3.655 \(\int \frac{\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{-\frac{3 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{2 c \sqrt{c+d x^2}}-\frac{a^2}{2 c x^2 \sqrt{c+d x^2}}-\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.283707, antiderivative size = 103, 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 \[ \frac{-\frac{3 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{2 c \sqrt{c+d x^2}}-\frac{a^2}{2 c x^2 \sqrt{c+d x^2}}-\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.6581, size = 92, normalized size = 0.89 \[ - \frac{a^{2}}{2 c x^{2} \sqrt{c + d x^{2}}} + \frac{a \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 c^{\frac{5}{2}}} - \frac{\frac{a d \left (3 a d - 4 b c\right )}{2} + b^{2} c^{2}}{c^{2} d \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(3/2),x)

[Out]

-a**2/(2*c*x**2*sqrt(c + d*x**2)) + a*(3*a*d - 4*b*c)*atanh(sqrt(c + d*x**2)/sqr
t(c))/(2*c**(5/2)) - (a*d*(3*a*d - 4*b*c)/2 + b**2*c**2)/(c**2*d*sqrt(c + d*x**2
))

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Mathematica [A]  time = 0.250312, size = 105, normalized size = 1.02 \[ \frac{-\sqrt{c} \sqrt{c+d x^2} \left (\frac{a^2}{x^2}+\frac{2 (b c-a d)^2}{d \left (c+d x^2\right )}\right )+a (3 a d-4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )-a \log (x) (3 a d-4 b c)}{2 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 135, normalized size = 1.3 \[ -{\frac{{b}^{2}}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{{a}^{2}}{2\,c{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{3\,{a}^{2}d}{2\,{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{3\,{a}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{ab}{c\sqrt{d{x}^{2}+c}}}-2\,{\frac{ab}{{c}^{3/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^3/(d*x^2+c)^(3/2),x)

[Out]

-b^2/d/(d*x^2+c)^(1/2)-1/2*a^2/c/x^2/(d*x^2+c)^(1/2)-3/2*a^2*d/c^2/(d*x^2+c)^(1/
2)+3/2*a^2*d/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+2*a*b/c/(d*x^2+c)^(1/
2)-2*a*b/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)

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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((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243675, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (a^{2} c d +{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} +{\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} +{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{4 \,{\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )} \sqrt{c}}, -\frac{{\left (a^{2} c d +{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} +{\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} +{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{2 \,{\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(3/2),x)

[Out]

Integral((a + b*x**2)**2/(x**3*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.258964, size = 189, normalized size = 1.83 \[ \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c} c^{2}} - \frac{2 \,{\left (d x^{2} + c\right )} b^{2} c^{2} - 2 \, b^{2} c^{3} - 4 \,{\left (d x^{2} + c\right )} a b c d + 4 \, a b c^{2} d + 3 \,{\left (d x^{2} + c\right )} a^{2} d^{2} - 2 \, a^{2} c d^{2}}{2 \,{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} - \sqrt{d x^{2} + c} c\right )} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^3),x, algorithm="giac")

[Out]

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