3.653 \(\int \frac{x^{15}}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}+\frac{a x^8 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]

[Out]

(a*x^8*Sqrt[c + d*x^4])/(4*b*(b*c - a*d)*(a + b*x^4)) - (Sqrt[c + d*x^4]*(4*b^2*
c^2 + 8*a*b*c*d - 15*a^2*d^2 - b*d*(2*b*c - 5*a*d)*x^4))/(12*b^3*d^2*(b*c - a*d)
) - (a^2*(6*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(4*
b^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.495988, antiderivative size = 175, 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 \[ -\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}+\frac{a x^8 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^15/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

(a*x^8*Sqrt[c + d*x^4])/(4*b*(b*c - a*d)*(a + b*x^4)) - (Sqrt[c + d*x^4]*(4*b^2*
c^2 + 8*a*b*c*d - 15*a^2*d^2 - b*d*(2*b*c - 5*a*d)*x^4))/(12*b^3*d^2*(b*c - a*d)
) - (a^2*(6*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(4*
b^(7/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 41.7839, size = 158, normalized size = 0.9 \[ \frac{a^{2} \left (5 a d - 6 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{a x^{8} \sqrt{c + d x^{4}}}{4 b \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{4}} \left (\frac{15 a^{2} d^{2}}{4} - 2 a b c d - b^{2} c^{2} - \frac{b d x^{4} \left (5 a d - 2 b c\right )}{4}\right )}{3 b^{3} d^{2} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

a**2*(5*a*d - 6*b*c)*atan(sqrt(b)*sqrt(c + d*x**4)/sqrt(a*d - b*c))/(4*b**(7/2)*
(a*d - b*c)**(3/2)) - a*x**8*sqrt(c + d*x**4)/(4*b*(a + b*x**4)*(a*d - b*c)) - s
qrt(c + d*x**4)*(15*a**2*d**2/4 - 2*a*b*c*d - b**2*c**2 - b*d*x**4*(5*a*d - 2*b*
c)/4)/(3*b**3*d**2*(a*d - b*c))

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Mathematica [A]  time = 0.551468, size = 130, normalized size = 0.74 \[ \frac{1}{4} \left (\frac{\sqrt{c+d x^4} \left (\frac{3 a^3}{\left (a+b x^4\right ) (b c-a d)}-\frac{4 (3 a d+b c)}{d^2}+\frac{2 b x^4}{d}\right )}{3 b^3}+\frac{a^2 (5 a d-6 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{b^{7/2} (b c-a d)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^15/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

((Sqrt[c + d*x^4]*((-4*(b*c + 3*a*d))/d^2 + (2*b*x^4)/d + (3*a^3)/((b*c - a*d)*(
a + b*x^4))))/(3*b^3) + (a^2*(-6*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/
Sqrt[b*c - a*d]])/(b^(7/2)*(b*c - a*d)^(3/2)))/4

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Maple [B]  time = 0.05, size = 923, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

1/6/b^2*(d*x^4+c)^(1/2)/d*x^4-1/3/b^2*(d*x^4+c)^(1/2)/d^2*c-1/b^3*a/d*(d*x^4+c)^
(1/2)-3/4*a^2/b^4/(-(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)))-3/4
*a^2/b^4/(-(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^4*
(-a*b)^(1/2)/(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^3/b^4*d/(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^4*(-a*b)^(1/
2)/(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^3/b^4*d/(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256738, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{8} - 4 \, a b^{2} c^{2} - 8 \, a^{2} b c d + 15 \, a^{3} d^{2} - 2 \,{\left (2 \, b^{3} c^{2} + 3 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d} + 3 \,{\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} +{\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right )}{24 \,{\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} +{\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{8} - 4 \, a b^{2} c^{2} - 8 \, a^{2} b c d + 15 \, a^{3} d^{2} - 2 \,{\left (2 \, b^{3} c^{2} + 3 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d} - 3 \,{\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} +{\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right )}{12 \,{\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} +{\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

[1/24*(2*(2*(b^3*c*d - a*b^2*d^2)*x^8 - 4*a*b^2*c^2 - 8*a^2*b*c*d + 15*a^3*d^2 -
2*(2*b^3*c^2 + 3*a*b^2*c*d - 5*a^2*b*d^2)*x^4)*sqrt(d*x^4 + c)*sqrt(b^2*c - a*b
*d) + 3*(6*a^3*b*c*d^2 - 5*a^4*d^3 + (6*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4)*log(((
b*d*x^4 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) - 2*sqrt(d*x^4 + c)*(b^2*c - a*b*d))/
(b*x^4 + a)))/((a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^4)*sqrt(b^
2*c - a*b*d)), 1/12*((2*(b^3*c*d - a*b^2*d^2)*x^8 - 4*a*b^2*c^2 - 8*a^2*b*c*d +
15*a^3*d^2 - 2*(2*b^3*c^2 + 3*a*b^2*c*d - 5*a^2*b*d^2)*x^4)*sqrt(d*x^4 + c)*sqrt
(-b^2*c + a*b*d) - 3*(6*a^3*b*c*d^2 - 5*a^4*d^3 + (6*a^2*b^2*c*d^2 - 5*a^3*b*d^3
)*x^4)*arctan(-(b*c - a*d)/(sqrt(d*x^4 + c)*sqrt(-b^2*c + a*b*d))))/((a*b^4*c*d^
2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^4)*sqrt(-b^2*c + a*b*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**15/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221737, size = 243, normalized size = 1.39 \[ \frac{\sqrt{d x^{4} + c} a^{3} d}{4 \,{\left (b^{4} c - a b^{3} d\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac{{\left (6 \, a^{2} b c - 5 \, a^{3} d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{4} c - a b^{3} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{{\left (d x^{4} + c\right )}^{\frac{3}{2}} b^{4} d^{4} - 3 \, \sqrt{d x^{4} + c} b^{4} c d^{4} - 6 \, \sqrt{d x^{4} + c} a b^{3} d^{5}}{6 \, b^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

1/4*sqrt(d*x^4 + c)*a^3*d/((b^4*c - a*b^3*d)*((d*x^4 + c)*b - b*c + a*d)) + 1/4*
(6*a^2*b*c - 5*a^3*d)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c - a
*b^3*d)*sqrt(-b^2*c + a*b*d)) + 1/6*((d*x^4 + c)^(3/2)*b^4*d^4 - 3*sqrt(d*x^4 +
c)*b^4*c*d^4 - 6*sqrt(d*x^4 + c)*a*b^3*d^5)/(b^6*d^6)