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

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

[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)

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Rubi [A]  time = 0.0701192, antiderivative size = 80, 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} \[ -\frac{b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 b \log (c+d x)}{a^3 d}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d}-\frac{1}{3 a^2 d (c+d x)^3} \]

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 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 )^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*}

Mathematica [A]  time = 0.0689235, 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]

-(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])/(3*a^3*d)

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Maple [A]  time = 0.019, size = 119, normalized size = 1.5 \begin{align*} -{\frac{1}{3\,{a}^{2}d \left ( dx+c \right ) ^{3}}}-2\,{\frac{b\ln \left ( dx+c \right ) }{{a}^{3}d}}-{\frac{b}{3\,{a}^{2}d \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}+{\frac{2\,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}^{3}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)^2,x)

[Out]

-1/3/a^2/d/(d*x+c)^3-2*b*ln(d*x+c)/a^3/d-1/3*b/a^2/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*b/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 = 1.03445, size = 300, normalized size = 3.75 \begin{align*} -\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} \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)^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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Fricas [B]  time = 1.793, size = 896, normalized size = 11.2 \begin{align*} -\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 )}} \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)^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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Sympy [B]  time = 161.726, size = 248, normalized size = 3.1 \begin{align*} - \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} \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)**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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Giac [A]  time = 1.14314, size = 88, normalized size = 1.1 \begin{align*} \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} \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)^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)