\(\int \frac {(c+d x)^3}{(a+b x)^7} \, dx\) [1270]

Optimal result

Integrand size = 15, antiderivative size = 92 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac {3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac {d^3}{3 b^4 (a+b x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac {3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac {(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac {d^3}{3 b^4 (a+b x)^3} \]

[In]

Int[(c + d*x)^3/(a + b*x)^7,x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^3}{b^3 (a+b x)^7}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^6}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^5}+\frac {d^3}{b^3 (a+b x)^4}\right ) \, dx \\ & = -\frac {(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac {3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac {d^3}{3 b^4 (a+b x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {a^3 d^3+3 a^2 b d^2 (c+2 d x)+3 a b^2 d \left (2 c^2+6 c d x+5 d^2 x^2\right )+b^3 \left (10 c^3+36 c^2 d x+45 c d^2 x^2+20 d^3 x^3\right )}{60 b^4 (a+b x)^6} \]

[In]

Integrate[(c + d*x)^3/(a + b*x)^7,x]

[Out]

-1/60*(a^3*d^3 + 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*d*(2*c^2 + 6*c*d*x + 5*d^2*x^2) + b^3*(10*c^3 + 36*c^2*d*x
+ 45*c*d^2*x^2 + 20*d^3*x^3))/(b^4*(a + b*x)^6)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20

method	result	size
risch	\(\frac {-\frac {d^{3} x^{3}}{3 b}-\frac {d^{2} \left (a d +3 b c \right ) x^{2}}{4 b^{2}}-\frac {d \left (a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) x}{10 b^{3}}-\frac {a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}}{60 b^{4}}}{\left (b x +a \right )^{6}}\)	\(110\)
gosper	\(-\frac {20 d^{3} x^{3} b^{3}+15 x^{2} a \,b^{2} d^{3}+45 x^{2} b^{3} c \,d^{2}+6 x \,a^{2} b \,d^{3}+18 x a \,b^{2} c \,d^{2}+36 x \,b^{3} c^{2} d +a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}}{60 b^{4} \left (b x +a \right )^{6}}\)	\(115\)
default	\(-\frac {d^{3}}{3 b^{4} \left (b x +a \right )^{3}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{6 b^{4} \left (b x +a \right )^{6}}+\frac {3 d^{2} \left (a d -b c \right )}{4 b^{4} \left (b x +a \right )^{4}}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{5 b^{4} \left (b x +a \right )^{5}}\)	\(122\)
parallelrisch	\(\frac {-20 d^{3} x^{3} b^{5}-15 a \,b^{4} d^{3} x^{2}-45 b^{5} c \,d^{2} x^{2}-6 a^{2} b^{3} d^{3} x -18 a \,b^{4} c \,d^{2} x -36 b^{5} c^{2} d x -a^{3} b^{2} d^{3}-3 a^{2} b^{3} c \,d^{2}-6 a \,b^{4} c^{2} d -10 b^{5} c^{3}}{60 b^{6} \left (b x +a \right )^{6}}\)	\(123\)
norman	\(\frac {-\frac {d^{3} x^{3}}{3 b}+\frac {\left (-a \,b^{2} d^{3}-3 b^{3} c \,d^{2}\right ) x^{2}}{4 b^{4}}+\frac {\left (-a^{2} b^{2} d^{3}-3 a \,b^{3} c \,d^{2}-6 b^{4} c^{2} d \right ) x}{10 b^{5}}+\frac {-a^{3} b^{2} d^{3}-3 a^{2} b^{3} c \,d^{2}-6 a \,b^{4} c^{2} d -10 b^{5} c^{3}}{60 b^{6}}}{\left (b x +a \right )^{6}}\)	\(132\)

[In]

int((d*x+c)^3/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

(-1/3/b*d^3*x^3-1/4/b^2*d^2*(a*d+3*b*c)*x^2-1/10/b^3*d*(a^2*d^2+3*a*b*c*d+6*b^2*c^2)*x-1/60/b^4*(a^3*d^3+3*a^2
*b*c*d^2+6*a*b^2*c^2*d+10*b^3*c^3))/(b*x+a)^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (84) = 168\).

Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \, {\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*d^3*x^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 15*(3*b^3*c*d^2 + a*b^2*d^3)*x^
2 + 6*(6*b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 +
15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (83) = 166\).

Time = 1.49 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=\frac {- a^{3} d^{3} - 3 a^{2} b c d^{2} - 6 a b^{2} c^{2} d - 10 b^{3} c^{3} - 20 b^{3} d^{3} x^{3} + x^{2} \left (- 15 a b^{2} d^{3} - 45 b^{3} c d^{2}\right ) + x \left (- 6 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 36 b^{3} c^{2} d\right )}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \]

[In]

integrate((d*x+c)**3/(b*x+a)**7,x)

[Out]

(-a**3*d**3 - 3*a**2*b*c*d**2 - 6*a*b**2*c**2*d - 10*b**3*c**3 - 20*b**3*d**3*x**3 + x**2*(-15*a*b**2*d**3 - 4
5*b**3*c*d**2) + x*(-6*a**2*b*d**3 - 18*a*b**2*c*d**2 - 36*b**3*c**2*d))/(60*a**6*b**4 + 360*a**5*b**5*x + 900
*a**4*b**6*x**2 + 1200*a**3*b**7*x**3 + 900*a**2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (84) = 168\).

Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \, {\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/60*(20*b^3*d^3*x^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 15*(3*b^3*c*d^2 + a*b^2*d^3)*x^
2 + 6*(6*b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 +
15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {20 \, b^{3} d^{3} x^{3} + 45 \, b^{3} c d^{2} x^{2} + 15 \, a b^{2} d^{3} x^{2} + 36 \, b^{3} c^{2} d x + 18 \, a b^{2} c d^{2} x + 6 \, a^{2} b d^{3} x + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}}{60 \, {\left (b x + a\right )}^{6} b^{4}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/60*(20*b^3*d^3*x^3 + 45*b^3*c*d^2*x^2 + 15*a*b^2*d^3*x^2 + 36*b^3*c^2*d*x + 18*a*b^2*c*d^2*x + 6*a^2*b*d^3*
x + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^6*b^4)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.79 \[ \int \frac {(c+d x)^3}{(a+b x)^7} \, dx=-\frac {\frac {a^3\,d^3+3\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d+10\,b^3\,c^3}{60\,b^4}+\frac {d^3\,x^3}{3\,b}+\frac {d\,x\,\left (a^2\,d^2+3\,a\,b\,c\,d+6\,b^2\,c^2\right )}{10\,b^3}+\frac {d^2\,x^2\,\left (a\,d+3\,b\,c\right )}{4\,b^2}}{a^6+6\,a^5\,b\,x+15\,a^4\,b^2\,x^2+20\,a^3\,b^3\,x^3+15\,a^2\,b^4\,x^4+6\,a\,b^5\,x^5+b^6\,x^6} \]

[In]

int((c + d*x)^3/(a + b*x)^7,x)

[Out]

-((a^3*d^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2)/(60*b^4) + (d^3*x^3)/(3*b) + (d*x*(a^2*d^2 + 6*b^2*c^
2 + 3*a*b*c*d))/(10*b^3) + (d^2*x^2*(a*d + 3*b*c))/(4*b^2))/(a^6 + b^6*x^6 + 6*a*b^5*x^5 + 15*a^4*b^2*x^2 + 20
*a^3*b^3*x^3 + 15*a^2*b^4*x^4 + 6*a^5*b*x)