3.660 \(\int \frac{x^9}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.418507, 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 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 b^2 (b c-a d)^{3/2}}+\frac{a x^2 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{2 b^2 \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[x^9/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 50.2296, size = 122, normalized size = 0.87 \[ - \frac{\sqrt{a} \left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{4 b^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{a x^{2} \sqrt{c + d x^{4}}}{4 b \left (a + b x^{4}\right ) \left (a d - b c\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c + d x^{4}}} \right )}}{2 b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.317793, size = 135, normalized size = 0.96 \[ \frac{\frac{a b x^2 \sqrt{c+d x^4}}{\left (a+b x^4\right ) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{(b c-a d)^{3/2}}+\frac{2 \log \left (\sqrt{d} \sqrt{c+d x^4}+d x^2\right )}{\sqrt{d}}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

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

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Maple [B]  time = 0.019, size = 893, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(x^9/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)

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Fricas [A]  time = 0.618587, 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^9/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.230959, size = 205, normalized size = 1.45 \[ \frac{1}{4} \, c^{2}{\left (\frac{{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{{\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \sqrt{a b c - a^{2} d}} + \frac{a \sqrt{d + \frac{c}{x^{4}}}}{{\left (b^{2} c^{2} - a b c d\right )}{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d + \frac{c}{x^{4}}}}{\sqrt{-d}}\right )}{b^{2} c^{2} \sqrt{-d}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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