3.259 \(\int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{14}} \, dx\)

Optimal. Leaf size=165 \[ \frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac{3 c^4 \sqrt{b x^2+c x^4}}{256 b^3 x^3}+\frac{c^3 \sqrt{b x^2+c x^4}}{128 b^2 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{160 b x^7}-\frac{\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac{3 c \sqrt{b x^2+c x^4}}{80 x^9} \]

[Out]

(-3*c*Sqrt[b*x^2 + c*x^4])/(80*x^9) - (c^2*Sqrt[b*x^2 + c*x^4])/(160*b*x^7) + (c
^3*Sqrt[b*x^2 + c*x^4])/(128*b^2*x^5) - (3*c^4*Sqrt[b*x^2 + c*x^4])/(256*b^3*x^3
) - (b*x^2 + c*x^4)^(3/2)/(10*x^13) + (3*c^5*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*
x^4]])/(256*b^(7/2))

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Rubi [A]  time = 0.43497, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac{3 c^4 \sqrt{b x^2+c x^4}}{256 b^3 x^3}+\frac{c^3 \sqrt{b x^2+c x^4}}{128 b^2 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{160 b x^7}-\frac{\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac{3 c \sqrt{b x^2+c x^4}}{80 x^9} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(3/2)/x^14,x]

[Out]

(-3*c*Sqrt[b*x^2 + c*x^4])/(80*x^9) - (c^2*Sqrt[b*x^2 + c*x^4])/(160*b*x^7) + (c
^3*Sqrt[b*x^2 + c*x^4])/(128*b^2*x^5) - (3*c^4*Sqrt[b*x^2 + c*x^4])/(256*b^3*x^3
) - (b*x^2 + c*x^4)^(3/2)/(10*x^13) + (3*c^5*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*
x^4]])/(256*b^(7/2))

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Rubi in Sympy [A]  time = 47.7787, size = 150, normalized size = 0.91 \[ - \frac{3 c \sqrt{b x^{2} + c x^{4}}}{80 x^{9}} - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{10 x^{13}} - \frac{c^{2} \sqrt{b x^{2} + c x^{4}}}{160 b x^{7}} + \frac{c^{3} \sqrt{b x^{2} + c x^{4}}}{128 b^{2} x^{5}} - \frac{3 c^{4} \sqrt{b x^{2} + c x^{4}}}{256 b^{3} x^{3}} + \frac{3 c^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{256 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(3/2)/x**14,x)

[Out]

-3*c*sqrt(b*x**2 + c*x**4)/(80*x**9) - (b*x**2 + c*x**4)**(3/2)/(10*x**13) - c**
2*sqrt(b*x**2 + c*x**4)/(160*b*x**7) + c**3*sqrt(b*x**2 + c*x**4)/(128*b**2*x**5
) - 3*c**4*sqrt(b*x**2 + c*x**4)/(256*b**3*x**3) + 3*c**5*atanh(sqrt(b)*x/sqrt(b
*x**2 + c*x**4))/(256*b**(7/2))

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Mathematica [A]  time = 0.270946, size = 137, normalized size = 0.83 \[ -\frac{\sqrt{b+c x^2} \left (\sqrt{b} \sqrt{b+c x^2} \left (128 b^4+176 b^3 c x^2+8 b^2 c^2 x^4-10 b c^3 x^6+15 c^4 x^8\right )-15 c^5 x^{10} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+15 c^5 x^{10} \log (x)\right )}{1280 b^{7/2} x^9 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(3/2)/x^14,x]

[Out]

-(Sqrt[b + c*x^2]*(Sqrt[b]*Sqrt[b + c*x^2]*(128*b^4 + 176*b^3*c*x^2 + 8*b^2*c^2*
x^4 - 10*b*c^3*x^6 + 15*c^4*x^8) + 15*c^5*x^10*Log[x] - 15*c^5*x^10*Log[b + Sqrt
[b]*Sqrt[b + c*x^2]]))/(1280*b^(7/2)*x^9*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.043, size = 186, normalized size = 1.1 \[{\frac{1}{1280\,{x}^{13}{b}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 15\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{3/2}{x}^{10}{c}^{5}-5\, \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{10}{c}^{5}+5\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{8}{c}^{4}-15\,\sqrt{c{x}^{2}+b}{x}^{10}b{c}^{5}+10\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}b{c}^{3}-40\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{2}{c}^{2}+80\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{3}c-128\, \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{4} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^14,x)

[Out]

1/1280*(c*x^4+b*x^2)^(3/2)*(15*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*b^(3/2)*x^10*
c^5-5*(c*x^2+b)^(3/2)*x^10*c^5+5*(c*x^2+b)^(5/2)*x^8*c^4-15*(c*x^2+b)^(1/2)*x^10
*b*c^5+10*(c*x^2+b)^(5/2)*x^6*b*c^3-40*(c*x^2+b)^(5/2)*x^4*b^2*c^2+80*(c*x^2+b)^
(5/2)*x^2*b^3*c-128*(c*x^2+b)^(5/2)*b^4)/x^13/(c*x^2+b)^(3/2)/b^5

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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)/x^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.33097, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b} c^{5} x^{11} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \,{\left (15 \, b c^{4} x^{8} - 10 \, b^{2} c^{3} x^{6} + 8 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} + 128 \, b^{5}\right )} \sqrt{c x^{4} + b x^{2}}}{2560 \, b^{4} x^{11}}, -\frac{15 \, \sqrt{-b} c^{5} x^{11} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (15 \, b c^{4} x^{8} - 10 \, b^{2} c^{3} x^{6} + 8 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} + 128 \, b^{5}\right )} \sqrt{c x^{4} + b x^{2}}}{1280 \, b^{4} x^{11}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^14,x, algorithm="fricas")

[Out]

[1/2560*(15*sqrt(b)*c^5*x^11*log(-((c*x^3 + 2*b*x)*sqrt(b) + 2*sqrt(c*x^4 + b*x^
2)*b)/x^3) - 2*(15*b*c^4*x^8 - 10*b^2*c^3*x^6 + 8*b^3*c^2*x^4 + 176*b^4*c*x^2 +
128*b^5)*sqrt(c*x^4 + b*x^2))/(b^4*x^11), -1/1280*(15*sqrt(-b)*c^5*x^11*arctan(s
qrt(-b)*x/sqrt(c*x^4 + b*x^2)) + (15*b*c^4*x^8 - 10*b^2*c^3*x^6 + 8*b^3*c^2*x^4
+ 176*b^4*c*x^2 + 128*b^5)*sqrt(c*x^4 + b*x^2))/(b^4*x^11)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{14}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(3/2)/x**14,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)/x**14, x)

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GIAC/XCAS [A]  time = 0.34379, size = 149, normalized size = 0.9 \[ -\frac{1}{1280} \, c^{5}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 70 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 128 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} + 70 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3} - 15 \, \sqrt{c x^{2} + b} b^{4}}{b^{3} c^{5} x^{10}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^14,x, algorithm="giac")

[Out]

-1/1280*c^5*(15*arctan(sqrt(c*x^2 + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(c*x^2 + b
)^(9/2) - 70*(c*x^2 + b)^(7/2)*b + 128*(c*x^2 + b)^(5/2)*b^2 + 70*(c*x^2 + b)^(3
/2)*b^3 - 15*sqrt(c*x^2 + b)*b^4)/(b^3*c^5*x^10))*sign(x)