3.127 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=177 \[ \frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{128 b^{5/2}}-\frac{c^2 \sqrt{b x^2+c x^4} (8 b B-3 A c)}{128 b^2 x^3}-\frac{\left (b x^2+c x^4\right )^{3/2} (8 b B-3 A c)}{48 b x^9}-\frac{c \sqrt{b x^2+c x^4} (8 b B-3 A c)}{64 b x^5}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}} \]

[Out]

-(c*(8*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(64*b*x^5) - (c^2*(8*b*B - 3*A*c)*Sqrt[
b*x^2 + c*x^4])/(128*b^2*x^3) - ((8*b*B - 3*A*c)*(b*x^2 + c*x^4)^(3/2))/(48*b*x^
9) - (A*(b*x^2 + c*x^4)^(5/2))/(8*b*x^13) + (c^3*(8*b*B - 3*A*c)*ArcTanh[(Sqrt[b
]*x)/Sqrt[b*x^2 + c*x^4]])/(128*b^(5/2))

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Rubi [A]  time = 0.45588, antiderivative size = 177, 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 \[ \frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{128 b^{5/2}}-\frac{c^2 \sqrt{b x^2+c x^4} (8 b B-3 A c)}{128 b^2 x^3}-\frac{\left (b x^2+c x^4\right )^{3/2} (8 b B-3 A c)}{48 b x^9}-\frac{c \sqrt{b x^2+c x^4} (8 b B-3 A c)}{64 b x^5}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^12,x]

[Out]

-(c*(8*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(64*b*x^5) - (c^2*(8*b*B - 3*A*c)*Sqrt[
b*x^2 + c*x^4])/(128*b^2*x^3) - ((8*b*B - 3*A*c)*(b*x^2 + c*x^4)^(3/2))/(48*b*x^
9) - (A*(b*x^2 + c*x^4)^(5/2))/(8*b*x^13) + (c^3*(8*b*B - 3*A*c)*ArcTanh[(Sqrt[b
]*x)/Sqrt[b*x^2 + c*x^4]])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 37.6962, size = 162, normalized size = 0.92 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{8 b x^{13}} + \frac{c \left (3 A c - 8 B b\right ) \sqrt{b x^{2} + c x^{4}}}{64 b x^{5}} + \frac{\left (3 A c - 8 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{48 b x^{9}} + \frac{c^{2} \left (3 A c - 8 B b\right ) \sqrt{b x^{2} + c x^{4}}}{128 b^{2} x^{3}} - \frac{c^{3} \left (3 A c - 8 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{128 b^{\frac{5}{2}}} \]

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**12,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(8*b*x**13) + c*(3*A*c - 8*B*b)*sqrt(b*x**2 + c*x**4
)/(64*b*x**5) + (3*A*c - 8*B*b)*(b*x**2 + c*x**4)**(3/2)/(48*b*x**9) + c**2*(3*A
*c - 8*B*b)*sqrt(b*x**2 + c*x**4)/(128*b**2*x**3) - c**3*(3*A*c - 8*B*b)*atanh(s
qrt(b)*x/sqrt(b*x**2 + c*x**4))/(128*b**(5/2))

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Mathematica [A]  time = 0.290603, size = 175, normalized size = 0.99 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{b} \sqrt{b+c x^2} \left (A \left (48 b^3+72 b^2 c x^2+6 b c^2 x^4-9 c^3 x^6\right )+8 b B x^2 \left (8 b^2+14 b c x^2+3 c^2 x^4\right )\right )+3 c^3 x^8 \log (x) (8 b B-3 A c)+3 c^3 x^8 (3 A c-8 b B) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )\right )}{384 b^{5/2} x^9 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^12,x]

[Out]

-(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b]*Sqrt[b + c*x^2]*(8*b*B*x^2*(8*b^2 + 14*b*c*x^2
+ 3*c^2*x^4) + A*(48*b^3 + 72*b^2*c*x^2 + 6*b*c^2*x^4 - 9*c^3*x^6)) + 3*c^3*(8*b
*B - 3*A*c)*x^8*Log[x] + 3*c^3*(-8*b*B + 3*A*c)*x^8*Log[b + Sqrt[b]*Sqrt[b + c*x
^2]]))/(384*b^(5/2)*x^9*Sqrt[b + c*x^2])

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Maple [B]  time = 0.029, size = 316, normalized size = 1.8 \[ -{\frac{1}{384\,{x}^{11}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -3\,A{c}^{4} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{8}{b}^{5/2}+8\,B{c}^{3} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{8}{b}^{7/2}+3\,A{c}^{3} \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}{b}^{5/2}-9\,A{c}^{4}\sqrt{c{x}^{2}+b}{x}^{8}{b}^{7/2}-8\,B{c}^{2} \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}{b}^{7/2}+24\,B{c}^{3}\sqrt{c{x}^{2}+b}{x}^{8}{b}^{9/2}-24\,B{c}^{3}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{8}{b}^{5}+6\,A{c}^{2} \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{7/2}+9\,A{c}^{4}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{8}{b}^{4}-16\,Bc \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{9/2}-24\,Ac \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{9/2}+64\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{11/2}+48\,A \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{11/2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{13}{2}}}} \]

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^12,x)

[Out]

-1/384*(c*x^4+b*x^2)^(3/2)*(-3*A*c^4*(c*x^2+b)^(3/2)*x^8*b^(5/2)+8*B*c^3*(c*x^2+
b)^(3/2)*x^8*b^(7/2)+3*A*c^3*(c*x^2+b)^(5/2)*x^6*b^(5/2)-9*A*c^4*(c*x^2+b)^(1/2)
*x^8*b^(7/2)-8*B*c^2*(c*x^2+b)^(5/2)*x^6*b^(7/2)+24*B*c^3*(c*x^2+b)^(1/2)*x^8*b^
(9/2)-24*B*c^3*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^8*b^5+6*A*c^2*(c*x^2+b)^(5/
2)*x^4*b^(7/2)+9*A*c^4*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^8*b^4-16*B*c*(c*x^2
+b)^(5/2)*x^4*b^(9/2)-24*A*c*(c*x^2+b)^(5/2)*x^2*b^(9/2)+64*B*(c*x^2+b)^(5/2)*x^
2*b^(11/2)+48*A*(c*x^2+b)^(5/2)*b^(11/2))/x^11/(c*x^2+b)^(3/2)/b^(13/2)

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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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320504, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} \sqrt{b} x^{9} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \,{\left (3 \,{\left (8 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{6} + 48 \, A b^{4} + 2 \,{\left (56 \, B b^{3} c + 3 \, A b^{2} c^{2}\right )} x^{4} + 8 \,{\left (8 \, B b^{4} + 9 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{768 \, b^{3} x^{9}}, -\frac{3 \,{\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} \sqrt{-b} x^{9} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (3 \,{\left (8 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{6} + 48 \, A b^{4} + 2 \,{\left (56 \, B b^{3} c + 3 \, A b^{2} c^{2}\right )} x^{4} + 8 \,{\left (8 \, B b^{4} + 9 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{384 \, b^{3} x^{9}}\right ] \]

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^12,x, algorithm="fricas")

[Out]

[-1/768*(3*(8*B*b*c^3 - 3*A*c^4)*sqrt(b)*x^9*log(-((c*x^3 + 2*b*x)*sqrt(b) - 2*s
qrt(c*x^4 + b*x^2)*b)/x^3) + 2*(3*(8*B*b^2*c^2 - 3*A*b*c^3)*x^6 + 48*A*b^4 + 2*(
56*B*b^3*c + 3*A*b^2*c^2)*x^4 + 8*(8*B*b^4 + 9*A*b^3*c)*x^2)*sqrt(c*x^4 + b*x^2)
)/(b^3*x^9), -1/384*(3*(8*B*b*c^3 - 3*A*c^4)*sqrt(-b)*x^9*arctan(sqrt(-b)*x/sqrt
(c*x^4 + b*x^2)) + (3*(8*B*b^2*c^2 - 3*A*b*c^3)*x^6 + 48*A*b^4 + 2*(56*B*b^3*c +
3*A*b^2*c^2)*x^4 + 8*(8*B*b^4 + 9*A*b^3*c)*x^2)*sqrt(c*x^4 + b*x^2))/(b^3*x^9)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{12}}\, dx \]

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**12,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**12, x)

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GIAC/XCAS [A]  time = 0.334166, size = 289, normalized size = 1.63 \[ -\frac{\frac{3 \,{\left (8 \, B b c^{4}{\rm sign}\left (x\right ) - 3 \, A c^{5}{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{24 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} B b c^{4}{\rm sign}\left (x\right ) + 40 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} B b^{2} c^{4}{\rm sign}\left (x\right ) - 88 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B b^{3} c^{4}{\rm sign}\left (x\right ) + 24 \, \sqrt{c x^{2} + b} B b^{4} c^{4}{\rm sign}\left (x\right ) - 9 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} A c^{5}{\rm sign}\left (x\right ) + 33 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} A b c^{5}{\rm sign}\left (x\right ) + 33 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A b^{2} c^{5}{\rm sign}\left (x\right ) - 9 \, \sqrt{c x^{2} + b} A b^{3} c^{5}{\rm sign}\left (x\right )}{b^{2} c^{4} x^{8}}}{384 \, c} \]

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^12,x, algorithm="giac")

[Out]

-1/384*(3*(8*B*b*c^4*sign(x) - 3*A*c^5*sign(x))*arctan(sqrt(c*x^2 + b)/sqrt(-b))
/(sqrt(-b)*b^2) + (24*(c*x^2 + b)^(7/2)*B*b*c^4*sign(x) + 40*(c*x^2 + b)^(5/2)*B
*b^2*c^4*sign(x) - 88*(c*x^2 + b)^(3/2)*B*b^3*c^4*sign(x) + 24*sqrt(c*x^2 + b)*B
*b^4*c^4*sign(x) - 9*(c*x^2 + b)^(7/2)*A*c^5*sign(x) + 33*(c*x^2 + b)^(5/2)*A*b*
c^5*sign(x) + 33*(c*x^2 + b)^(3/2)*A*b^2*c^5*sign(x) - 9*sqrt(c*x^2 + b)*A*b^3*c
^5*sign(x))/(b^2*c^4*x^8))/c