3.40 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{17}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\left (b+c x^2\right )^4 (5 b B-A c)}{40 b^2 x^8}-\frac{A \left (b+c x^2\right )^4}{10 b x^{10}} \]

[Out]

-(A*(b + c*x^2)^4)/(10*b*x^10) - ((5*b*B - A*c)*(b + c*x^2)^4)/(40*b^2*x^8)

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Rubi [A]  time = 0.127253, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\left (b+c x^2\right )^4 (5 b B-A c)}{40 b^2 x^8}-\frac{A \left (b+c x^2\right )^4}{10 b x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^17,x]

[Out]

-(A*(b + c*x^2)^4)/(10*b*x^10) - ((5*b*B - A*c)*(b + c*x^2)^4)/(40*b^2*x^8)

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Rubi in Sympy [A]  time = 20.7863, size = 70, normalized size = 1.43 \[ - \frac{A b^{3}}{10 x^{10}} - \frac{B c^{3}}{2 x^{2}} - \frac{b^{2} \left (3 A c + B b\right )}{8 x^{8}} - \frac{b c \left (A c + B b\right )}{2 x^{6}} - \frac{c^{2} \left (A c + 3 B b\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**17,x)

[Out]

-A*b**3/(10*x**10) - B*c**3/(2*x**2) - b**2*(3*A*c + B*b)/(8*x**8) - b*c*(A*c +
B*b)/(2*x**6) - c**2*(A*c + 3*B*b)/(4*x**4)

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Mathematica [A]  time = 0.0345217, size = 78, normalized size = 1.59 \[ -\frac{A \left (4 b^3+15 b^2 c x^2+20 b c^2 x^4+10 c^3 x^6\right )+5 B x^2 \left (b^3+4 b^2 c x^2+6 b c^2 x^4+4 c^3 x^6\right )}{40 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^17,x]

[Out]

-(5*B*x^2*(b^3 + 4*b^2*c*x^2 + 6*b*c^2*x^4 + 4*c^3*x^6) + A*(4*b^3 + 15*b^2*c*x^
2 + 20*b*c^2*x^4 + 10*c^3*x^6))/(40*x^10)

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Maple [A]  time = 0.007, size = 66, normalized size = 1.4 \[ -{\frac{bc \left ( Ac+Bb \right ) }{2\,{x}^{6}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{4\,{x}^{4}}}-{\frac{A{b}^{3}}{10\,{x}^{10}}}-{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{8\,{x}^{8}}}-{\frac{B{c}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^3/x^17,x)

[Out]

-1/2*b*c*(A*c+B*b)/x^6-1/4*c^2*(A*c+3*B*b)/x^4-1/10*A*b^3/x^10-1/8*b^2*(3*A*c+B*
b)/x^8-1/2*B*c^3/x^2

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Maxima [A]  time = 1.37082, size = 101, normalized size = 2.06 \[ -\frac{20 \, B c^{3} x^{8} + 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 20 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + 4 \, A b^{3} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^17,x, algorithm="maxima")

[Out]

-1/40*(20*B*c^3*x^8 + 10*(3*B*b*c^2 + A*c^3)*x^6 + 20*(B*b^2*c + A*b*c^2)*x^4 +
4*A*b^3 + 5*(B*b^3 + 3*A*b^2*c)*x^2)/x^10

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Fricas [A]  time = 0.205925, size = 101, normalized size = 2.06 \[ -\frac{20 \, B c^{3} x^{8} + 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 20 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + 4 \, A b^{3} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^17,x, algorithm="fricas")

[Out]

-1/40*(20*B*c^3*x^8 + 10*(3*B*b*c^2 + A*c^3)*x^6 + 20*(B*b^2*c + A*b*c^2)*x^4 +
4*A*b^3 + 5*(B*b^3 + 3*A*b^2*c)*x^2)/x^10

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Sympy [A]  time = 7.52738, size = 80, normalized size = 1.63 \[ - \frac{4 A b^{3} + 20 B c^{3} x^{8} + x^{6} \left (10 A c^{3} + 30 B b c^{2}\right ) + x^{4} \left (20 A b c^{2} + 20 B b^{2} c\right ) + x^{2} \left (15 A b^{2} c + 5 B b^{3}\right )}{40 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**17,x)

[Out]

-(4*A*b**3 + 20*B*c**3*x**8 + x**6*(10*A*c**3 + 30*B*b*c**2) + x**4*(20*A*b*c**2
+ 20*B*b**2*c) + x**2*(15*A*b**2*c + 5*B*b**3))/(40*x**10)

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GIAC/XCAS [A]  time = 0.211149, size = 107, normalized size = 2.18 \[ -\frac{20 \, B c^{3} x^{8} + 30 \, B b c^{2} x^{6} + 10 \, A c^{3} x^{6} + 20 \, B b^{2} c x^{4} + 20 \, A b c^{2} x^{4} + 5 \, B b^{3} x^{2} + 15 \, A b^{2} c x^{2} + 4 \, A b^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^17,x, algorithm="giac")

[Out]

-1/40*(20*B*c^3*x^8 + 30*B*b*c^2*x^6 + 10*A*c^3*x^6 + 20*B*b^2*c*x^4 + 20*A*b*c^
2*x^4 + 5*B*b^3*x^2 + 15*A*b^2*c*x^2 + 4*A*b^3)/x^10