3.125 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=135 \[ \frac{3 c \sqrt{b x^2+c x^4} (A c+4 b B)}{8 b x}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{b}}-\frac{\left (b x^2+c x^4\right )^{3/2} (A c+4 b B)}{8 b x^5}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9} \]

[Out]

(3*c*(4*b*B + A*c)*Sqrt[b*x^2 + c*x^4])/(8*b*x) - ((4*b*B + A*c)*(b*x^2 + c*x^4)
^(3/2))/(8*b*x^5) - (A*(b*x^2 + c*x^4)^(5/2))/(4*b*x^9) - (3*c*(4*b*B + A*c)*Arc
Tanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(8*Sqrt[b])

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Rubi [A]  time = 0.354982, antiderivative size = 135, 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 \[ \frac{3 c \sqrt{b x^2+c x^4} (A c+4 b B)}{8 b x}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{b}}-\frac{\left (b x^2+c x^4\right )^{3/2} (A c+4 b B)}{8 b x^5}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^8,x]

[Out]

(3*c*(4*b*B + A*c)*Sqrt[b*x^2 + c*x^4])/(8*b*x) - ((4*b*B + A*c)*(b*x^2 + c*x^4)
^(3/2))/(8*b*x^5) - (A*(b*x^2 + c*x^4)^(5/2))/(4*b*x^9) - (3*c*(4*b*B + A*c)*Arc
Tanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(8*Sqrt[b])

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Rubi in Sympy [A]  time = 27.6024, size = 121, normalized size = 0.9 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{4 b x^{9}} + \frac{3 c \left (A c + 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{8 b x} - \frac{\left (A c + 4 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{8 b x^{5}} - \frac{3 c \left (A c + 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{8 \sqrt{b}} \]

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**8,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(4*b*x**9) + 3*c*(A*c + 4*B*b)*sqrt(b*x**2 + c*x**4)
/(8*b*x) - (A*c + 4*B*b)*(b*x**2 + c*x**4)**(3/2)/(8*b*x**5) - 3*c*(A*c + 4*B*b)
*atanh(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/(8*sqrt(b))

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Mathematica [A]  time = 0.186026, size = 133, normalized size = 0.99 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 c x^4 \log (x) (A c+4 b B)-\sqrt{b} \sqrt{b+c x^2} \left (2 A b+5 A c x^2+4 b B x^2-8 B c x^4\right )-3 c x^4 (A c+4 b B) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )\right )}{8 \sqrt{b} x^5 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^8,x]

[Out]

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

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Maple [A]  time = 0.017, size = 227, normalized size = 1.7 \[ -{\frac{1}{8\,{x}^{7}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -A{c}^{2} \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{4}\sqrt{b}+12\,Bc\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{3}-4\,Bc \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{4}{b}^{3/2}+Ac \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{2}\sqrt{b}-3\,A{c}^{2}\sqrt{c{x}^{2}+b}{x}^{4}{b}^{3/2}+3\,A{c}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{2}+4\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{3/2}-12\,Bc\sqrt{c{x}^{2}+b}{x}^{4}{b}^{5/2}+2\,A \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{3/2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]

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^8,x)

[Out]

-1/8*(c*x^4+b*x^2)^(3/2)*(-A*c^2*(c*x^2+b)^(3/2)*x^4*b^(1/2)+12*B*c*ln(2*(b^(1/2
)*(c*x^2+b)^(1/2)+b)/x)*x^4*b^3-4*B*c*(c*x^2+b)^(3/2)*x^4*b^(3/2)+A*c*(c*x^2+b)^
(5/2)*x^2*b^(1/2)-3*A*c^2*(c*x^2+b)^(1/2)*x^4*b^(3/2)+3*A*c^2*ln(2*(b^(1/2)*(c*x
^2+b)^(1/2)+b)/x)*x^4*b^2+4*B*(c*x^2+b)^(5/2)*x^2*b^(3/2)-12*B*c*(c*x^2+b)^(1/2)
*x^4*b^(5/2)+2*A*(c*x^2+b)^(5/2)*b^(3/2))/x^7/(c*x^2+b)^(3/2)/b^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250745, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B b c + A c^{2}\right )} \sqrt{b} x^{5} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \,{\left (8 \, B b c x^{4} - 2 \, A b^{2} -{\left (4 \, B b^{2} + 5 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{16 \, b x^{5}}, \frac{3 \,{\left (4 \, B b c + A c^{2}\right )} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (8 \, B b c x^{4} - 2 \, A b^{2} -{\left (4 \, B b^{2} + 5 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{8 \, b x^{5}}\right ] \]

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^8,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{8}}\, dx \]

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**8,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**8, x)

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GIAC/XCAS [A]  time = 0.250429, size = 196, normalized size = 1.45 \[ \frac{8 \, \sqrt{c x^{2} + b} B c^{2}{\rm sign}\left (x\right ) + \frac{3 \,{\left (4 \, B b c^{2}{\rm sign}\left (x\right ) + A c^{3}{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{4 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B b c^{2}{\rm sign}\left (x\right ) - 4 \, \sqrt{c x^{2} + b} B b^{2} c^{2}{\rm sign}\left (x\right ) + 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A c^{3}{\rm sign}\left (x\right ) - 3 \, \sqrt{c x^{2} + b} A b c^{3}{\rm sign}\left (x\right )}{c^{2} x^{4}}}{8 \, c} \]

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^8,x, algorithm="giac")

[Out]

1/8*(8*sqrt(c*x^2 + b)*B*c^2*sign(x) + 3*(4*B*b*c^2*sign(x) + A*c^3*sign(x))*arc
tan(sqrt(c*x^2 + b)/sqrt(-b))/sqrt(-b) - (4*(c*x^2 + b)^(3/2)*B*b*c^2*sign(x) -
4*sqrt(c*x^2 + b)*B*b^2*c^2*sign(x) + 5*(c*x^2 + b)^(3/2)*A*c^3*sign(x) - 3*sqrt
(c*x^2 + b)*A*b*c^3*sign(x))/(c^2*x^4))/c