3.113 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=104 \[ -\frac{A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}}+B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{B c \sqrt{b x^2+c x^4}}{x^2}-\frac{B \left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]

[Out]

-((B*c*Sqrt[b*x^2 + c*x^4])/x^2) - (B*(b*x^2 + c*x^4)^(3/2))/(3*x^6) - (A*(b*x^2
+ c*x^4)^(5/2))/(5*b*x^10) + B*c^(3/2)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4
]]

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Rubi [A]  time = 0.446767, antiderivative size = 104, 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 \[ -\frac{A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}}+B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{B c \sqrt{b x^2+c x^4}}{x^2}-\frac{B \left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^9,x]

[Out]

-((B*c*Sqrt[b*x^2 + c*x^4])/x^2) - (B*(b*x^2 + c*x^4)^(3/2))/(3*x^6) - (A*(b*x^2
+ c*x^4)^(5/2))/(5*b*x^10) + B*c^(3/2)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4
]]

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Rubi in Sympy [A]  time = 25.6657, size = 92, normalized size = 0.88 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{5 b x^{10}} + B c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )} - \frac{B c \sqrt{b x^{2} + c x^{4}}}{x^{2}} - \frac{B \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 x^{6}} \]

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**9,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(5*b*x**10) + B*c**(3/2)*atanh(sqrt(c)*x**2/sqrt(b*x
**2 + c*x**4)) - B*c*sqrt(b*x**2 + c*x**4)/x**2 - B*(b*x**2 + c*x**4)**(3/2)/(3*
x**6)

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Mathematica [A]  time = 0.184325, size = 112, normalized size = 1.08 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (15 b B c^{3/2} x^5 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-\sqrt{b+c x^2} \left (3 A \left (b+c x^2\right )^2+5 b B x^2 \left (b+4 c x^2\right )\right )\right )}{15 b x^6 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^9,x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-(Sqrt[b + c*x^2]*(3*A*(b + c*x^2)^2 + 5*b*B*x^2*(b + 4*
c*x^2))) + 15*b*B*c^(3/2)*x^5*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(15*b*x^6*Sqr
t[b + c*x^2])

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Maple [A]  time = 0.019, size = 142, normalized size = 1.4 \[ -{\frac{1}{15\,{x}^{8}{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -15\,B{c}^{3/2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){b}^{2}{x}^{5}-10\,B{c}^{2}{x}^{6} \left ( c{x}^{2}+b \right ) ^{3/2}+10\,Bc \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}-15\,B{c}^{2}{x}^{6}\sqrt{c{x}^{2}+b}b+5\,B \left ( c{x}^{2}+b \right ) ^{5/2}b{x}^{2}+3\,A \left ( c{x}^{2}+b \right ) ^{5/2}b \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

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^9,x)

[Out]

-1/15*(c*x^4+b*x^2)^(3/2)*(-15*B*c^(3/2)*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*b^2*x^5-1
0*B*c^2*x^6*(c*x^2+b)^(3/2)+10*B*c*(c*x^2+b)^(5/2)*x^4-15*B*c^2*x^6*(c*x^2+b)^(1
/2)*b+5*B*(c*x^2+b)^(5/2)*b*x^2+3*A*(c*x^2+b)^(5/2)*b)/x^8/(c*x^2+b)^(3/2)/b^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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24232, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B b c^{\frac{3}{2}} x^{6} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left ({\left (20 \, B b c + 3 \, A c^{2}\right )} x^{4} + 3 \, A b^{2} +{\left (5 \, B b^{2} + 6 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{30 \, b x^{6}}, \frac{15 \, B b \sqrt{-c} c x^{6} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}\right ) -{\left ({\left (20 \, B b c + 3 \, A c^{2}\right )} x^{4} + 3 \, A b^{2} +{\left (5 \, B b^{2} + 6 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, b x^{6}}\right ] \]

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^9,x, algorithm="fricas")

[Out]

[1/30*(15*B*b*c^(3/2)*x^6*log(-2*c*x^2 - b - 2*sqrt(c*x^4 + b*x^2)*sqrt(c)) - 2*
((20*B*b*c + 3*A*c^2)*x^4 + 3*A*b^2 + (5*B*b^2 + 6*A*b*c)*x^2)*sqrt(c*x^4 + b*x^
2))/(b*x^6), 1/15*(15*B*b*sqrt(-c)*c*x^6*arctan(c*x^2/(sqrt(c*x^4 + b*x^2)*sqrt(
-c))) - ((20*B*b*c + 3*A*c^2)*x^4 + 3*A*b^2 + (5*B*b^2 + 6*A*b*c)*x^2)*sqrt(c*x^
4 + b*x^2))/(b*x^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{9}}\, dx \]

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**9,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**9, x)

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GIAC/XCAS [A]  time = 0.471783, size = 343, normalized size = 3.3 \[ -\frac{1}{2} \, B c^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \,{\left (30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A c^{\frac{5}{2}}{\rm sign}\left (x\right ) - 90 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{2} c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 110 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{3} c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{2} c^{\frac{5}{2}}{\rm sign}\left (x\right ) - 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{4} c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 20 \, B b^{5} c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 3 \, A b^{4} c^{\frac{5}{2}}{\rm sign}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{5}} \]

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^9,x, algorithm="giac")

[Out]

-1/2*B*c^(3/2)*ln((sqrt(c)*x - sqrt(c*x^2 + b))^2)*sign(x) + 2/15*(30*(sqrt(c)*x
- sqrt(c*x^2 + b))^8*B*b*c^(3/2)*sign(x) + 15*(sqrt(c)*x - sqrt(c*x^2 + b))^8*A
*c^(5/2)*sign(x) - 90*(sqrt(c)*x - sqrt(c*x^2 + b))^6*B*b^2*c^(3/2)*sign(x) + 11
0*(sqrt(c)*x - sqrt(c*x^2 + b))^4*B*b^3*c^(3/2)*sign(x) + 30*(sqrt(c)*x - sqrt(c
*x^2 + b))^4*A*b^2*c^(5/2)*sign(x) - 70*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^4*c^
(3/2)*sign(x) + 20*B*b^5*c^(3/2)*sign(x) + 3*A*b^4*c^(5/2)*sign(x))/((sqrt(c)*x
- sqrt(c*x^2 + b))^2 - b)^5