3.1008 \(\int \frac{(\frac{b c}{d}+b x)^4}{(c+d x)^3} \, dx\)

Optimal. Leaf size=23 \[ \frac{b^4 c x}{d^4}+\frac{b^4 x^2}{2 d^3} \]

[Out]

(b^4*c*x)/d^4 + (b^4*x^2)/(2*d^3)

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Rubi [A]  time = 0.0049852, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {21} \[ \frac{b^4 c x}{d^4}+\frac{b^4 x^2}{2 d^3} \]

Antiderivative was successfully verified.

[In]

Int[((b*c)/d + b*x)^4/(c + d*x)^3,x]

[Out]

(b^4*c*x)/d^4 + (b^4*x^2)/(2*d^3)

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])

Rubi steps

\begin{align*} \int \frac{\left (\frac{b c}{d}+b x\right )^4}{(c+d x)^3} \, dx &=\frac{b^4 \int (c+d x) \, dx}{d^4}\\ &=\frac{b^4 c x}{d^4}+\frac{b^4 x^2}{2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.001033, size = 19, normalized size = 0.83 \[ \frac{b^4 \left (c x+\frac{d x^2}{2}\right )}{d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((b*c)/d + b*x)^4/(c + d*x)^3,x]

[Out]

(b^4*(c*x + (d*x^2)/2))/d^4

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Maple [A]  time = 0.001, size = 18, normalized size = 0.8 \begin{align*}{\frac{{b}^{4}}{{d}^{4}} \left ( cx+{\frac{d{x}^{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*c/d+b*x)^4/(d*x+c)^3,x)

[Out]

b^4/d^4*(c*x+1/2*d*x^2)

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Maxima [A]  time = 1.02181, size = 28, normalized size = 1.22 \begin{align*} \frac{b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^4/(d*x+c)^3,x, algorithm="maxima")

[Out]

1/2*(b^4*d*x^2 + 2*b^4*c*x)/d^4

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Fricas [A]  time = 1.78849, size = 45, normalized size = 1.96 \begin{align*} \frac{b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^4/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(b^4*d*x^2 + 2*b^4*c*x)/d^4

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Sympy [A]  time = 0.100628, size = 20, normalized size = 0.87 \begin{align*} \frac{b^{4} c x}{d^{4}} + \frac{b^{4} x^{2}}{2 d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)**4/(d*x+c)**3,x)

[Out]

b**4*c*x/d**4 + b**4*x**2/(2*d**3)

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Giac [A]  time = 1.05597, size = 28, normalized size = 1.22 \begin{align*} \frac{b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^4/(d*x+c)^3,x, algorithm="giac")

[Out]

1/2*(b^4*d*x^2 + 2*b^4*c*x)/d^4