3.10 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^7} \, dx\)

Optimal. Leaf size=29 \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0713066, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.82072, size = 31, normalized size = 1.07 \[ - \frac{A b}{4 x^{4}} + \frac{B c \log{\left (x^{2} \right )}}{2} - \frac{\frac{A c}{2} + \frac{B b}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0297427, size = 31, normalized size = 1.07 \[ \frac{-A c-b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^7,x]

[Out]

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

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Maple [A]  time = 0.009, size = 28, normalized size = 1. \[ Bc\ln \left ( x \right ) -{\frac{Ab}{4\,{x}^{4}}}-{\frac{Ac}{2\,{x}^{2}}}-{\frac{Bb}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.37416, size = 41, normalized size = 1.41 \[ \frac{1}{2} \, B c \log \left (x^{2}\right ) - \frac{2 \,{\left (B b + A c\right )} x^{2} + A b}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.242909, size = 42, normalized size = 1.45 \[ \frac{4 \, B c x^{4} \log \left (x\right ) - 2 \,{\left (B b + A c\right )} x^{2} - A b}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.96944, size = 27, normalized size = 0.93 \[ B c \log{\left (x \right )} - \frac{A b + x^{2} \left (2 A c + 2 B b\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209331, size = 53, normalized size = 1.83 \[ \frac{1}{2} \, B c{\rm ln}\left (x^{2}\right ) - \frac{3 \, B c x^{4} + 2 \, B b x^{2} + 2 \, A c x^{2} + A b}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/x^7,x, algorithm="giac")

[Out]

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