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

Optimal. Leaf size=80 \[ -\frac{a^2 \sqrt{c+d x^2}}{2 c x^2}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d} \]

[Out]

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

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Rubi [A]  time = 0.226071, antiderivative size = 80, 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 \[ -\frac{a^2 \sqrt{c+d x^2}}{2 c x^2}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.1397, size = 68, normalized size = 0.85 \[ - \frac{a^{2} \sqrt{c + d x^{2}}}{2 c x^{2}} + \frac{a \left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}}} + \frac{b^{2} \sqrt{c + d x^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.174317, size = 92, normalized size = 1.15 \[ \sqrt{c+d x^2} \left (\frac{b^2}{d}-\frac{a^2}{2 c x^2}\right )+\frac{a (a d-4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{2 c^{3/2}}-\frac{a \log (x) (a d-4 b c)}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 100, normalized size = 1.3 \[{\frac{{b}^{2}}{d}\sqrt{d{x}^{2}+c}}-{\frac{{a}^{2}}{2\,c{x}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{{a}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{ab}{\sqrt{c}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^3/(d*x^2+c)^(1/2),x)

[Out]

b^2*(d*x^2+c)^(1/2)/d-1/2*a^2*(d*x^2+c)^(1/2)/c/x^2+1/2*a^2*d/c^(3/2)*ln((2*c+2*
c^(1/2)*(d*x^2+c)^(1/2))/x)-2*a*b/c^(1/2)*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/(sqrt(d*x^2 + c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260459, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \,{\left (2 \, b^{2} c x^{2} - a^{2} d\right )} \sqrt{d x^{2} + c} \sqrt{c}}{4 \, c^{\frac{3}{2}} d x^{2}}, -\frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (2 \, b^{2} c x^{2} - a^{2} d\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{2 \, \sqrt{-c} c d x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 42.8166, size = 99, normalized size = 1.24 \[ - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 c x} + \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2 c^{\frac{3}{2}}} - \frac{2 a b \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{\sqrt{c}} + b^{2} \left (\begin{cases} \frac{x^{2}}{2 \sqrt{c}} & \text{for}\: d = 0 \\\frac{\sqrt{c + d x^{2}}}{d} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(1/2),x)

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(2*c*x) + a**2*d*asinh(sqrt(c)/(sqrt(d)*x))/(
2*c**(3/2)) - 2*a*b*asinh(sqrt(c)/(sqrt(d)*x))/sqrt(c) + b**2*Piecewise((x**2/(2
*sqrt(c)), Eq(d, 0)), (sqrt(c + d*x**2)/d, True))

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GIAC/XCAS [A]  time = 0.231535, size = 109, normalized size = 1.36 \[ \frac{2 \, \sqrt{d x^{2} + c} b^{2} - \frac{\sqrt{d x^{2} + c} a^{2} d}{c x^{2}} + \frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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