\(\int \frac {(c+d x)^3}{(a+b x)^8} \, dx\) [1271]

Optimal result

Integrand size = 15, antiderivative size = 92 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac {d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac {3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac {d^3}{4 b^4 (a+b x)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (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} \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac {d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac {(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac {d^3}{4 b^4 (a+b x)^4} \]

[In]

Int[(c + d*x)^3/(a + b*x)^8,x]

[Out]

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

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)^8}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^7}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^6}+\frac {d^3}{b^3 (a+b x)^5}\right ) \, dx \\ & = -\frac {(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac {d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac {3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac {d^3}{4 b^4 (a+b x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {a^3 d^3+a^2 b d^2 (4 c+7 d x)+a b^2 d \left (10 c^2+28 c d x+21 d^2 x^2\right )+b^3 \left (20 c^3+70 c^2 d x+84 c d^2 x^2+35 d^3 x^3\right )}{140 b^4 (a+b x)^7} \]

[In]

Integrate[(c + d*x)^3/(a + b*x)^8,x]

[Out]

-1/140*(a^3*d^3 + a^2*b*d^2*(4*c + 7*d*x) + a*b^2*d*(10*c^2 + 28*c*d*x + 21*d^2*x^2) + b^3*(20*c^3 + 70*c^2*d*
x + 84*c*d^2*x^2 + 35*d^3*x^3))/(b^4*(a + b*x)^7)

Maple [A] (verified)

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

method	result	size
risch	\(\frac {-\frac {d^{3} x^{3}}{4 b}-\frac {3 d^{2} \left (a d +4 b c \right ) x^{2}}{20 b^{2}}-\frac {d \left (a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}\right ) x}{20 b^{3}}-\frac {a^{3} d^{3}+4 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d +20 b^{3} c^{3}}{140 b^{4}}}{\left (b x +a \right )^{7}}\)	\(110\)
gosper	\(-\frac {35 d^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} d^{3}+84 x^{2} b^{3} c \,d^{2}+7 x \,a^{2} b \,d^{3}+28 x a \,b^{2} c \,d^{2}+70 x \,b^{3} c^{2} d +a^{3} d^{3}+4 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d +20 b^{3} c^{3}}{140 b^{4} \left (b x +a \right )^{7}}\)	\(115\)
default	\(-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{4} \left (b x +a \right )^{6}}-\frac {d^{3}}{4 b^{4} \left (b x +a \right )^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{7 b^{4} \left (b x +a \right )^{7}}+\frac {3 d^{2} \left (a d -b c \right )}{5 b^{4} \left (b x +a \right )^{5}}\)	\(122\)
parallelrisch	\(\frac {-35 d^{3} x^{3} b^{6}-21 a \,b^{5} d^{3} x^{2}-84 b^{6} c \,d^{2} x^{2}-7 a^{2} b^{4} d^{3} x -28 a \,b^{5} c \,d^{2} x -70 b^{6} c^{2} d x -a^{3} b^{3} d^{3}-4 a^{2} b^{4} c \,d^{2}-10 a \,b^{5} c^{2} d -20 c^{3} b^{6}}{140 b^{7} \left (b x +a \right )^{7}}\)	\(123\)
norman	\(\frac {-\frac {d^{3} x^{3}}{4 b}+\frac {3 \left (-a \,b^{3} d^{3}-4 b^{4} c \,d^{2}\right ) x^{2}}{20 b^{5}}+\frac {\left (-a^{2} b^{3} d^{3}-4 a \,b^{4} c \,d^{2}-10 b^{5} c^{2} d \right ) x}{20 b^{6}}+\frac {-a^{3} b^{3} d^{3}-4 a^{2} b^{4} c \,d^{2}-10 a \,b^{5} c^{2} d -20 c^{3} b^{6}}{140 b^{7}}}{\left (b x +a \right )^{7}}\)	\(132\)

[In]

int((d*x+c)^3/(b*x+a)^8,x,method=_RETURNVERBOSE)

[Out]

(-1/4/b*d^3*x^3-3/20/b^2*d^2*(a*d+4*b*c)*x^2-1/20/b^3*d*(a^2*d^2+4*a*b*c*d+10*b^2*c^2)*x-1/140/b^4*(a^3*d^3+4*
a^2*b*c*d^2+10*a*b^2*c^2*d+20*b^3*c^3))/(b*x+a)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (84) = 168\).

Time = 0.22 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \, {\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \, {\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*d^3*x^3 + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2 + a^3*d^3 + 21*(4*b^3*c*d^2 + a*b^2*d^3)*
x^2 + 7*(10*b^3*c^2*d + 4*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^11*x^7 + 7*a*b^10*x^6 + 21*a^2*b^9*x^5 + 35*a^3*b^8*x
^4 + 35*a^4*b^7*x^3 + 21*a^5*b^6*x^2 + 7*a^6*b^5*x + a^7*b^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (82) = 164\).

Time = 1.83 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.13 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=\frac {- a^{3} d^{3} - 4 a^{2} b c d^{2} - 10 a b^{2} c^{2} d - 20 b^{3} c^{3} - 35 b^{3} d^{3} x^{3} + x^{2} \left (- 21 a b^{2} d^{3} - 84 b^{3} c d^{2}\right ) + x \left (- 7 a^{2} b d^{3} - 28 a b^{2} c d^{2} - 70 b^{3} c^{2} d\right )}{140 a^{7} b^{4} + 980 a^{6} b^{5} x + 2940 a^{5} b^{6} x^{2} + 4900 a^{4} b^{7} x^{3} + 4900 a^{3} b^{8} x^{4} + 2940 a^{2} b^{9} x^{5} + 980 a b^{10} x^{6} + 140 b^{11} x^{7}} \]

[In]

integrate((d*x+c)**3/(b*x+a)**8,x)

[Out]

(-a**3*d**3 - 4*a**2*b*c*d**2 - 10*a*b**2*c**2*d - 20*b**3*c**3 - 35*b**3*d**3*x**3 + x**2*(-21*a*b**2*d**3 -
84*b**3*c*d**2) + x*(-7*a**2*b*d**3 - 28*a*b**2*c*d**2 - 70*b**3*c**2*d))/(140*a**7*b**4 + 980*a**6*b**5*x + 2
940*a**5*b**6*x**2 + 4900*a**4*b**7*x**3 + 4900*a**3*b**8*x**4 + 2940*a**2*b**9*x**5 + 980*a*b**10*x**6 + 140*
b**11*x**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (84) = 168\).

Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \, {\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \, {\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*d^3*x^3 + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2 + a^3*d^3 + 21*(4*b^3*c*d^2 + a*b^2*d^3)*
x^2 + 7*(10*b^3*c^2*d + 4*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^11*x^7 + 7*a*b^10*x^6 + 21*a^2*b^9*x^5 + 35*a^3*b^8*x
^4 + 35*a^4*b^7*x^3 + 21*a^5*b^6*x^2 + 7*a^6*b^5*x + a^7*b^4)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {35 \, b^{3} d^{3} x^{3} + 84 \, b^{3} c d^{2} x^{2} + 21 \, a b^{2} d^{3} x^{2} + 70 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 7 \, a^{2} b d^{3} x + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3}}{140 \, {\left (b x + a\right )}^{7} b^{4}} \]

[In]

integrate((d*x+c)^3/(b*x+a)^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*d^3*x^3 + 84*b^3*c*d^2*x^2 + 21*a*b^2*d^3*x^2 + 70*b^3*c^2*d*x + 28*a*b^2*c*d^2*x + 7*a^2*b*d^3
*x + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^7*b^4)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^3}{(a+b x)^8} \, dx=-\frac {\frac {a^3\,d^3+4\,a^2\,b\,c\,d^2+10\,a\,b^2\,c^2\,d+20\,b^3\,c^3}{140\,b^4}+\frac {d^3\,x^3}{4\,b}+\frac {d\,x\,\left (a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2\right )}{20\,b^3}+\frac {3\,d^2\,x^2\,\left (a\,d+4\,b\,c\right )}{20\,b^2}}{a^7+7\,a^6\,b\,x+21\,a^5\,b^2\,x^2+35\,a^4\,b^3\,x^3+35\,a^3\,b^4\,x^4+21\,a^2\,b^5\,x^5+7\,a\,b^6\,x^6+b^7\,x^7} \]

[In]

int((c + d*x)^3/(a + b*x)^8,x)

[Out]

-((a^3*d^3 + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2)/(140*b^4) + (d^3*x^3)/(4*b) + (d*x*(a^2*d^2 + 10*b^2
*c^2 + 4*a*b*c*d))/(20*b^3) + (3*d^2*x^2*(a*d + 4*b*c))/(20*b^2))/(a^7 + b^7*x^7 + 7*a*b^6*x^6 + 21*a^5*b^2*x^
2 + 35*a^4*b^3*x^3 + 35*a^3*b^4*x^4 + 21*a^2*b^5*x^5 + 7*a^6*b*x)