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

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

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log \left (a+b (c+d x)^4\right )}{4 b d} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+b (c+d x)^4\right )}{4 b d} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^3}{a+b (c+d x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.88, size = 54, normalized size = 2.45 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.19, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end {gather*}

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)

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maple [B]  time = 0.00, size = 55, normalized size = 2.50 \begin {gather*} \frac {\ln \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 {gather*}

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 = 0.55, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end {gather*}

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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mupad [B]  time = 1.26, size = 54, normalized size = 2.45 \begin {gather*} \frac {\ln \left (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+a\right )}{4\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.44, size = 56, normalized size = 2.55 \begin {gather*} \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 {gather*}

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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