3.2882 \(\int \frac {1}{(c+d x) (a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=82 \[ -\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d}+\frac {\log (c+d x)}{a^3 d}+\frac {1}{3 a^2 d \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d}+\frac {\log (c+d x)}{a^3 d}+\frac {1}{6 a d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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

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Mathematica [A]  time = 0.06, size = 63, normalized size = 0.77 \[ \frac {\frac {a \left (2 \left (a+b (c+d x)^3\right )+a\right )}{\left (a+b (c+d x)^3\right )^2}-2 \log \left (a+b (c+d x)^3\right )+6 \log (c+d x)}{6 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.14, size = 467, normalized size = 5.70 \[ \frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{2} d^{7} x^{6} + 6 \, a^{3} b^{2} c d^{6} x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} x^{4} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} x^{2} + 6 \, {\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} x + {\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 + 6*a*b*c^2*d*x + 2*a*b*c^3 + 3*a^2 - 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 +
15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*
(b^2*c^5 + a*b*c^2)*d*x + a^2)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) + 6*(b^2*d^6*x^6 + 6*b
^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*
c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*log(d*x + c))/(a^3*b^2*d^7*x^6 + 6*a^3*b^2*c*d^6*x^5 + 15*a^3*b^
2*c^2*d^5*x^4 + 2*(10*a^3*b^2*c^3 + a^4*b)*d^4*x^3 + 3*(5*a^3*b^2*c^4 + 2*a^4*b*c)*d^3*x^2 + 6*(a^3*b^2*c^5 +
a^4*b*c^2)*d^2*x + (a^3*b^2*c^6 + 2*a^4*b*c^3 + a^5)*d)

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giac [A]  time = 0.24, size = 144, normalized size = 1.76 \[ -\frac {\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} + \frac {\log \left ({\left | d x + c \right |}\right )}{a^{3} d} + \frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}}{6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 283, normalized size = 3.45 \[ \frac {b \,d^{2} x^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2}}+\frac {b c d \,x^{2}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2}}+\frac {b \,c^{2} x}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2}}+\frac {b \,c^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} d}+\frac {1}{2 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a d}+\frac {\ln \left (d x +c \right )}{a^{3} d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(d*x+c)/a^3/d+1/3*b/a^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+b/a^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+b/a^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+1/3*b/a^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+1/2/a/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2/d-1/3/a^
3/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 = 0.66, size = 245, normalized size = 2.99 \[ \frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \, {\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \, {\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x + {\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} + \frac {\log \left (d x + c\right )}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + 3*a)/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^
2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^
5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*d) - 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)/(a^3*d) + log(d*x + c)/(a^3*d)

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mupad [B]  time = 1.87, size = 230, normalized size = 2.80 \[ \frac {\frac {2\,b\,c^3+3\,a}{6\,a^2\,d}+\frac {b\,d^2\,x^3}{3\,a^2}+\frac {b\,c^2\,x}{a^2}+\frac {b\,c\,d\,x^2}{a^2}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d}+\frac {\ln \left (c+d\,x\right )}{a^3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*a + 2*b*c^3)/(6*a^2*d) + (b*d^2*x^3)/(3*a^2) + (b*c^2*x)/a^2 + (b*c*d*x^2)/a^2)/(x^3*(20*b^2*c^3*d^3 + 2*a
*b*d^3) + x^2*(15*b^2*c^4*d^2 + 6*a*b*c*d^2) + a^2 + x*(6*b^2*c^5*d + 6*a*b*c^2*d) + b^2*c^6 + b^2*d^6*x^6 + 2
*a*b*c^3 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4) - log(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2)/(3
*a^3*d) + log(c + d*x)/(a^3*d)

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sympy [B]  time = 4.76, size = 269, normalized size = 3.28 \[ \frac {3 a + 2 b c^{3} + 6 b c^{2} d x + 6 b c d^{2} x^{2} + 2 b d^{3} x^{3}}{6 a^{4} d + 12 a^{3} b c^{3} d + 6 a^{2} b^{2} c^{6} d + 90 a^{2} b^{2} c^{2} d^{5} x^{4} + 36 a^{2} b^{2} c d^{6} x^{5} + 6 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (12 a^{3} b d^{4} + 120 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (36 a^{3} b c d^{3} + 90 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (36 a^{3} b c^{2} d^{2} + 36 a^{2} b^{2} c^{5} d^{2}\right )} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{3} d} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a + 2*b*c**3 + 6*b*c**2*d*x + 6*b*c*d**2*x**2 + 2*b*d**3*x**3)/(6*a**4*d + 12*a**3*b*c**3*d + 6*a**2*b**2*c
**6*d + 90*a**2*b**2*c**2*d**5*x**4 + 36*a**2*b**2*c*d**6*x**5 + 6*a**2*b**2*d**7*x**6 + x**3*(12*a**3*b*d**4
+ 120*a**2*b**2*c**3*d**4) + x**2*(36*a**3*b*c*d**3 + 90*a**2*b**2*c**4*d**3) + x*(36*a**3*b*c**2*d**2 + 36*a*
*2*b**2*c**5*d**2)) + log(c/d + x)/(a**3*d) - log(3*c**2*x/d**2 + 3*c*x**2/d + x**3 + (a + b*c**3)/(b*d**3))/(
3*a**3*d)

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