3.2914 \(\int \frac{(c+d x)^3}{a+b (c+d x)^4} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]

[Out]

Log[a + b*(c + d*x)^4]/(4*b*d)

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Rubi [A]  time = 0.0242018, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {372, 260} \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*(c + d*x)^4]/(4*b*d)

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0087351, size = 22, normalized size = 1. \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*(c + d*x)^4]/(4*b*d)

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Maple [B]  time = 0.002, size = 55, normalized size = 2.5 \begin{align*}{\frac{\ln \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) }{4\,bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/b/d*ln(b*d^4*x^4+4*b*c*d^3*x^3+6*b*c^2*d^2*x^2+4*b*c^3*d*x+b*c^4+a)

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Maxima [A]  time = 0.973005, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log((d*x + c)^4*b + a)/(b*d)

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Fricas [B]  time = 1.22035, size = 116, normalized size = 5.27 \begin{align*} \frac{\log \left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}{4 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a)/(b*d)

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Sympy [B]  time = 0.467771, size = 56, normalized size = 2.55 \begin{align*} \frac{\log{\left (a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4} \right )}}{4 b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x**3 + b*d**4*x**4)/(4*b*d)

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Giac [A]  time = 1.19299, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log((d*x + c)^4*b + a)/(b*d)