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

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 = 0.0036141, 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, Rules used = {21, 32} \[ \frac{b^5 (c+d x)^3}{3 d^6} \]

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)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (\frac{b c}{d}+b x\right )^5}{(c+d x)^3} \, dx &=\frac{b^5 \int (c+d x)^2 \, dx}{d^5}\\ &=\frac{b^5 (c+d x)^3}{3 d^6}\\ \end{align*}

Mathematica [A]  time = 0.0024827, 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 = 0.003, size = 16, normalized size = 0.9 \begin{align*}{\frac{{b}^{5} \left ( dx+c \right ) ^{3}}{3\,{d}^{6}}} \end{align*}

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 [B]  time = 1.03419, size = 47, normalized size = 2.76 \begin{align*} \frac{b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^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 [B]  time = 1.73966, size = 72, normalized size = 4.24 \begin{align*} \frac{b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^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 [B]  time = 0.107849, size = 34, normalized size = 2. \begin{align*} \frac{b^{5} c^{2} x}{d^{5}} + \frac{b^{5} c x^{2}}{d^{4}} + \frac{b^{5} x^{3}}{3 d^{3}} \end{align*}

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 [B]  time = 1.06061, size = 47, normalized size = 2.76 \begin{align*} \frac{b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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