3.124 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=133 \[ \frac{\sqrt{b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac{1}{2} \sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )+\frac{\left (b x^2+c x^4\right )^{3/2} (3 A c+2 b B)}{6 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7} \]

[Out]

((2*b*B + 3*A*c)*Sqrt[b*x^2 + c*x^4])/(2*x) + ((2*b*B + 3*A*c)*(b*x^2 + c*x^4)^(
3/2))/(6*b*x^3) - (A*(b*x^2 + c*x^4)^(5/2))/(2*b*x^7) - (Sqrt[b]*(2*b*B + 3*A*c)
*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/2

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Rubi [A]  time = 0.354519, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac{1}{2} \sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )+\frac{\left (b x^2+c x^4\right )^{3/2} (3 A c+2 b B)}{6 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^6,x]

[Out]

((2*b*B + 3*A*c)*Sqrt[b*x^2 + c*x^4])/(2*x) + ((2*b*B + 3*A*c)*(b*x^2 + c*x^4)^(
3/2))/(6*b*x^3) - (A*(b*x^2 + c*x^4)^(5/2))/(2*b*x^7) - (Sqrt[b]*(2*b*B + 3*A*c)
*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/2

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Rubi in Sympy [A]  time = 27.4787, size = 116, normalized size = 0.87 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{2 b x^{7}} - \frac{\sqrt{b} \left (3 A c + 2 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2} + \frac{\left (\frac{3 A c}{2} + B b\right ) \sqrt{b x^{2} + c x^{4}}}{x} + \frac{\left (3 A c + 2 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**6,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(2*b*x**7) - sqrt(b)*(3*A*c + 2*B*b)*atanh(sqrt(b)*x
/sqrt(b*x**2 + c*x**4))/2 + (3*A*c/2 + B*b)*sqrt(b*x**2 + c*x**4)/x + (3*A*c + 2
*B*b)*(b*x**2 + c*x**4)**(3/2)/(6*b*x**3)

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Mathematica [A]  time = 0.184673, size = 132, normalized size = 0.99 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 \sqrt{b} x^2 \log (x) (3 A c+2 b B)-3 \sqrt{b} x^2 (3 A c+2 b B) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+\sqrt{b+c x^2} \left (-3 A b+6 A c x^2+8 b B x^2+2 B c x^4\right )\right )}{6 x^3 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^6,x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b + c*x^2]*(-3*A*b + 8*b*B*x^2 + 6*A*c*x^2 + 2*B*c*
x^4) + 3*Sqrt[b]*(2*b*B + 3*A*c)*x^2*Log[x] - 3*Sqrt[b]*(2*b*B + 3*A*c)*x^2*Log[
b + Sqrt[b]*Sqrt[b + c*x^2]]))/(6*x^3*Sqrt[b + c*x^2])

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Maple [A]  time = 0.016, size = 172, normalized size = 1.3 \[ -{\frac{1}{6\,b{x}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 6\,B{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}+9\,A{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) c{x}^{2}-3\,Ac \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}-2\,B \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}b+3\,A \left ( c{x}^{2}+b \right ) ^{5/2}-9\,Ac\sqrt{c{x}^{2}+b}{x}^{2}b-6\,B\sqrt{c{x}^{2}+b}{b}^{2}{x}^{2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^6,x)

[Out]

-1/6*(c*x^4+b*x^2)^(3/2)*(6*B*b^(5/2)*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^2+9*
A*b^(3/2)*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*c*x^2-3*A*c*(c*x^2+b)^(3/2)*x^2-2*
B*(c*x^2+b)^(3/2)*x^2*b+3*A*(c*x^2+b)^(5/2)-9*A*c*(c*x^2+b)^(1/2)*x^2*b-6*B*(c*x
^2+b)^(1/2)*b^2*x^2)/x^5/(c*x^2+b)^(3/2)/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((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250294, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, B b + 3 \, A c\right )} \sqrt{b} x^{3} \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \,{\left (2 \, B c x^{4} + 2 \,{\left (4 \, B b + 3 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt{c x^{4} + b x^{2}}}{12 \, x^{3}}, -\frac{3 \,{\left (2 \, B b + 3 \, A c\right )} \sqrt{-b} x^{3} \arctan \left (\frac{b x}{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}\right ) -{\left (2 \, B c x^{4} + 2 \,{\left (4 \, B b + 3 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt{c x^{4} + b x^{2}}}{6 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^6,x, algorithm="fricas")

[Out]

[1/12*(3*(2*B*b + 3*A*c)*sqrt(b)*x^3*log(-(c*x^3 + 2*b*x - 2*sqrt(c*x^4 + b*x^2)
*sqrt(b))/x^3) + 2*(2*B*c*x^4 + 2*(4*B*b + 3*A*c)*x^2 - 3*A*b)*sqrt(c*x^4 + b*x^
2))/x^3, -1/6*(3*(2*B*b + 3*A*c)*sqrt(-b)*x^3*arctan(b*x/(sqrt(c*x^4 + b*x^2)*sq
rt(-b))) - (2*B*c*x^4 + 2*(4*B*b + 3*A*c)*x^2 - 3*A*b)*sqrt(c*x^4 + b*x^2))/x^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**6,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**6, x)

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GIAC/XCAS [A]  time = 0.253153, size = 155, normalized size = 1.17 \[ \frac{2 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B c{\rm sign}\left (x\right ) + 6 \, \sqrt{c x^{2} + b} B b c{\rm sign}\left (x\right ) + 6 \, \sqrt{c x^{2} + b} A c^{2}{\rm sign}\left (x\right ) - \frac{3 \, \sqrt{c x^{2} + b} A b c{\rm sign}\left (x\right )}{x^{2}} + \frac{3 \,{\left (2 \, B b^{2} c{\rm sign}\left (x\right ) + 3 \, A b c^{2}{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^6,x, algorithm="giac")

[Out]

1/6*(2*(c*x^2 + b)^(3/2)*B*c*sign(x) + 6*sqrt(c*x^2 + b)*B*b*c*sign(x) + 6*sqrt(
c*x^2 + b)*A*c^2*sign(x) - 3*sqrt(c*x^2 + b)*A*b*c*sign(x)/x^2 + 3*(2*B*b^2*c*si
gn(x) + 3*A*b*c^2*sign(x))*arctan(sqrt(c*x^2 + b)/sqrt(-b))/sqrt(-b))/c