3.1036 \(\int \frac{a+b x}{(a c-b c x)^5} \, dx\)

Optimal. Leaf size=38 \[ \frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3} \]

[Out]

a/(2*b*c^5*(a - b*x)^4) - 1/(3*b*c^5*(a - b*x)^3)

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Rubi [A]  time = 0.044134, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(a*c - b*c*x)^5,x]

[Out]

a/(2*b*c^5*(a - b*x)^4) - 1/(3*b*c^5*(a - b*x)^3)

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Rubi in Sympy [A]  time = 11.4608, size = 29, normalized size = 0.76 \[ \frac{a}{2 b c^{5} \left (a - b x\right )^{4}} - \frac{1}{3 b c^{5} \left (a - b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(-b*c*x+a*c)**5,x)

[Out]

a/(2*b*c**5*(a - b*x)**4) - 1/(3*b*c**5*(a - b*x)**3)

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Mathematica [A]  time = 0.0160516, size = 24, normalized size = 0.63 \[ \frac{a+2 b x}{6 b c^5 (a-b x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(a*c - b*c*x)^5,x]

[Out]

(a + 2*b*x)/(6*b*c^5*(a - b*x)^4)

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \[{\frac{1}{{c}^{5}} \left ({\frac{1}{3\,b \left ( bx-a \right ) ^{3}}}+{\frac{a}{2\,b \left ( bx-a \right ) ^{4}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b*c*x+a*c)^5,x)

[Out]

1/c^5*(1/3/b/(b*x-a)^3+1/2*a/b/(b*x-a)^4)

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Maxima [A]  time = 1.34531, size = 90, normalized size = 2.37 \[ \frac{2 \, b x + a}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x + a)/(b^5*c^5*x^4 - 4*a*b^4*c^5*x^3 + 6*a^2*b^3*c^5*x^2 - 4*a^3*b^2*c
^5*x + a^4*b*c^5)

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Fricas [A]  time = 0.214355, size = 90, normalized size = 2.37 \[ \frac{2 \, b x + a}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x + a)/(b^5*c^5*x^4 - 4*a*b^4*c^5*x^3 + 6*a^2*b^3*c^5*x^2 - 4*a^3*b^2*c
^5*x + a^4*b*c^5)

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Sympy [A]  time = 2.04459, size = 70, normalized size = 1.84 \[ \frac{a + 2 b x}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(-b*c*x+a*c)**5,x)

[Out]

(a + 2*b*x)/(6*a**4*b*c**5 - 24*a**3*b**2*c**5*x + 36*a**2*b**3*c**5*x**2 - 24*a
*b**4*c**5*x**3 + 6*b**5*c**5*x**4)

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GIAC/XCAS [A]  time = 0.20687, size = 54, normalized size = 1.42 \[ \frac{a}{2 \,{\left (b c x - a c\right )}^{4} b c} + \frac{1}{3 \,{\left (b c x - a c\right )}^{3} b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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