3.755 \(\int \frac{\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=168 \[ -\frac{(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} b^2}-\frac{c \sqrt{c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x \left (a+b x^2\right )}+\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.537071, antiderivative size = 168, 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 \[ -\frac{(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} b^2}-\frac{c \sqrt{c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x \left (a+b x^2\right )}+\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 73.8086, size = 144, normalized size = 0.86 \[ \frac{d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{b^{2}} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b x \left (a + b x^{2}\right )} + \frac{c \sqrt{c + d x^{2}} \left (a d - 3 b c\right )}{2 a^{2} b x} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (2 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{5}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(5/2)/x**2/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.244004, size = 150, normalized size = 0.89 \[ -\frac{(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} b^2}+\sqrt{c+d x^2} \left (-\frac{x (b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac{c^2}{a^2 x}\right )+\frac{d^{5/2} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)^2),x]

[Out]

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

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Maple [B]  time = 0.035, size = 7529, normalized size = 44.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(5/2)/x^2/(b*x^2+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^2),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^2), x)

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Fricas [A]  time = 0.711887, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^2),x, algorithm="fricas")

[Out]

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

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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((d*x**2+c)**(5/2)/x**2/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.602174, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^2),x, algorithm="giac")

[Out]

sage0*x