3.31 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^8} \, dx\)

Optimal. Leaf size=65 \[ -\frac{A b^3}{x}+b^2 x (3 A c+b B)+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7 \]

[Out]

-((A*b^3)/x) + b^2*(b*B + 3*A*c)*x + b*c*(b*B + A*c)*x^3 + (c^2*(3*b*B + A*c)*x^
5)/5 + (B*c^3*x^7)/7

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Rubi [A]  time = 0.125101, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A b^3}{x}+b^2 x (3 A c+b B)+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^8,x]

[Out]

-((A*b^3)/x) + b^2*(b*B + 3*A*c)*x + b*c*(b*B + A*c)*x^3 + (c^2*(3*b*B + A*c)*x^
5)/5 + (B*c^3*x^7)/7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A b^{3}}{x} + \frac{B c^{3} x^{7}}{7} + b c x^{3} \left (A c + B b\right ) + \frac{c^{2} x^{5} \left (A c + 3 B b\right )}{5} + \frac{b^{2} \left (3 A c + B b\right ) \int B\, dx}{B} \]

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**8,x)

[Out]

-A*b**3/x + B*c**3*x**7/7 + b*c*x**3*(A*c + B*b) + c**2*x**5*(A*c + 3*B*b)/5 + b
**2*(3*A*c + B*b)*Integral(B, x)/B

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Mathematica [A]  time = 0.0404254, size = 65, normalized size = 1. \[ -\frac{A b^3}{x}+b^2 x (3 A c+b B)+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^8,x]

[Out]

-((A*b^3)/x) + b^2*(b*B + 3*A*c)*x + b*c*(b*B + A*c)*x^3 + (c^2*(3*b*B + A*c)*x^
5)/5 + (B*c^3*x^7)/7

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Maple [A]  time = 0.006, size = 71, normalized size = 1.1 \[{\frac{B{c}^{3}{x}^{7}}{7}}+{\frac{A{x}^{5}{c}^{3}}{5}}+{\frac{3\,B{x}^{5}b{c}^{2}}{5}}+A{x}^{3}b{c}^{2}+B{x}^{3}{b}^{2}c+3\,Ax{b}^{2}c+Bx{b}^{3}-{\frac{A{b}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^3/x^8,x)

[Out]

1/7*B*c^3*x^7+1/5*A*x^5*c^3+3/5*B*x^5*b*c^2+A*x^3*b*c^2+B*x^3*b^2*c+3*A*x*b^2*c+
B*x*b^3-A*b^3/x

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Maxima [A]  time = 1.37591, size = 93, normalized size = 1.43 \[ \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} +{\left (B b^{2} c + A b c^{2}\right )} x^{3} - \frac{A b^{3}}{x} +{\left (B b^{3} + 3 \, A b^{2} c\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^8,x, algorithm="maxima")

[Out]

1/7*B*c^3*x^7 + 1/5*(3*B*b*c^2 + A*c^3)*x^5 + (B*b^2*c + A*b*c^2)*x^3 - A*b^3/x
+ (B*b^3 + 3*A*b^2*c)*x

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Fricas [A]  time = 0.199088, size = 101, normalized size = 1.55 \[ \frac{5 \, B c^{3} x^{8} + 7 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 35 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} - 35 \, A b^{3} + 35 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^8,x, algorithm="fricas")

[Out]

1/35*(5*B*c^3*x^8 + 7*(3*B*b*c^2 + A*c^3)*x^6 + 35*(B*b^2*c + A*b*c^2)*x^4 - 35*
A*b^3 + 35*(B*b^3 + 3*A*b^2*c)*x^2)/x

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Sympy [A]  time = 0.631553, size = 68, normalized size = 1.05 \[ - \frac{A b^{3}}{x} + \frac{B c^{3} x^{7}}{7} + x^{5} \left (\frac{A c^{3}}{5} + \frac{3 B b c^{2}}{5}\right ) + x^{3} \left (A b c^{2} + B b^{2} c\right ) + x \left (3 A b^{2} c + B b^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**8,x)

[Out]

-A*b**3/x + B*c**3*x**7/7 + x**5*(A*c**3/5 + 3*B*b*c**2/5) + x**3*(A*b*c**2 + B*
b**2*c) + x*(3*A*b**2*c + B*b**3)

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GIAC/XCAS [A]  time = 0.206951, size = 95, normalized size = 1.46 \[ \frac{1}{7} \, B c^{3} x^{7} + \frac{3}{5} \, B b c^{2} x^{5} + \frac{1}{5} \, A c^{3} x^{5} + B b^{2} c x^{3} + A b c^{2} x^{3} + B b^{3} x + 3 \, A b^{2} c x - \frac{A b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^8,x, algorithm="giac")

[Out]

1/7*B*c^3*x^7 + 3/5*B*b*c^2*x^5 + 1/5*A*c^3*x^5 + B*b^2*c*x^3 + A*b*c^2*x^3 + B*
b^3*x + 3*A*b^2*c*x - A*b^3/x