\(\int \frac {a+b x}{(c+d x)^8} \, dx\) [1370]

Optimal result

Integrand size = 13, antiderivative size = 38 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=\frac {b c-a d}{7 d^2 (c+d x)^7}-\frac {b}{6 d^2 (c+d x)^6} \]

[Out]

1/7*(-a*d+b*c)/d^2/(d*x+c)^7-1/6*b/d^2/(d*x+c)^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {a+b x}{(c+d x)^8} \, dx=\frac {b c-a d}{7 d^2 (c+d x)^7}-\frac {b}{6 d^2 (c+d x)^6} \]

[In]

Int[(a + b*x)/(c + d*x)^8,x]

[Out]

(b*c - a*d)/(7*d^2*(c + d*x)^7) - b/(6*d^2*(c + d*x)^6)

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}{d (c+d x)^8}+\frac {b}{d (c+d x)^7}\right ) \, dx \\ & = \frac {b c-a d}{7 d^2 (c+d x)^7}-\frac {b}{6 d^2 (c+d x)^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=-\frac {6 a d+b (c+7 d x)}{42 d^2 (c+d x)^7} \]

[In]

Integrate[(a + b*x)/(c + d*x)^8,x]

[Out]

-1/42*(6*a*d + b*(c + 7*d*x))/(d^2*(c + d*x)^7)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68

method	result	size
gosper	\(-\frac {7 b d x +6 a d +b c}{42 d^{2} \left (d x +c \right )^{7}}\)	\(26\)
risch	\(\frac {-\frac {b x}{6 d}-\frac {6 a d +b c}{42 d^{2}}}{\left (d x +c \right )^{7}}\)	\(30\)
parallelrisch	\(\frac {-7 b x \,d^{6}-6 a \,d^{6}-b c \,d^{5}}{42 d^{7} \left (d x +c \right )^{7}}\)	\(34\)
default	\(-\frac {b}{6 d^{2} \left (d x +c \right )^{6}}-\frac {a d -b c}{7 d^{2} \left (d x +c \right )^{7}}\)	\(35\)
norman	\(\frac {-\frac {b x}{6 d}-\frac {6 a \,d^{6}+b c \,d^{5}}{42 d^{7}}}{\left (d x +c \right )^{7}}\)	\(35\)

[In]

int((b*x+a)/(d*x+c)^8,x,method=_RETURNVERBOSE)

[Out]

-1/42/d^2*(7*b*d*x+6*a*d+b*c)/(d*x+c)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (34) = 68\).

Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=-\frac {7 \, b d x + b c + 6 \, a d}{42 \, {\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \]

[In]

integrate((b*x+a)/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/42*(7*b*d*x + b*c + 6*a*d)/(d^9*x^7 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d^6*x^4 + 35*c^4*d^5*x^3 + 21*c
^5*d^4*x^2 + 7*c^6*d^3*x + c^7*d^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (32) = 64\).

Time = 0.44 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.63 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=\frac {- 6 a d - b c - 7 b d x}{42 c^{7} d^{2} + 294 c^{6} d^{3} x + 882 c^{5} d^{4} x^{2} + 1470 c^{4} d^{5} x^{3} + 1470 c^{3} d^{6} x^{4} + 882 c^{2} d^{7} x^{5} + 294 c d^{8} x^{6} + 42 d^{9} x^{7}} \]

[In]

integrate((b*x+a)/(d*x+c)**8,x)

[Out]

(-6*a*d - b*c - 7*b*d*x)/(42*c**7*d**2 + 294*c**6*d**3*x + 882*c**5*d**4*x**2 + 1470*c**4*d**5*x**3 + 1470*c**
3*d**6*x**4 + 882*c**2*d**7*x**5 + 294*c*d**8*x**6 + 42*d**9*x**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (34) = 68\).

Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=-\frac {7 \, b d x + b c + 6 \, a d}{42 \, {\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \]

[In]

integrate((b*x+a)/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/42*(7*b*d*x + b*c + 6*a*d)/(d^9*x^7 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d^6*x^4 + 35*c^4*d^5*x^3 + 21*c
^5*d^4*x^2 + 7*c^6*d^3*x + c^7*d^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=-\frac {7 \, b d x + b c + 6 \, a d}{42 \, {\left (d x + c\right )}^{7} d^{2}} \]

[In]

integrate((b*x+a)/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/42*(7*b*d*x + b*c + 6*a*d)/((d*x + c)^7*d^2)

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int \frac {a+b x}{(c+d x)^8} \, dx=-\frac {\frac {6\,a\,d+b\,c}{42\,d^2}+\frac {b\,x}{6\,d}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \]

[In]

int((a + b*x)/(c + d*x)^8,x)

[Out]

-((6*a*d + b*c)/(42*d^2) + (b*x)/(6*d))/(c^7 + d^7*x^7 + 7*c*d^6*x^6 + 21*c^5*d^2*x^2 + 35*c^4*d^3*x^3 + 35*c^
3*d^4*x^4 + 21*c^2*d^5*x^5 + 7*c^6*d*x)