3.712 \(\int \frac{1}{x^4 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=110 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac{\sqrt{c+d x^2}}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.373361, antiderivative size = 110, 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 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac{\sqrt{c+d x^2}}{3 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.2618, size = 95, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{2}}}{3 a c x^{3}} + \frac{\sqrt{c + d x^{2}} \left (2 a d + 3 b c\right )}{3 a^{2} c^{2} x} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.17042, size = 96, normalized size = 0.87 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} \left (-a c+2 a d x^2+3 b c x^2\right )}{3 a^2 c^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 379, normalized size = 3.5 \[ -{\frac{1}{3\,ac{x}^{3}}\sqrt{d{x}^{2}+c}}+{\frac{2\,d}{3\,a{c}^{2}x}\sqrt{d{x}^{2}+c}}+{\frac{b}{{a}^{2}cx}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{{b}^{2}}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(d*x^2+c)^(1/2)/a/c/x^3+2/3/a*d/c^2/x*(d*x^2+c)^(1/2)+b/a^2/c/x*(d*x^2+c)^(
1/2)-1/2*b^2/a^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-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)))
+1/2*b^2/a^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+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. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4), x)

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Fricas [A]  time = 0.324493, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{2} x^{3} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt{-a b c + a^{2} d}{\left ({\left (3 \, b c + 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{12 \, \sqrt{-a b c + a^{2} d} a^{2} c^{2} x^{3}}, \frac{3 \, b^{2} c^{2} x^{3} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right ) + 2 \, \sqrt{a b c - a^{2} d}{\left ({\left (3 \, b c + 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{6 \, \sqrt{a b c - a^{2} d} a^{2} c^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="fricas")

[Out]

[1/12*(3*b^2*c^2*x^3*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)*sqrt(-a*b*c + a^2*d) + 4*((a*b^2*c^2 - 3*a^2*b*c*d +
2*a^3*d^2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2
+ a^2)) + 4*sqrt(-a*b*c + a^2*d)*((3*b*c + 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c))/(
sqrt(-a*b*c + a^2*d)*a^2*c^2*x^3), 1/6*(3*b^2*c^2*x^3*arctan(1/2*((b*c - 2*a*d)*
x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)) + 2*sqrt(a*b*c - a^2*d)*((3*
b*c + 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c))/(sqrt(a*b*c - a^2*d)*a^2*c^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*(a + b*x**2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.739166, size = 263, normalized size = 2.39 \[ -\frac{1}{3} \, d^{\frac{5}{2}}{\left (\frac{3 \, b^{2} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a^{2} d^{2}} + \frac{2 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + 2 \, a c d\right )}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="giac")

[Out]

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