3.58 \(\int \frac{(a+b x) (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=33 \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{b B}{2 x^2} \]

[Out]

-(a*A)/(4*x^4) - (A*b + a*B)/(3*x^3) - (b*B)/(2*x^2)

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Rubi [A]  time = 0.0437458, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{b B}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/x^5,x]

[Out]

-(a*A)/(4*x^4) - (A*b + a*B)/(3*x^3) - (b*B)/(2*x^2)

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Rubi in Sympy [A]  time = 9.42148, size = 31, normalized size = 0.94 \[ - \frac{A a}{4 x^{4}} - \frac{B b}{2 x^{2}} - \frac{\frac{A b}{3} + \frac{B a}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/x**5,x)

[Out]

-A*a/(4*x**4) - B*b/(2*x**2) - (A*b/3 + B*a/3)/x**3

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Mathematica [A]  time = 0.0139481, size = 29, normalized size = 0.88 \[ -\frac{3 a A+4 a B x+4 A b x+6 b B x^2}{12 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/x^5,x]

[Out]

-(3*a*A + 4*A*b*x + 4*a*B*x + 6*b*B*x^2)/(12*x^4)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Bb}{2\,{x}^{2}}}-{\frac{Ab+Ba}{3\,{x}^{3}}}-{\frac{Aa}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.34802, size = 36, normalized size = 1.09 \[ -\frac{6 \, B b x^{2} + 3 \, A a + 4 \,{\left (B a + A b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^5,x, algorithm="maxima")

[Out]

-1/12*(6*B*b*x^2 + 3*A*a + 4*(B*a + A*b)*x)/x^4

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Fricas [A]  time = 0.193922, size = 36, normalized size = 1.09 \[ -\frac{6 \, B b x^{2} + 3 \, A a + 4 \,{\left (B a + A b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^5,x, algorithm="fricas")

[Out]

-1/12*(6*B*b*x^2 + 3*A*a + 4*(B*a + A*b)*x)/x^4

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Sympy [A]  time = 2.02201, size = 31, normalized size = 0.94 \[ - \frac{3 A a + 6 B b x^{2} + x \left (4 A b + 4 B a\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/x**5,x)

[Out]

-(3*A*a + 6*B*b*x**2 + x*(4*A*b + 4*B*a))/(12*x**4)

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GIAC/XCAS [A]  time = 0.375601, size = 36, normalized size = 1.09 \[ -\frac{6 \, B b x^{2} + 4 \, B a x + 4 \, A b x + 3 \, A a}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^5,x, algorithm="giac")

[Out]

-1/12*(6*B*b*x^2 + 4*B*a*x + 4*A*b*x + 3*A*a)/x^4