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

Optimal. Leaf size=132 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^2 (b c-a d)^{3/2}}+\frac{a x \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b^2 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.325883, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^2 (b c-a d)^{3/2}}+\frac{a x \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b^2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.118, size = 116, normalized size = 0.88 \[ - \frac{\sqrt{a} \left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 b^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{a x \sqrt{c + d x^{2}}}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.318985, size = 129, normalized size = 0.98 \[ \frac{\frac{a b x \sqrt{c+d x^2}}{\left (a+b x^2\right ) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{3/2}}+\frac{2 \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 846, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.614859, 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^4/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

[1/8*(4*sqrt(d*x^2 + c)*a*b*sqrt(d)*x + (3*a*b*c - 2*a^2*d + (3*b^2*c - 2*a*b*d)
*x^2)*sqrt(d)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 +
a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x
^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a
*b*x^2 + a^2)) + 4*(a*b*c - a^2*d + (b^2*c - a*b*d)*x^2)*log(-2*sqrt(d*x^2 + c)*
d*x - (2*d*x^2 + c)*sqrt(d)))/((a*b^3*c - a^2*b^2*d + (b^4*c - a*b^3*d)*x^2)*sqr
t(d)), 1/8*(4*sqrt(d*x^2 + c)*a*b*sqrt(-d)*x + (3*a*b*c - 2*a^2*d + (3*b^2*c - 2
*a*b*d)*x^2)*sqrt(-d)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2
)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^
2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x
^4 + 2*a*b*x^2 + a^2)) + 8*(a*b*c - a^2*d + (b^2*c - a*b*d)*x^2)*arctan(sqrt(-d)
*x/sqrt(d*x^2 + c)))/((a*b^3*c - a^2*b^2*d + (b^4*c - a*b^3*d)*x^2)*sqrt(-d)), 1
/4*(2*sqrt(d*x^2 + c)*a*b*sqrt(d)*x - (3*a*b*c - 2*a^2*d + (3*b^2*c - 2*a*b*d)*x
^2)*sqrt(d)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(d*x^2
+ c)*(b*c - a*d)*x*sqrt(a/(b*c - a*d)))) + 2*(a*b*c - a^2*d + (b^2*c - a*b*d)*x
^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/((a*b^3*c - a^2*b^2*d +
(b^4*c - a*b^3*d)*x^2)*sqrt(d)), 1/4*(2*sqrt(d*x^2 + c)*a*b*sqrt(-d)*x - (3*a*b
*c - 2*a^2*d + (3*b^2*c - 2*a*b*d)*x^2)*sqrt(-d)*sqrt(a/(b*c - a*d))*arctan(1/2*
((b*c - 2*a*d)*x^2 - a*c)/(sqrt(d*x^2 + c)*(b*c - a*d)*x*sqrt(a/(b*c - a*d)))) +
4*(a*b*c - a^2*d + (b^2*c - a*b*d)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((a
*b^3*c - a^2*b^2*d + (b^4*c - a*b^3*d)*x^2)*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**4/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.571823, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x