3.2876 \(\int \frac {1}{(c+d x)^4 (a+b (c+d x)^3)^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac {2 b \log (c+d x)}{a^3 d}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d}-\frac {b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {1}{3 a^2 d (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 80, 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} \[ -\frac {b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 b \log (c+d x)}{a^3 d}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d}-\frac {1}{3 a^2 d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^4*(a + b*(c + d*x)^3)^2),x]

[Out]

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

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Mathematica [A]  time = 0.08, size = 60, normalized size = 0.75 \[ -\frac {a \left (\frac {b}{a+b (c+d x)^3}+\frac {1}{(c+d x)^3}\right )-2 b \log \left (a+b (c+d x)^3\right )+6 b \log (c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^4*(a + b*(c + d*x)^3)^2),x]

[Out]

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

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fricas [B]  time = 0.81, size = 431, normalized size = 5.39 \[ -\frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \, {\left (a^{3} b d^{7} x^{6} + 6 \, a^{3} b c d^{6} x^{5} + 15 \, a^{3} b c^{2} d^{5} x^{4} + {\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} x^{2} + 3 \, {\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} x + {\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

-1/3*(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 + a^2 - 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 1
5*b^2*c^2*d^4*x^4 + b^2*c^6 + (20*b^2*c^3 + a*b)*d^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2*b^2*
c^5 + a*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) + 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 + (20*b^2*c^3 + a*b)*d^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2
*b^2*c^5 + a*b*c^2)*d*x)*log(d*x + c))/(a^3*b*d^7*x^6 + 6*a^3*b*c*d^6*x^5 + 15*a^3*b*c^2*d^5*x^4 + (20*a^3*b*c
^3 + a^4)*d^4*x^3 + 3*(5*a^3*b*c^4 + a^4*c)*d^3*x^2 + 3*(2*a^3*b*c^5 + a^4*c^2)*d^2*x + (a^3*b*c^6 + a^4*c^3)*
d)

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giac [A]  time = 0.22, size = 65, normalized size = 0.81 \[ \frac {2 \, b \log \left ({\left | -b - \frac {a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{3} d} + \frac {b^{2}}{3 \, a^{3} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )} d} - \frac {1}{3 \, {\left (d x + c\right )}^{3} a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 119, normalized size = 1.49 \[ -\frac {b}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a^{2} d}-\frac {2 b \ln \left (d x +c \right )}{a^{3} d}+\frac {2 b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{3} d}-\frac {1}{3 \left (d x +c \right )^{3} a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x)

[Out]

-1/3/a^2/d/(d*x+c)^3-2*b*ln(d*x+c)/a^3/d-1/3/a^2*b/d/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+2/3/a^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 [B]  time = 0.68, size = 222, normalized size = 2.78 \[ -\frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \, {\left (a^{2} b d^{7} x^{6} + 6 \, a^{2} b c d^{6} x^{5} + 15 \, a^{2} b c^{2} d^{5} x^{4} + {\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} x^{2} + 3 \, {\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} x + {\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d\right )}} + \frac {2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} - \frac {2 \, b \log \left (d x + c\right )}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

-1/3*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + a)/(a^2*b*d^7*x^6 + 6*a^2*b*c*d^6*x^5 + 15*a^2*b*c
^2*d^5*x^4 + (20*a^2*b*c^3 + a^3)*d^4*x^3 + 3*(5*a^2*b*c^4 + a^3*c)*d^3*x^2 + 3*(2*a^2*b*c^5 + a^3*c^2)*d^2*x
+ (a^2*b*c^6 + a^3*c^3)*d) + 2/3*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^3*d) - 2*b*log(
d*x + c)/(a^3*d)

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mupad [B]  time = 1.94, size = 211, normalized size = 2.64 \[ \frac {2\,b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d}-\frac {\frac {2\,b\,c^3+a}{3\,a^2\,d}+\frac {2\,b\,d^2\,x^3}{3\,a^2}+\frac {2\,b\,c^2\,x}{a^2}+\frac {2\,b\,c\,d\,x^2}{a^2}}{x\,\left (6\,b\,d\,c^5+3\,a\,d\,c^2\right )+x^3\,\left (20\,b\,c^3\,d^3+a\,d^3\right )+a\,c^3+b\,c^6+x^2\,\left (15\,b\,c^4\,d^2+3\,a\,c\,d^2\right )+b\,d^6\,x^6+15\,b\,c^2\,d^4\,x^4+6\,b\,c\,d^5\,x^5}-\frac {2\,b\,\ln \left (c+d\,x\right )}{a^3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*(c + d*x)^3)^2*(c + d*x)^4),x)

[Out]

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

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sympy [B]  time = 3.58, size = 250, normalized size = 3.12 \[ \frac {- a - 2 b c^{3} - 6 b c^{2} d x - 6 b c d^{2} x^{2} - 2 b d^{3} x^{3}}{3 a^{3} c^{3} d + 3 a^{2} b c^{6} d + 45 a^{2} b c^{2} d^{5} x^{4} + 18 a^{2} b c d^{6} x^{5} + 3 a^{2} b d^{7} x^{6} + x^{3} \left (3 a^{3} d^{4} + 60 a^{2} b c^{3} d^{4}\right ) + x^{2} \left (9 a^{3} c d^{3} + 45 a^{2} b c^{4} d^{3}\right ) + x \left (9 a^{3} c^{2} d^{2} + 18 a^{2} b c^{5} d^{2}\right )} - \frac {2 b \log {\left (\frac {c}{d} + x \right )}}{a^{3} d} + \frac {2 b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3)**2,x)

[Out]

(-a - 2*b*c**3 - 6*b*c**2*d*x - 6*b*c*d**2*x**2 - 2*b*d**3*x**3)/(3*a**3*c**3*d + 3*a**2*b*c**6*d + 45*a**2*b*
c**2*d**5*x**4 + 18*a**2*b*c*d**6*x**5 + 3*a**2*b*d**7*x**6 + x**3*(3*a**3*d**4 + 60*a**2*b*c**3*d**4) + x**2*
(9*a**3*c*d**3 + 45*a**2*b*c**4*d**3) + x*(9*a**3*c**2*d**2 + 18*a**2*b*c**5*d**2)) - 2*b*log(c/d + x)/(a**3*d
) + 2*b*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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