3.2867 \(\int \frac{1}{(c+d x)^4 (a+b (c+d x)^3)} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b \log (c+d x)}{a^2 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}-\frac{1}{3 a d (c+d x)^3} \]

[Out]

-1/(3*a*d*(c + d*x)^3) - (b*Log[c + d*x])/(a^2*d) + (b*Log[a + b*(c + d*x)^3])/(3*a^2*d)

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Rubi [A]  time = 0.0414471, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {372, 266, 44} \[ -\frac{b \log (c+d x)}{a^2 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}-\frac{1}{3 a d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

-1/(3*a*d*(c + d*x)^3) - (b*Log[c + d*x])/(a^2*d) + (b*Log[a + b*(c + d*x)^3])/(3*a^2*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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^4 \left (a+b (c+d x)^3\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d}\\ &=-\frac{1}{3 a d (c+d x)^3}-\frac{b \log (c+d x)}{a^2 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0212032, size = 44, normalized size = 0.79 \[ \frac{b \log \left (a+b (c+d x)^3\right )-\frac{a}{(c+d x)^3}-3 b \log (c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

(-(a/(c + d*x)^3) - 3*b*Log[c + d*x] + b*Log[a + b*(c + d*x)^3])/(3*a^2*d)

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Maple [A]  time = 0.007, size = 75, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,ad \left ( dx+c \right ) ^{3}}}-{\frac{b\ln \left ( dx+c \right ) }{{a}^{2}d}}+{\frac{b\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{2}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^4/(a+b*(d*x+c)^3),x)

[Out]

-1/3/a/d/(d*x+c)^3-b*ln(d*x+c)/a^2/d+1/3/a^2*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 = 1.17117, size = 132, normalized size = 2.36 \begin{align*} -\frac{1}{3 \,{\left (a d^{4} x^{3} + 3 \, a c d^{3} x^{2} + 3 \, a c^{2} d^{2} x + a c^{3} d\right )}} + \frac{b \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 \, a^{2} d} - \frac{b \log \left (d x + c\right )}{a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.5038, size = 333, normalized size = 5.95 \begin{align*} \frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \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 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} d^{4} x^{3} + 3 \, a^{2} c d^{3} x^{2} + 3 \, a^{2} c^{2} d^{2} x + a^{2} c^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.94532, size = 100, normalized size = 1.79 \begin{align*} - \frac{1}{3 a c^{3} d + 9 a c^{2} d^{2} x + 9 a c d^{3} x^{2} + 3 a d^{4} x^{3}} - \frac{b \log{\left (\frac{c}{d} + x \right )}}{a^{2} d} + \frac{b \log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3),x)

[Out]

-1/(3*a*c**3*d + 9*a*c**2*d**2*x + 9*a*c*d**3*x**2 + 3*a*d**4*x**3) - b*log(c/d + x)/(a**2*d) + b*log(3*c**2*x
/d**2 + 3*c*x**2/d + x**3 + (a + b*c**3)/(b*d**3))/(3*a**2*d)

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Giac [A]  time = 1.13702, size = 55, normalized size = 0.98 \begin{align*} \frac{b \log \left ({\left | -b - \frac{a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{2} d} - \frac{1}{3 \,{\left (d x + c\right )}^{3} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

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