\(\int \frac {(c+d x)^2}{(a+b (c+d x)^3)^3} \, dx\) [2879]

Optimal result

Integrand size = 21, antiderivative size = 23 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {379, 267} \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[In]

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

[Out]

-1/6*1/(b*d*(a + b*(c + d*x)^3)^2)

Rule 267

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

Rule 379

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[In]

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

[Out]

-1/6*1/(b*d*(a + b*(c + d*x)^3)^2)

Maple [A] (verified)

Time = 3.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {1}{6 b d \left (a +b \left (d x +c \right )^{3}\right )^{2}}\)	\(22\)
gosper	\(-\frac {1}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2} b d}\)	\(44\)
default	\(-\frac {1}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2} b d}\)	\(44\)
norman	\(-\frac {1}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2} b d}\)	\(44\)
risch	\(-\frac {1}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2} b d}\)	\(44\)
parallelrisch	\(-\frac {1}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2} b d}\)	\(44\)

[In]

int((d*x+c)^2/(a+b*(d*x+c)^3)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 5.78 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6 \, {\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 2 \, {\left (10 \, b^{3} c^{3} + a b^{2}\right )} d^{4} x^{3} + 3 \, {\left (5 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{3} x^{2} + 6 \, {\left (b^{3} c^{5} + a b^{2} c^{2}\right )} d^{2} x + {\left (b^{3} c^{6} + 2 \, a b^{2} c^{3} + a^{2} b\right )} d\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (19) = 38\).

Time = 1.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.65 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=- \frac {1}{6 a^{2} b d + 12 a b^{2} c^{3} d + 6 b^{3} c^{6} d + 90 b^{3} c^{2} d^{5} x^{4} + 36 b^{3} c d^{6} x^{5} + 6 b^{3} d^{7} x^{6} + x^{3} \cdot \left (12 a b^{2} d^{4} + 120 b^{3} c^{3} d^{4}\right ) + x^{2} \cdot \left (36 a b^{2} c d^{3} + 90 b^{3} c^{4} d^{3}\right ) + x \left (36 a b^{2} c^{2} d^{2} + 36 b^{3} c^{5} d^{2}\right )} \]

[In]

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

[Out]

-1/(6*a**2*b*d + 12*a*b**2*c**3*d + 6*b**3*c**6*d + 90*b**3*c**2*d**5*x**4 + 36*b**3*c*d**6*x**5 + 6*b**3*d**7
*x**6 + x**3*(12*a*b**2*d**4 + 120*b**3*c**3*d**4) + x**2*(36*a*b**2*c*d**3 + 90*b**3*c**4*d**3) + x*(36*a*b**
2*c**2*d**2 + 36*b**3*c**5*d**2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6 \, {\left ({\left (d x + c\right )}^{3} b + a\right )}^{2} b d} \]

[In]

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

[Out]

-1/6/(((d*x + c)^3*b + a)^2*b*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6 \, {\left ({\left (d x + c\right )}^{3} b + a\right )}^{2} b d} \]

[In]

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

[Out]

-1/6/(((d*x + c)^3*b + a)^2*b*d)

Mupad [B] (verification not implemented)

Time = 5.67 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.70 \[ \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{6\,b\,d\,\left (x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4\right )} \]

[In]

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

[Out]

-1/(6*b*d*(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))