3.2885 \(\int \frac{1}{(c+d x)^4 (a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac{b}{6 a^2 d \left (a+b (c+d x)^3\right )^2}-\frac{3 b \log (c+d x)}{a^4 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{a^4 d}-\frac{1}{3 a^3 d (c+d x)^3} \]

[Out]

-1/(3*a^3*d*(c + d*x)^3) - b/(6*a^2*d*(a + b*(c + d*x)^3)^2) - (2*b)/(3*a^3*d*(a + b*(c + d*x)^3)) - (3*b*Log[
c + d*x])/(a^4*d) + (b*Log[a + b*(c + d*x)^3])/(a^4*d)

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Rubi [A]  time = 0.0951593, antiderivative size = 101, 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} \[ -\frac{2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac{b}{6 a^2 d \left (a+b (c+d x)^3\right )^2}-\frac{3 b \log (c+d x)}{a^4 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{a^4 d}-\frac{1}{3 a^3 d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*a^3*d*(c + d*x)^3) - b/(6*a^2*d*(a + b*(c + d*x)^3)^2) - (2*b)/(3*a^3*d*(a + b*(c + d*x)^3)) - (3*b*Log[
c + d*x])/(a^4*d) + (b*Log[a + b*(c + d*x)^3])/(a^4*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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^2}-\frac{3 b}{a^4 x}+\frac{b^2}{a^2 (a+b x)^3}+\frac{2 b^2}{a^3 (a+b x)^2}+\frac{3 b^2}{a^4 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d}\\ &=-\frac{1}{3 a^3 d (c+d x)^3}-\frac{b}{6 a^2 d \left (a+b (c+d x)^3\right )^2}-\frac{2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac{3 b \log (c+d x)}{a^4 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{a^4 d}\\ \end{align*}

Mathematica [A]  time = 0.106545, size = 80, normalized size = 0.79 \[ \frac{a \left (-\frac{4 b}{a+b (c+d x)^3}-\frac{a b}{\left (a+b (c+d x)^3\right )^2}-\frac{2}{(c+d x)^3}\right )+6 b \log \left (a+b (c+d x)^3\right )-18 b \log (c+d x)}{6 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-2/(c + d*x)^3 - (a*b)/(a + b*(c + d*x)^3)^2 - (4*b)/(a + b*(c + d*x)^3)) - 18*b*Log[c + d*x] + 6*b*Log[a
+ b*(c + d*x)^3])/(6*a^4*d)

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Maple [B]  time = 0.029, size = 311, normalized size = 3.1 \begin{align*} -{\frac{1}{3\,{a}^{3}d \left ( dx+c \right ) ^{3}}}-3\,{\frac{b\ln \left ( dx+c \right ) }{{a}^{4}d}}-{\frac{2\,{d}^{2}{b}^{2}{x}^{3}}{3\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-2\,{\frac{{b}^{2}cd{x}^{2}}{{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-2\,{\frac{{b}^{2}{c}^{2}x}{{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{2\,{b}^{2}{c}^{3}}{3\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{5\,b}{6\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{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 ) }{{a}^{4}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)^3,x)

[Out]

-1/3/a^3/d/(d*x+c)^3-3*b*ln(d*x+c)/a^4/d-2/3/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d^2*x^3-2
/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c*d*x^2-2/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*
x+b*c^3+a)^2*c^2*x-2/3/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2/d*c^3-5/6/a^2*b/(b*d^3*x^3+3*b*
c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2/d+1/a^4*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 [B]  time = 1.16545, size = 591, normalized size = 5.85 \begin{align*} -\frac{6 \, b^{2} d^{6} x^{6} + 36 \, b^{2} c d^{5} x^{5} + 90 \, b^{2} c^{2} d^{4} x^{4} + 6 \, b^{2} c^{6} + 3 \,{\left (40 \, b^{2} c^{3} + 3 \, a b\right )} d^{3} x^{3} + 9 \, a b c^{3} + 9 \,{\left (10 \, b^{2} c^{4} + 3 \, a b c\right )} d^{2} x^{2} + 9 \,{\left (4 \, b^{2} c^{5} + 3 \, a b c^{2}\right )} d x + 2 \, a^{2}}{6 \,{\left (a^{3} b^{2} d^{10} x^{9} + 9 \, a^{3} b^{2} c d^{9} x^{8} + 36 \, a^{3} b^{2} c^{2} d^{8} x^{7} + 2 \,{\left (42 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{7} x^{6} + 6 \,{\left (21 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{6} x^{5} + 6 \,{\left (21 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{5} x^{4} +{\left (84 \, a^{3} b^{2} c^{6} + 40 \, a^{4} b c^{3} + a^{5}\right )} d^{4} x^{3} + 3 \,{\left (12 \, a^{3} b^{2} c^{7} + 10 \, a^{4} b c^{4} + a^{5} c\right )} d^{3} x^{2} + 3 \,{\left (3 \, a^{3} b^{2} c^{8} + 4 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d^{2} x +{\left (a^{3} b^{2} c^{9} + 2 \, a^{4} b c^{6} + a^{5} c^{3}\right )} 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 )}{a^{4} d} - \frac{3 \, b \log \left (d x + c\right )}{a^{4} 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)^3,x, algorithm="maxima")

[Out]

-1/6*(6*b^2*d^6*x^6 + 36*b^2*c*d^5*x^5 + 90*b^2*c^2*d^4*x^4 + 6*b^2*c^6 + 3*(40*b^2*c^3 + 3*a*b)*d^3*x^3 + 9*a
*b*c^3 + 9*(10*b^2*c^4 + 3*a*b*c)*d^2*x^2 + 9*(4*b^2*c^5 + 3*a*b*c^2)*d*x + 2*a^2)/(a^3*b^2*d^10*x^9 + 9*a^3*b
^2*c*d^9*x^8 + 36*a^3*b^2*c^2*d^8*x^7 + 2*(42*a^3*b^2*c^3 + a^4*b)*d^7*x^6 + 6*(21*a^3*b^2*c^4 + 2*a^4*b*c)*d^
6*x^5 + 6*(21*a^3*b^2*c^5 + 5*a^4*b*c^2)*d^5*x^4 + (84*a^3*b^2*c^6 + 40*a^4*b*c^3 + a^5)*d^4*x^3 + 3*(12*a^3*b
^2*c^7 + 10*a^4*b*c^4 + a^5*c)*d^3*x^2 + 3*(3*a^3*b^2*c^8 + 4*a^4*b*c^5 + a^5*c^2)*d^2*x + (a^3*b^2*c^9 + 2*a^
4*b*c^6 + a^5*c^3)*d) + 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^4*d) - 3*b*log(d*x + c)/
(a^4*d)

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Fricas [B]  time = 2.6307, size = 1828, normalized size = 18.1 \begin{align*} \text{result too large to display} \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)^3,x, algorithm="fricas")

[Out]

-1/6*(6*a*b^2*d^6*x^6 + 36*a*b^2*c*d^5*x^5 + 90*a*b^2*c^2*d^4*x^4 + 6*a*b^2*c^6 + 3*(40*a*b^2*c^3 + 3*a^2*b)*d
^3*x^3 + 9*a^2*b*c^3 + 9*(10*a*b^2*c^4 + 3*a^2*b*c)*d^2*x^2 + 2*a^3 + 9*(4*a*b^2*c^5 + 3*a^2*b*c^2)*d*x - 6*(b
^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 2*(42*b^3*c^3 + a*b^2)*d^6*x^6 + b^3*c^9 + 6*(21*b^3*c^4 +
2*a*b^2*c)*d^5*x^5 + 2*a*b^2*c^6 + 6*(21*b^3*c^5 + 5*a*b^2*c^2)*d^4*x^4 + (84*b^3*c^6 + 40*a*b^2*c^3 + a^2*b)
*d^3*x^3 + a^2*b*c^3 + 3*(12*b^3*c^7 + 10*a*b^2*c^4 + a^2*b*c)*d^2*x^2 + 3*(3*b^3*c^8 + 4*a*b^2*c^5 + a^2*b*c^
2)*d*x)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) + 18*(b^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*
c^2*d^7*x^7 + 2*(42*b^3*c^3 + a*b^2)*d^6*x^6 + b^3*c^9 + 6*(21*b^3*c^4 + 2*a*b^2*c)*d^5*x^5 + 2*a*b^2*c^6 + 6*
(21*b^3*c^5 + 5*a*b^2*c^2)*d^4*x^4 + (84*b^3*c^6 + 40*a*b^2*c^3 + a^2*b)*d^3*x^3 + a^2*b*c^3 + 3*(12*b^3*c^7 +
10*a*b^2*c^4 + a^2*b*c)*d^2*x^2 + 3*(3*b^3*c^8 + 4*a*b^2*c^5 + a^2*b*c^2)*d*x)*log(d*x + c))/(a^4*b^2*d^10*x^
9 + 9*a^4*b^2*c*d^9*x^8 + 36*a^4*b^2*c^2*d^8*x^7 + 2*(42*a^4*b^2*c^3 + a^5*b)*d^7*x^6 + 6*(21*a^4*b^2*c^4 + 2*
a^5*b*c)*d^6*x^5 + 6*(21*a^4*b^2*c^5 + 5*a^5*b*c^2)*d^5*x^4 + (84*a^4*b^2*c^6 + 40*a^5*b*c^3 + a^6)*d^4*x^3 +
3*(12*a^4*b^2*c^7 + 10*a^5*b*c^4 + a^6*c)*d^3*x^2 + 3*(3*a^4*b^2*c^8 + 4*a^5*b*c^5 + a^6*c^2)*d^2*x + (a^4*b^2
*c^9 + 2*a^5*b*c^6 + a^6*c^3)*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.1532, size = 108, normalized size = 1.07 \begin{align*} \frac{b \log \left ({\left | -b - \frac{a}{{\left (d x + c\right )}^{3}} \right |}\right )}{a^{4} d} + \frac{5 \, b^{3} + \frac{6 \, a b^{2}}{{\left (d x + c\right )}^{3}}}{6 \, a^{4}{\left (b + \frac{a}{{\left (d x + c\right )}^{3}}\right )}^{2} d} - \frac{1}{3 \,{\left (d x + c\right )}^{3} a^{3} 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)^3,x, algorithm="giac")

[Out]

b*log(abs(-b - a/(d*x + c)^3))/(a^4*d) + 1/6*(5*b^3 + 6*a*b^2/(d*x + c)^3)/(a^4*(b + a/(d*x + c)^3)^2*d) - 1/3
/((d*x + c)^3*a^3*d)