3.2861 \(\int \frac{(c+d x)^2}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

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Rubi [A]  time = 0.0213417, 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} \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*(c + d*x)^3),x]

[Out]

Log[a + b*(c + d*x)^3]/(3*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)^2}{a+b (c+d x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\log \left (a+b (c+d x)^3\right )}{3 b d}\\ \end{align*}

Mathematica [A]  time = 0.0065988, size = 22, normalized size = 1. \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*(c + d*x)^3),x]

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

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Maple [B]  time = 0.002, size = 43, normalized size = 2. \begin{align*}{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+b*(d*x+c)^3),x)

[Out]

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

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Maxima [A]  time = 1.16176, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left ({\left (d x + c\right )}^{3} b + a\right )}{3 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+b*(d*x+c)^3),x, algorithm="maxima")

[Out]

1/3*log((d*x + c)^3*b + a)/(b*d)

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Fricas [B]  time = 1.4815, size = 92, normalized size = 4.18 \begin{align*} \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.410921, size = 42, normalized size = 1.91 \begin{align*} \frac{\log{\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+b*(d*x+c)**3),x)

[Out]

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

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Giac [A]  time = 1.12219, size = 28, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left |{\left (d x + c\right )}^{3} b + a \right |}\right )}{3 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

1/3*log(abs((d*x + c)^3*b + a))/(b*d)