3.264 \(\int \frac{\sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.362906, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (a d+2 b c)}{2 c \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.7992, size = 124, normalized size = 0.7 \[ - \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{2 c x^{2} \left (a + b x^{2}\right )} + \frac{\sqrt{c + d x^{2}} \left (\frac{a d}{2} + b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}}}{c \left (a + b x^{2}\right )} - \frac{\left (\frac{a d}{2} + b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c} \left (a + b x^{2}\right )} \]

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**3,x)

[Out]

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

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Mathematica [A]  time = 0.111068, size = 109, normalized size = 0.62 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{c} \left (2 b x^2-a\right ) \sqrt{c+d x^2}+x^2 \log (x) (a d+2 b c)-x^2 (a d+2 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )\right )}{2 \sqrt{c} x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^3,x]

[Out]

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

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Maple [A]  time = 0.015, size = 139, normalized size = 0.8 \[ -{\frac{1}{ \left ( 2\,b{x}^{2}+2\,a \right ){x}^{2}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 2\,{c}^{2}\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ) b{x}^{2}+ad\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ){x}^{2}c-ad\sqrt{d{x}^{2}+c}{x}^{2}\sqrt{c}-2\,\sqrt{d{x}^{2}+c}b{x}^{2}{c}^{3/2}+a \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{c} \right ){c}^{-{\frac{3}{2}}}} \]

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^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287476, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b c + a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) + 2 \,{\left (2 \, b x^{2} - a\right )} \sqrt{d x^{2} + c} \sqrt{c}}{4 \, \sqrt{c} x^{2}}, -\frac{{\left (2 \, b c + a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (2 \, b x^{2} - a\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{2 \, \sqrt{-c} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27651, size = 135, normalized size = 0.76 \[ \frac{2 \, \sqrt{d x^{2} + c} b d{\rm sign}\left (b x^{2} + a\right ) + \frac{{\left (2 \, b c d{\rm sign}\left (b x^{2} + a\right ) + a d^{2}{\rm sign}\left (b x^{2} + a\right )\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{\sqrt{d x^{2} + c} a d{\rm sign}\left (b x^{2} + a\right )}{x^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*sqrt(d*x^2 + c)*b*d*sign(b*x^2 + a) + (2*b*c*d*sign(b*x^2 + a) + a*d^2*si
gn(b*x^2 + a))*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c) - sqrt(d*x^2 + c)*a*d*s
ign(b*x^2 + a)/x^2)/d