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

Optimal. Leaf size=82 \[ -\frac{a^2 \sqrt{c+d x^2}}{c x}-\frac{b (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.14273, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \sqrt{c+d x^2}}{c x}-\frac{b (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.1221, size = 70, normalized size = 0.85 \[ - \frac{a^{2} \sqrt{c + d x^{2}}}{c x} + \frac{b^{2} x \sqrt{c + d x^{2}}}{2 d} + \frac{b \left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0958954, size = 76, normalized size = 0.93 \[ \sqrt{c+d x^2} \left (\frac{b^2 x}{2 d}-\frac{a^2}{c x}\right )-\frac{b (b c-4 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{2 d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 88, normalized size = 1.1 \[{\frac{{b}^{2}x}{2\,d}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}}{cx}\sqrt{d{x}^{2}+c}}+2\,{\frac{ab\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{\sqrt{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257743, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} x \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) - 2 \,{\left (b^{2} c x^{2} - 2 \, a^{2} d\right )} \sqrt{d x^{2} + c} \sqrt{d}}{4 \, c d^{\frac{3}{2}} x}, -\frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (b^{2} c x^{2} - 2 \, a^{2} d\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{2 \, c \sqrt{-d} d x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.679, size = 155, normalized size = 1.89 \[ - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c} + 2 a b \left (\begin{cases} \frac{\sqrt{- \frac{c}{d}} \operatorname{asin}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d < 0 \\\frac{\sqrt{\frac{c}{d}} \operatorname{asinh}{\left (x \sqrt{\frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d > 0 \\\frac{\sqrt{- \frac{c}{d}} \operatorname{acosh}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{- c}} & \text{for}\: d > 0 \wedge c < 0 \end{cases}\right ) + \frac{b^{2} \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}}}{2 d} - \frac{b^{2} c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/c + 2*a*b*Piecewise((sqrt(-c/d)*asin(x*sqrt(-
d/c))/sqrt(c), (c > 0) & (d < 0)), (sqrt(c/d)*asinh(x*sqrt(d/c))/sqrt(c), (c > 0
) & (d > 0)), (sqrt(-c/d)*acosh(x*sqrt(-d/c))/sqrt(-c), (d > 0) & (c < 0))) + b*
*2*sqrt(c)*x*sqrt(1 + d*x**2/c)/(2*d) - b**2*c*asinh(sqrt(d)*x/sqrt(c))/(2*d**(3
/2))

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GIAC/XCAS [A]  time = 0.248599, size = 126, normalized size = 1.54 \[ \frac{\sqrt{d x^{2} + c} b^{2} x}{2 \, d} + \frac{2 \, a^{2} \sqrt{d}}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} + \frac{{\left (b^{2} c \sqrt{d} - 4 \, a b d^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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