b^3*(b*c - a*d)^(3/2)) - ((b*c + 4*a*d)
*ArcTanh[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]])/(4*b^3*d^(3/2))

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Rubi [A]  time = 0.806463, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac{(4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{4 b^3 d^{3/2}}+\frac{x^2 \sqrt{c+d x^4} (b c-2 a d)}{4 b^2 d (b c-a d)}+\frac{a x^6 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^13/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

((b*c - 2*a*d)*x^2*Sqrt[c + d*x^4])/(4*b^2*d*(b*c - a*d)) + (a*x^6*Sqrt[c + d*x^
4])/(4*b*(b*c - a*d)*(a + b*x^4)) + (a^(3/2)*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c -
a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*b^3*(b*c - a*d)^(3/2)) - ((b*c + 4*a*d)
*ArcTanh[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]])/(4*b^3*d^(3/2))

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Rubi in Sympy [A]  time = 84.7295, size = 167, normalized size = 0.87 \[ \frac{a^{\frac{3}{2}} \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{4 b^{3} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{a x^{6} \sqrt{c + d x^{4}}}{4 b \left (a + b x^{4}\right ) \left (a d - b c\right )} + \frac{x^{2} \sqrt{c + d x^{4}} \left (2 a d - b c\right )}{4 b^{2} d \left (a d - b c\right )} - \frac{\left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c + d x^{4}}} \right )}}{4 b^{3} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

a**(3/2)*(4*a*d - 5*b*c)*atanh(x**2*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**4)))/
(4*b**3*(a*d - b*c)**(3/2)) - a*x**6*sqrt(c + d*x**4)/(4*b*(a + b*x**4)*(a*d - b
*c)) + x**2*sqrt(c + d*x**4)*(2*a*d - b*c)/(4*b**2*d*(a*d - b*c)) - (4*a*d + b*c
)*atanh(sqrt(d)*x**2/sqrt(c + d*x**4))/(4*b**3*d**(3/2))

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Mathematica [A]  time = 0.631068, size = 150, normalized size = 0.79 \[ \frac{\frac{a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{(b c-a d)^{3/2}}+b x^2 \sqrt{c+d x^4} \left (\frac{a^2}{\left (a+b x^4\right ) (a d-b c)}+\frac{1}{d}\right )-\frac{(4 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^4}+d x^2\right )}{d^{3/2}}}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

(b*x^2*Sqrt[c + d*x^4]*(d^(-1) + a^2/((-(b*c) + a*d)*(a + b*x^4))) + (a^(3/2)*(5
*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*c - a*
d)^(3/2) - ((b*c + 4*a*d)*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]])/d^(3/2))/(4*b^3)

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Maple [B]  time = 0.045, size = 953, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

1/4/b^2*x^2/d*(d*x^4+c)^(1/2)-1/4/b^2*c/d^(3/2)*ln(x^2*d^(1/2)+(d*x^4+c)^(1/2))-
1/b^3*a*ln(x^2*d^(1/2)+(d*x^4+c)^(1/2))/d^(1/2)-5/8*a^2/b^3/(-a*b)^(1/2)/(-(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)))+5/8*a^2/b^3/(-a*b)^(1/2)/(-(
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^2/b^3/(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^4*d*(-a*b)^(1/2)/(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^2/b^3/(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^4*d*(-a*b)^(1/2)/(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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(x^13/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)

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Fricas [A]  time = 1.02205, 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^13/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

[1/16*((5*a^2*b*c*d - 4*a^3*d^2 + (5*a*b^2*c*d - 4*a^2*b*d^2)*x^4)*sqrt(d)*sqrt(
-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^
2*c*d)*x^4 + a^2*c^2 + 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b*c^2 - a^2
*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^8 + 2*a*b*x^4 + a^2)) +
4*((b^3*c - a*b^2*d)*x^6 + (a*b^2*c - 2*a^2*b*d)*x^2)*sqrt(d*x^4 + c)*sqrt(d) +
2*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c*d - 4*a^2*b*d^2)*x
^4)*log(2*sqrt(d*x^4 + c)*d*x^2 - (2*d*x^4 + c)*sqrt(d)))/((a*b^4*c*d - a^2*b^3*
d^2 + (b^5*c*d - a*b^4*d^2)*x^4)*sqrt(d)), 1/16*((5*a^2*b*c*d - 4*a^3*d^2 + (5*a
*b^2*c*d - 4*a^2*b*d^2)*x^4)*sqrt(-d)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b
*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 + 4*((b^2*c^2 -
3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b*c^2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b
*c - a*d)))/(b^2*x^8 + 2*a*b*x^4 + a^2)) + 4*((b^3*c - a*b^2*d)*x^6 + (a*b^2*c -
2*a^2*b*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-d) - 4*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d
^2 + (b^3*c^2 + 3*a*b^2*c*d - 4*a^2*b*d^2)*x^4)*arctan(sqrt(-d)*x^2/sqrt(d*x^4 +
c)))/((a*b^4*c*d - a^2*b^3*d^2 + (b^5*c*d - a*b^4*d^2)*x^4)*sqrt(-d)), 1/8*((5*
a^2*b*c*d - 4*a^3*d^2 + (5*a*b^2*c*d - 4*a^2*b*d^2)*x^4)*sqrt(d)*sqrt(a/(b*c - a
*d))*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*(b*c - a*d)*x^2*sqrt(
a/(b*c - a*d)))) + 2*((b^3*c - a*b^2*d)*x^6 + (a*b^2*c - 2*a^2*b*d)*x^2)*sqrt(d*
x^4 + c)*sqrt(d) + (a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c*d
- 4*a^2*b*d^2)*x^4)*log(2*sqrt(d*x^4 + c)*d*x^2 - (2*d*x^4 + c)*sqrt(d)))/((a*b
^4*c*d - a^2*b^3*d^2 + (b^5*c*d - a*b^4*d^2)*x^4)*sqrt(d)), 1/8*((5*a^2*b*c*d -
4*a^3*d^2 + (5*a*b^2*c*d - 4*a^2*b*d^2)*x^4)*sqrt(-d)*sqrt(a/(b*c - a*d))*arctan
(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*(b*c - a*d)*x^2*sqrt(a/(b*c - a*
d)))) + 2*((b^3*c - a*b^2*d)*x^6 + (a*b^2*c - 2*a^2*b*d)*x^2)*sqrt(d*x^4 + c)*sq
rt(-d) - 2*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c*d - 4*a^2
*b*d^2)*x^4)*arctan(sqrt(-d)*x^2/sqrt(d*x^4 + c)))/((a*b^4*c*d - a^2*b^3*d^2 + (
b^5*c*d - a*b^4*d^2)*x^4)*sqrt(-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**13/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.367785, size = 238, normalized size = 1.25 \[ -\frac{a^{2} c \sqrt{d + \frac{c}{x^{4}}}}{4 \,{\left (b^{3} c - a b^{2} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}} - \frac{{\left (5 \, a^{2} b c - 4 \, a^{3} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \,{\left (b^{4} c - a b^{3} d\right )} \sqrt{a b c - a^{2} d}} + \frac{\sqrt{d x^{4} + c} x^{2}}{4 \, b^{2} d} + \frac{{\left (b c + 4 \, a d\right )} \arctan \left (\frac{\sqrt{d + \frac{c}{x^{4}}}}{\sqrt{-d}}\right )}{4 \, b^{3} \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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