3.111 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=128 \[ \frac{\left (b x^2+c x^4\right )^{3/2} (4 A c+b B)}{4 b x^2}+\frac{3}{8} \sqrt{b x^2+c x^4} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6} \]

[Out]

(3*(b*B + 4*A*c)*Sqrt[b*x^2 + c*x^4])/8 + ((b*B + 4*A*c)*(b*x^2 + c*x^4)^(3/2))/
(4*b*x^2) - (A*(b*x^2 + c*x^4)^(5/2))/(b*x^6) + (3*b*(b*B + 4*A*c)*ArcTanh[(Sqrt
[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(8*Sqrt[c])

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Rubi [A]  time = 0.490199, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (b x^2+c x^4\right )^{3/2} (4 A c+b B)}{4 b x^2}+\frac{3}{8} \sqrt{b x^2+c x^4} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^5,x]

[Out]

(3*(b*B + 4*A*c)*Sqrt[b*x^2 + c*x^4])/8 + ((b*B + 4*A*c)*(b*x^2 + c*x^4)^(3/2))/
(4*b*x^2) - (A*(b*x^2 + c*x^4)^(5/2))/(b*x^6) + (3*b*(b*B + 4*A*c)*ArcTanh[(Sqrt
[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(8*Sqrt[c])

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Rubi in Sympy [A]  time = 28.1598, size = 117, normalized size = 0.91 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{b x^{6}} + \frac{3 b \left (4 A c + B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{8 \sqrt{c}} + \left (\frac{3 A c}{2} + \frac{3 B b}{8}\right ) \sqrt{b x^{2} + c x^{4}} + \frac{\left (4 A c + B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{4 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**5,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(b*x**6) + 3*b*(4*A*c + B*b)*atanh(sqrt(c)*x**2/sqrt
(b*x**2 + c*x**4))/(8*sqrt(c)) + (3*A*c/2 + 3*B*b/8)*sqrt(b*x**2 + c*x**4) + (4*
A*c + B*b)*(b*x**2 + c*x**4)**(3/2)/(4*b*x**2)

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Mathematica [A]  time = 0.191365, size = 109, normalized size = 0.85 \[ \frac{3 b x \sqrt{b+c x^2} (4 A c+b B) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )+\sqrt{c} \left (b+c x^2\right ) \left (-8 A b+4 A c x^2+5 b B x^2+2 B c x^4\right )}{8 \sqrt{c} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^5,x]

[Out]

(Sqrt[c]*(b + c*x^2)*(-8*A*b + 5*b*B*x^2 + 4*A*c*x^2 + 2*B*c*x^4) + 3*b*(b*B + 4
*A*c)*x*Sqrt[b + c*x^2]*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]])/(8*Sqrt[c]*Sqrt[x^2*
(b + c*x^2)])

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Maple [A]  time = 0.017, size = 174, normalized size = 1.4 \[{\frac{1}{8\,b{x}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 8\,A{c}^{3/2}{x}^{2} \left ( c{x}^{2}+b \right ) ^{3/2}+2\,B{x}^{2} \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}b+12\,Ac{b}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ) x-8\,A \left ( c{x}^{2}+b \right ) ^{5/2}\sqrt{c}+12\,A{c}^{3/2}{x}^{2}\sqrt{c{x}^{2}+b}b+3\,B{b}^{2}{x}^{2}\sqrt{c{x}^{2}+b}\sqrt{c}+3\,B{b}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ) x \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \]

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^5,x)

[Out]

1/8*(c*x^4+b*x^2)^(3/2)*(8*A*c^(3/2)*x^2*(c*x^2+b)^(3/2)+2*B*x^2*(c*x^2+b)^(3/2)
*c^(1/2)*b+12*A*c*b^2*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*x-8*A*(c*x^2+b)^(5/2)*c^(1/2
)+12*A*c^(3/2)*x^2*(c*x^2+b)^(1/2)*b+3*B*b^2*x^2*(c*x^2+b)^(1/2)*c^(1/2)+3*B*b^3
*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*x)/x^4/(c*x^2+b)^(3/2)/c^(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((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247775, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B b^{2} + 4 \, A b c\right )} \sqrt{c} x^{2} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) + 2 \,{\left (2 \, B c^{2} x^{4} - 8 \, A b c +{\left (5 \, B b c + 4 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{16 \, c x^{2}}, -\frac{3 \,{\left (B b^{2} + 4 \, A b c\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) -{\left (2 \, B c^{2} x^{4} - 8 \, A b c +{\left (5 \, B b c + 4 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{8 \, c x^{2}}\right ] \]

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^5,x, algorithm="fricas")

[Out]

[1/16*(3*(B*b^2 + 4*A*b*c)*sqrt(c)*x^2*log(-(2*c*x^2 + b)*sqrt(c) - 2*sqrt(c*x^4
+ b*x^2)*c) + 2*(2*B*c^2*x^4 - 8*A*b*c + (5*B*b*c + 4*A*c^2)*x^2)*sqrt(c*x^4 +
b*x^2))/(c*x^2), -1/8*(3*(B*b^2 + 4*A*b*c)*sqrt(-c)*x^2*arctan(sqrt(-c)*x^2/sqrt
(c*x^4 + b*x^2)) - (2*B*c^2*x^4 - 8*A*b*c + (5*B*b*c + 4*A*c^2)*x^2)*sqrt(c*x^4
+ b*x^2))/(c*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{5}}\, dx \]

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**5,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**5, x)

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GIAC/XCAS [A]  time = 0.259976, size = 170, normalized size = 1.33 \[ \frac{2 \, A b^{2} \sqrt{c}{\rm sign}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b} + \frac{1}{8} \,{\left (2 \, B c x^{2}{\rm sign}\left (x\right ) + \frac{5 \, B b c^{2}{\rm sign}\left (x\right ) + 4 \, A c^{3}{\rm sign}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + b} x - \frac{3 \,{\left (B b^{2} \sqrt{c}{\rm sign}\left (x\right ) + 4 \, A b c^{\frac{3}{2}}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right )}{16 \, c} \]

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^5,x, algorithm="giac")

[Out]

2*A*b^2*sqrt(c)*sign(x)/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b) + 1/8*(2*B*c*x^2*s
ign(x) + (5*B*b*c^2*sign(x) + 4*A*c^3*sign(x))/c^2)*sqrt(c*x^2 + b)*x - 3/16*(B*
b^2*sqrt(c)*sign(x) + 4*A*b*c^(3/2)*sign(x))*ln((sqrt(c)*x - sqrt(c*x^2 + b))^2)
/c