3.257 \(\int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=109 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}-\frac{c \sqrt{b x^2+c x^4}}{8 x^5} \]

[Out]

-(c*Sqrt[b*x^2 + c*x^4])/(8*x^5) - (c^2*Sqrt[b*x^2 + c*x^4])/(16*b*x^3) - (b*x^2
+ c*x^4)^(3/2)/(6*x^9) + (c^3*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(16*b^(
3/2))

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Rubi [A]  time = 0.252975, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}-\frac{c \sqrt{b x^2+c x^4}}{8 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(3/2)/x^10,x]

[Out]

-(c*Sqrt[b*x^2 + c*x^4])/(8*x^5) - (c^2*Sqrt[b*x^2 + c*x^4])/(16*b*x^3) - (b*x^2
+ c*x^4)^(3/2)/(6*x^9) + (c^3*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(16*b^(
3/2))

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Rubi in Sympy [A]  time = 29.4524, size = 94, normalized size = 0.86 \[ - \frac{c \sqrt{b x^{2} + c x^{4}}}{8 x^{5}} - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 x^{9}} - \frac{c^{2} \sqrt{b x^{2} + c x^{4}}}{16 b x^{3}} + \frac{c^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{16 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(3/2)/x**10,x)

[Out]

-c*sqrt(b*x**2 + c*x**4)/(8*x**5) - (b*x**2 + c*x**4)**(3/2)/(6*x**9) - c**2*sqr
t(b*x**2 + c*x**4)/(16*b*x**3) + c**3*atanh(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/(16
*b**(3/2))

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Mathematica [A]  time = 0.113159, size = 115, normalized size = 1.06 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{b} \sqrt{b+c x^2} \left (8 b^2+14 b c x^2+3 c^2 x^4\right )-3 c^3 x^6 \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+3 c^3 x^6 \log (x)\right )}{48 b^{3/2} x^7 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(3/2)/x^10,x]

[Out]

-(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b]*Sqrt[b + c*x^2]*(8*b^2 + 14*b*c*x^2 + 3*c^2*x^4
) + 3*c^3*x^6*Log[x] - 3*c^3*x^6*Log[b + Sqrt[b]*Sqrt[b + c*x^2]]))/(48*b^(3/2)*
x^7*Sqrt[b + c*x^2])

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Maple [A]  time = 0.014, size = 145, normalized size = 1.3 \[{\frac{1}{48\,{x}^{9}{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{3/2}{x}^{6}{c}^{3}- \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{6}{c}^{3}+ \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{4}{c}^{2}-3\,\sqrt{c{x}^{2}+b}{x}^{6}b{c}^{3}+2\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}bc-8\, \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^10,x)

[Out]

1/48*(c*x^4+b*x^2)^(3/2)*(3*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*b^(3/2)*x^6*c^3-
(c*x^2+b)^(3/2)*x^6*c^3+(c*x^2+b)^(5/2)*x^4*c^2-3*(c*x^2+b)^(1/2)*x^6*b*c^3+2*(c
*x^2+b)^(5/2)*x^2*b*c-8*(c*x^2+b)^(5/2)*b^2)/x^9/(c*x^2+b)^(3/2)/b^3

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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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285315, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{b} c^{3} x^{7} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \,{\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, b^{2} x^{7}}, -\frac{3 \, \sqrt{-b} c^{3} x^{7} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, b^{2} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^10,x, algorithm="fricas")

[Out]

[1/96*(3*sqrt(b)*c^3*x^7*log(-((c*x^3 + 2*b*x)*sqrt(b) + 2*sqrt(c*x^4 + b*x^2)*b
)/x^3) - 2*(3*b*c^2*x^4 + 14*b^2*c*x^2 + 8*b^3)*sqrt(c*x^4 + b*x^2))/(b^2*x^7),
-1/48*(3*sqrt(-b)*c^3*x^7*arctan(sqrt(-b)*x/sqrt(c*x^4 + b*x^2)) + (3*b*c^2*x^4
+ 14*b^2*c*x^2 + 8*b^3)*sqrt(c*x^4 + b*x^2))/(b^2*x^7)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(3/2)/x**10,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)/x**10, x)

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GIAC/XCAS [A]  time = 0.303543, size = 111, normalized size = 1.02 \[ -\frac{1}{48} \, c^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} + 8 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{c x^{2} + b} b^{2}}{b c^{3} x^{6}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^10,x, algorithm="giac")

[Out]

-1/48*c^3*(3*arctan(sqrt(c*x^2 + b)/sqrt(-b))/(sqrt(-b)*b) + (3*(c*x^2 + b)^(5/2
) + 8*(c*x^2 + b)^(3/2)*b - 3*sqrt(c*x^2 + b)*b^2)/(b*c^3*x^6))*sign(x)