3.1019 \(\int \frac{(a+b x)^5}{a c+b c x} \, dx\)

Optimal. Leaf size=17 \[ \frac{(a+b x)^5}{5 b c} \]

[Out]

(a + b*x)^5/(5*b*c)

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Rubi [A]  time = 0.01148, 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 \[ \frac{(a+b x)^5}{5 b c} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^5/(5*b*c)

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Rubi in Sympy [A]  time = 4.3615, size = 10, normalized size = 0.59 \[ \frac{\left (a + b x\right )^{5}}{5 b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**5/(5*b*c)

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Mathematica [A]  time = 0.00239059, size = 17, normalized size = 1. \[ \frac{(a+b x)^5}{5 b c} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^5/(5*b*c)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \[{\frac{ \left ( bx+a \right ) ^{5}}{5\,bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c),x)

[Out]

1/5*(b*x+a)^5/b/c

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Maxima [A]  time = 1.34641, size = 65, normalized size = 3.82 \[ \frac{b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

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Fricas [A]  time = 0.191581, size = 65, normalized size = 3.82 \[ \frac{b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

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Sympy [A]  time = 0.19211, size = 51, normalized size = 3. \[ \frac{a^{4} x}{c} + \frac{2 a^{3} b x^{2}}{c} + \frac{2 a^{2} b^{2} x^{3}}{c} + \frac{a b^{3} x^{4}}{c} + \frac{b^{4} x^{5}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207892, size = 85, normalized size = 5. \[ \frac{b^{4} c^{4} x^{5} + 5 \, a b^{3} c^{4} x^{4} + 10 \, a^{2} b^{2} c^{4} x^{3} + 10 \, a^{3} b c^{4} x^{2} + 5 \, a^{4} c^{4} x}{5 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b^4*c^4*x^5 + 5*a*b^3*c^4*x^4 + 10*a^2*b^2*c^4*x^3 + 10*a^3*b*c^4*x^2 + 5*a
^4*c^4*x)/c^5