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

Optimal. Leaf size=51 \[ -\frac{A b^2}{4 x^4}-\frac{b (2 A c+b B)}{2 x^2}+c \log (x) (A c+2 b B)+\frac{1}{2} B c^2 x^2 \]

[Out]

-(A*b^2)/(4*x^4) - (b*(b*B + 2*A*c))/(2*x^2) + (B*c^2*x^2)/2 + c*(2*b*B + A*c)*L
og[x]

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Rubi [A]  time = 0.136238, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{A b^2}{4 x^4}-\frac{b (2 A c+b B)}{2 x^2}+c \log (x) (A c+2 b B)+\frac{1}{2} B c^2 x^2 \]

Antiderivative was successfully verified.

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

[Out]

-(A*b^2)/(4*x^4) - (b*(b*B + 2*A*c))/(2*x^2) + (B*c^2*x^2)/2 + c*(2*b*B + A*c)*L
og[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A b^{2}}{4 x^{4}} - \frac{b \left (2 A c + B b\right )}{2 x^{2}} + \frac{c^{2} \int ^{x^{2}} B\, dx}{2} + \frac{c \left (A c + 2 B b\right ) \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*b**2/(4*x**4) - b*(2*A*c + B*b)/(2*x**2) + c**2*Integral(B, (x, x**2))/2 + c*
(A*c + 2*B*b)*log(x**2)/2

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Mathematica [A]  time = 0.0568923, size = 50, normalized size = 0.98 \[ c \log (x) (A c+2 b B)-\frac{A b \left (b+4 c x^2\right )+2 B x^2 \left (b^2-c^2 x^4\right )}{4 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(A*b*(b + 4*c*x^2) + 2*B*x^2*(b^2 - c^2*x^4))/(4*x^4) + c*(2*b*B + A*c)*Log[x]

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Maple [A]  time = 0.01, size = 51, normalized size = 1. \[{\frac{B{c}^{2}{x}^{2}}{2}}+A\ln \left ( x \right ){c}^{2}+2\,B\ln \left ( x \right ) bc-{\frac{{b}^{2}A}{4\,{x}^{4}}}-{\frac{Abc}{{x}^{2}}}-{\frac{{b}^{2}B}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*c^2*x^2+A*ln(x)*c^2+2*B*ln(x)*b*c-1/4*A*b^2/x^4-b/x^2*A*c-1/2*B*b^2/x^2

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Maxima [A]  time = 1.36992, size = 73, normalized size = 1.43 \[ \frac{1}{2} \, B c^{2} x^{2} + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} \log \left (x^{2}\right ) - \frac{A b^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^9,x, algorithm="maxima")

[Out]

1/2*B*c^2*x^2 + 1/2*(2*B*b*c + A*c^2)*log(x^2) - 1/4*(A*b^2 + 2*(B*b^2 + 2*A*b*c
)*x^2)/x^4

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Fricas [A]  time = 0.206328, size = 74, normalized size = 1.45 \[ \frac{2 \, B c^{2} x^{6} + 4 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} \log \left (x\right ) - A b^{2} - 2 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^9,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.29304, size = 49, normalized size = 0.96 \[ \frac{B c^{2} x^{2}}{2} + c \left (A c + 2 B b\right ) \log{\left (x \right )} - \frac{A b^{2} + x^{2} \left (4 A b c + 2 B b^{2}\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c**2*x**2/2 + c*(A*c + 2*B*b)*log(x) - (A*b**2 + x**2*(4*A*b*c + 2*B*b**2))/(4
*x**4)

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GIAC/XCAS [A]  time = 0.208659, size = 97, normalized size = 1.9 \[ \frac{1}{2} \, B c^{2} x^{2} + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )}{\rm ln}\left (x^{2}\right ) - \frac{6 \, B b c x^{4} + 3 \, A c^{2} x^{4} + 2 \, B b^{2} x^{2} + 4 \, A b c x^{2} + A b^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^9,x, algorithm="giac")

[Out]

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