3.1007 \(\int \frac{\left (\frac{b c}{d}+b x\right )^5}{(c+d x)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac{b^5 (c+d x)^3}{3 d^6} \]

[Out]

(b^5*(c + d*x)^3)/(3*d^6)

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Rubi [A]  time = 0.0118669, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b^5 (c+d x)^3}{3 d^6} \]

Antiderivative was successfully verified.

[In]  Int[((b*c)/d + b*x)^5/(c + d*x)^3,x]

[Out]

(b^5*(c + d*x)^3)/(3*d^6)

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Rubi in Sympy [A]  time = 4.47594, size = 14, normalized size = 0.82 \[ \frac{b^{5} \left (c + d x\right )^{3}}{3 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*c/d+b*x)**5/(d*x+c)**3,x)

[Out]

b**5*(c + d*x)**3/(3*d**6)

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Mathematica [A]  time = 0.00385356, size = 17, normalized size = 1. \[ \frac{b^5 (c+d x)^3}{3 d^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((b*c)/d + b*x)^5/(c + d*x)^3,x]

[Out]

(b^5*(c + d*x)^3)/(3*d^6)

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Maple [A]  time = 0., size = 16, normalized size = 0.9 \[{\frac{{b}^{5} \left ( dx+c \right ) ^{3}}{3\,{d}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*c/d+b*x)^5/(d*x+c)^3,x)

[Out]

1/3*b^5*(d*x+c)^3/d^6

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Maxima [A]  time = 1.34347, size = 47, normalized size = 2.76 \[ \frac{b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + b*c/d)^5/(d*x + c)^3,x, algorithm="maxima")

[Out]

1/3*(b^5*d^2*x^3 + 3*b^5*c*d*x^2 + 3*b^5*c^2*x)/d^5

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Fricas [A]  time = 0.191766, size = 47, normalized size = 2.76 \[ \frac{b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + b*c/d)^5/(d*x + c)^3,x, algorithm="fricas")

[Out]

1/3*(b^5*d^2*x^3 + 3*b^5*c*d*x^2 + 3*b^5*c^2*x)/d^5

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Sympy [A]  time = 0.256357, size = 34, normalized size = 2. \[ \frac{b^{5} c^{2} x}{d^{5}} + \frac{b^{5} c x^{2}}{d^{4}} + \frac{b^{5} x^{3}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c/d+b*x)**5/(d*x+c)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207103, size = 54, normalized size = 3.18 \[ \frac{b^{5} d^{12} x^{3} + 3 \, b^{5} c d^{11} x^{2} + 3 \, b^{5} c^{2} d^{10} x}{3 \, d^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + b*c/d)^5/(d*x + c)^3,x, algorithm="giac")

[Out]

1/3*(b^5*d^12*x^3 + 3*b^5*c*d^11*x^2 + 3*b^5*c^2*d^10*x)/d^15