\(\int \frac {(c+d x)^3}{(a+b x)^6} \, dx\) [1269]

Optimal result

Integrand size = 15, antiderivative size = 58 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=-\frac {(c+d x)^4}{5 (b c-a d) (a+b x)^5}+\frac {d (c+d x)^4}{20 (b c-a d)^2 (a+b x)^4} \]

[Out]

-1/5*(d*x+c)^4/(-a*d+b*c)/(b*x+a)^5+1/20*d*(d*x+c)^4/(-a*d+b*c)^2/(b*x+a)^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=\frac {d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac {(c+d x)^4}{5 (a+b x)^5 (b c-a d)} \]

[In]

Int[(c + d*x)^3/(a + b*x)^6,x]

[Out]

-1/5*(c + d*x)^4/((b*c - a*d)*(a + b*x)^5) + (d*(c + d*x)^4)/(20*(b*c - a*d)^2*(a + b*x)^4)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^4}{5 (b c-a d) (a+b x)^5}-\frac {d \int \frac {(c+d x)^3}{(a+b x)^5} \, dx}{5 (b c-a d)} \\ & = -\frac {(c+d x)^4}{5 (b c-a d) (a+b x)^5}+\frac {d (c+d x)^4}{20 (b c-a d)^2 (a+b x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=-\frac {a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )}{20 b^4 (a+b x)^5} \]

[In]

Integrate[(c + d*x)^3/(a + b*x)^6,x]

[Out]

-1/20*(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a*b^2*d*(3*c^2 + 10*c*d*x + 10*d^2*x^2) + b^3*(4*c^3 + 15*c^2*d*x +
20*c*d^2*x^2 + 10*d^3*x^3))/(b^4*(a + b*x)^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(54)=108\).

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.90

method	result	size
risch	\(\frac {-\frac {d^{3} x^{3}}{2 b}-\frac {d^{2} \left (a d +2 b c \right ) x^{2}}{2 b^{2}}-\frac {d \left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) x}{4 b^{3}}-\frac {a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}}{20 b^{4}}}{\left (b x +a \right )^{5}}\)	\(110\)
gosper	\(-\frac {10 d^{3} x^{3} b^{3}+10 x^{2} a \,b^{2} d^{3}+20 x^{2} b^{3} c \,d^{2}+5 x \,a^{2} b \,d^{3}+10 x a \,b^{2} c \,d^{2}+15 x \,b^{3} c^{2} d +a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}}{20 b^{4} \left (b x +a \right )^{5}}\)	\(115\)
default	\(\frac {d^{2} \left (a d -b c \right )}{b^{4} \left (b x +a \right )^{3}}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 b^{4} \left (b x +a \right )^{4}}-\frac {d^{3}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{5 b^{4} \left (b x +a \right )^{5}}\)	\(121\)
parallelrisch	\(\frac {-10 d^{3} x^{3} b^{4}-10 a \,b^{3} d^{3} x^{2}-20 b^{4} c \,d^{2} x^{2}-5 a^{2} b^{2} d^{3} x -10 a \,b^{3} c \,d^{2} x -15 b^{4} c^{2} d x -a^{3} b \,d^{3}-2 a^{2} b^{2} c \,d^{2}-3 a \,b^{3} c^{2} d -4 b^{4} c^{3}}{20 b^{5} \left (b x +a \right )^{5}}\)	\(121\)
norman	\(\frac {-\frac {d^{3} x^{3}}{2 b}+\frac {\left (-a b \,d^{3}-2 b^{2} c \,d^{2}\right ) x^{2}}{2 b^{3}}+\frac {\left (-a^{2} b \,d^{3}-2 a \,b^{2} c \,d^{2}-3 b^{3} c^{2} d \right ) x}{4 b^{4}}+\frac {-a^{3} b \,d^{3}-2 a^{2} b^{2} c \,d^{2}-3 a \,b^{3} c^{2} d -4 b^{4} c^{3}}{20 b^{5}}}{\left (b x +a \right )^{5}}\)	\(126\)

[In]

int((d*x+c)^3/(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (54) = 108\).

Time = 0.21 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=-\frac {10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \, {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*d^3*x^3 + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3 + 10*(2*b^3*c*d^2 + a*b^2*d^3)*x^2
+ 5*(3*b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (46) = 92\).

Time = 1.14 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.97 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=\frac {- a^{3} d^{3} - 2 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 4 b^{3} c^{3} - 10 b^{3} d^{3} x^{3} + x^{2} \left (- 10 a b^{2} d^{3} - 20 b^{3} c d^{2}\right ) + x \left (- 5 a^{2} b d^{3} - 10 a b^{2} c d^{2} - 15 b^{3} c^{2} d\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \]

[In]

integrate((d*x+c)**3/(b*x+a)**6,x)

[Out]

(-a**3*d**3 - 2*a**2*b*c*d**2 - 3*a*b**2*c**2*d - 4*b**3*c**3 - 10*b**3*d**3*x**3 + x**2*(-10*a*b**2*d**3 - 20
*b**3*c*d**2) + x*(-5*a**2*b*d**3 - 10*a*b**2*c*d**2 - 15*b**3*c**2*d))/(20*a**5*b**4 + 100*a**4*b**5*x + 200*
a**3*b**6*x**2 + 200*a**2*b**7*x**3 + 100*a*b**8*x**4 + 20*b**9*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (54) = 108\).

Time = 0.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=-\frac {10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \, {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/20*(10*b^3*d^3*x^3 + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3 + 10*(2*b^3*c*d^2 + a*b^2*d^3)*x^2
+ 5*(3*b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (54) = 108\).

Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.97 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=-\frac {10 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c d^{2} x^{2} + 10 \, a b^{2} d^{3} x^{2} + 15 \, b^{3} c^{2} d x + 10 \, a b^{2} c d^{2} x + 5 \, a^{2} b d^{3} x + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}}{20 \, {\left (b x + a\right )}^{5} b^{4}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^6,x, algorithm="giac")

[Out]

-1/20*(10*b^3*d^3*x^3 + 20*b^3*c*d^2*x^2 + 10*a*b^2*d^3*x^2 + 15*b^3*c^2*d*x + 10*a*b^2*c*d^2*x + 5*a^2*b*d^3*
x + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^5*b^4)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^3}{(a+b x)^6} \, dx=\frac {{\left (c+d\,x\right )}^4\,\left (5\,a\,d-4\,b\,c+b\,d\,x\right )}{20\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^5} \]

[In]

int((c + d*x)^3/(a + b*x)^6,x)

[Out]

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