3.263 \(\int \frac{\sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx\)

Optimal. Leaf size=177 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac{x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (2 a d+b c)}{2 c \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.287868, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac{x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (2 a d+b c)}{2 c \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.751, size = 128, normalized size = 0.72 \[ - \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{c x \left (a + b x^{2}\right )} + \frac{\left (2 a d + b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 \sqrt{d} \left (a + b x^{2}\right )} + \frac{x \sqrt{c + d x^{2}} \left (2 a d + b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}}}{2 c \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.080046, size = 93, normalized size = 0.53 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{d} \left (b x^2-2 a\right ) \sqrt{c+d x^2}+x (2 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{2 \sqrt{d} x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(Sqrt[d]*(-2*a + b*x^2)*Sqrt[c + d*x^2] + (b*c + 2*a*d)*x*L
og[d*x + Sqrt[d]*Sqrt[c + d*x^2]]))/(2*Sqrt[d]*x*(a + b*x^2))

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Maple [A]  time = 0.013, size = 128, normalized size = 0.7 \[{\frac{1}{ \left ( 2\,b{x}^{2}+2\,a \right ) cx}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 2\,ad\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) cx+2\,a{d}^{3/2}{x}^{2}\sqrt{d{x}^{2}+c}+b{x}^{2}\sqrt{d{x}^{2}+c}\sqrt{d}c-2\,a \left ( d{x}^{2}+c \right ) ^{3/2}\sqrt{d}+b{c}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) x \right ){\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289159, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + 2 \, a d\right )} x \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (b x^{2} - 2 \, a\right )} \sqrt{d x^{2} + c} \sqrt{d}}{4 \, \sqrt{d} x}, \frac{{\left (b c + 2 \, a d\right )} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (b x^{2} - 2 \, a\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{2 \, \sqrt{-d} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282439, size = 157, normalized size = 0.89 \[ \frac{1}{2} \, \sqrt{d x^{2} + c} b x{\rm sign}\left (b x^{2} + a\right ) + \frac{2 \, a c \sqrt{d}{\rm sign}\left (b x^{2} + a\right )}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} - \frac{{\left (b c \sqrt{d}{\rm sign}\left (b x^{2} + a\right ) + 2 \, a d^{\frac{3}{2}}{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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