3.1026 \(\int \frac{(a+b x)^5}{(a c+b c x)^8} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{2 b c^8 (a+b x)^2} \]

[Out]

-1/(2*b*c^8*(a + b*x)^2)

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Rubi [A]  time = 0.0122736, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{1}{2 b c^8 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + b*c*x)^8,x]

[Out]

-1/(2*b*c^8*(a + b*x)^2)

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Rubi in Sympy [A]  time = 4.1028, size = 15, normalized size = 0.88 \[ - \frac{1}{2 b c^{8} \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(b*c*x+a*c)**8,x)

[Out]

-1/(2*b*c**8*(a + b*x)**2)

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Mathematica [A]  time = 0.00597728, size = 17, normalized size = 1. \[ -\frac{1}{2 b c^8 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + b*c*x)^8,x]

[Out]

-1/(2*b*c^8*(a + b*x)^2)

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[ -{\frac{1}{2\,b{c}^{8} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^8,x)

[Out]

-1/2/b/c^8/(b*x+a)^2

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Maxima [A]  time = 1.33122, size = 45, normalized size = 2.65 \[ -\frac{1}{2 \,{\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^8,x, algorithm="maxima")

[Out]

-1/2/(b^3*c^8*x^2 + 2*a*b^2*c^8*x + a^2*b*c^8)

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Fricas [A]  time = 0.190956, size = 45, normalized size = 2.65 \[ -\frac{1}{2 \,{\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^8,x, algorithm="fricas")

[Out]

-1/2/(b^3*c^8*x^2 + 2*a*b^2*c^8*x + a^2*b*c^8)

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Sympy [A]  time = 1.555, size = 36, normalized size = 2.12 \[ - \frac{1}{2 a^{2} b c^{8} + 4 a b^{2} c^{8} x + 2 b^{3} c^{8} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(b*c*x+a*c)**8,x)

[Out]

-1/(2*a**2*b*c**8 + 4*a*b**2*c**8*x + 2*b**3*c**8*x**2)

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GIAC/XCAS [A]  time = 0.208155, size = 20, normalized size = 1.18 \[ -\frac{1}{2 \,{\left (b x + a\right )}^{2} b c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^8,x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b*c^8)