3.95 \(\int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^5} \, dx\)

Optimal. Leaf size=80 \[ -\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}-\frac{B \sqrt{b x^2+c x^4}}{x^2}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right ) \]

[Out]

-((B*Sqrt[b*x^2 + c*x^4])/x^2) - (A*(b*x^2 + c*x^4)^(3/2))/(3*b*x^6) + B*Sqrt[c]
*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]]

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Rubi [A]  time = 0.369491, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}-\frac{B \sqrt{b x^2+c x^4}}{x^2}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^5,x]

[Out]

-((B*Sqrt[b*x^2 + c*x^4])/x^2) - (A*(b*x^2 + c*x^4)^(3/2))/(3*b*x^6) + B*Sqrt[c]
*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]]

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Rubi in Sympy [A]  time = 21.7098, size = 70, normalized size = 0.88 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 b x^{6}} + B \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )} - \frac{B \sqrt{b x^{2} + c x^{4}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**5,x)

[Out]

-A*(b*x**2 + c*x**4)**(3/2)/(3*b*x**6) + B*sqrt(c)*atanh(sqrt(c)*x**2/sqrt(b*x**
2 + c*x**4)) - B*sqrt(b*x**2 + c*x**4)/x**2

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Mathematica [A]  time = 0.11725, size = 101, normalized size = 1.26 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 b B \sqrt{c} x^3 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-\sqrt{b+c x^2} \left (A \left (b+c x^2\right )+3 b B x^2\right )\right )}{3 b x^4 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^5,x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-(Sqrt[b + c*x^2]*(3*b*B*x^2 + A*(b + c*x^2))) + 3*b*B*S
qrt[c]*x^3*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(3*b*x^4*Sqrt[b + c*x^2])

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Maple [A]  time = 0.017, size = 100, normalized size = 1.3 \[ -{\frac{1}{3\,b{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -3\,B\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){x}^{3}b-3\,Bc{x}^{4}\sqrt{c{x}^{2}+b}+3\,B \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}+A \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^5,x)

[Out]

-1/3*(c*x^4+b*x^2)^(1/2)*(-3*B*c^(1/2)*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*x^3*b-3*B*c
*x^4*(c*x^2+b)^(1/2)+3*B*(c*x^2+b)^(3/2)*x^2+A*(c*x^2+b)^(3/2))/x^4/(c*x^2+b)^(1
/2)/b

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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(c*x^4 + b*x^2)*(B*x^2 + A)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236005, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B b \sqrt{c} x^{4} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}{\left ({\left (3 \, B b + A c\right )} x^{2} + A b\right )}}{6 \, b x^{4}}, \frac{3 \, B b \sqrt{-c} x^{4} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}\right ) - \sqrt{c x^{4} + b x^{2}}{\left ({\left (3 \, B b + A c\right )} x^{2} + A b\right )}}{3 \, b x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^5,x, algorithm="fricas")

[Out]

[1/6*(3*B*b*sqrt(c)*x^4*log(-2*c*x^2 - b - 2*sqrt(c*x^4 + b*x^2)*sqrt(c)) - 2*sq
rt(c*x^4 + b*x^2)*((3*B*b + A*c)*x^2 + A*b))/(b*x^4), 1/3*(3*B*b*sqrt(-c)*x^4*ar
ctan(c*x^2/(sqrt(c*x^4 + b*x^2)*sqrt(-c))) - sqrt(c*x^4 + b*x^2)*((3*B*b + A*c)*
x^2 + A*b))/(b*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**5,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))*(A + B*x**2)/x**5, x)

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GIAC/XCAS [A]  time = 0.318962, size = 220, normalized size = 2.75 \[ -\frac{1}{2} \, B \sqrt{c}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b \sqrt{c}{\rm sign}\left (x\right ) + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A c^{\frac{3}{2}}{\rm sign}\left (x\right ) - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{2} \sqrt{c}{\rm sign}\left (x\right ) + 3 \, B b^{3} \sqrt{c}{\rm sign}\left (x\right ) + A b^{2} c^{\frac{3}{2}}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^5,x, algorithm="giac")

[Out]

-1/2*B*sqrt(c)*ln((sqrt(c)*x - sqrt(c*x^2 + b))^2)*sign(x) + 2/3*(3*(sqrt(c)*x -
sqrt(c*x^2 + b))^4*B*b*sqrt(c)*sign(x) + 3*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*c^
(3/2)*sign(x) - 6*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^2*sqrt(c)*sign(x) + 3*B*b^
3*sqrt(c)*sign(x) + A*b^2*c^(3/2)*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)
^3