3.1364 \(\int \frac{(a+b x)^7}{(c+d x)^8} \, dx\)

Optimal. Leaf size=194 \[ \frac{7 b^6 (b c-a d)}{d^8 (c+d x)}-\frac{21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac{35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac{35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac{21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac{7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac{(b c-a d)^7}{7 d^8 (c+d x)^7}+\frac{b^7 \log (c+d x)}{d^8} \]

[Out]

(b*c - a*d)^7/(7*d^8*(c + d*x)^7) - (7*b*(b*c - a*d)^6)/(6*d^8*(c + d*x)^6) + (21*b^2*(b*c - a*d)^5)/(5*d^8*(c
+ d*x)^5) - (35*b^3*(b*c - a*d)^4)/(4*d^8*(c + d*x)^4) + (35*b^4*(b*c - a*d)^3)/(3*d^8*(c + d*x)^3) - (21*b^5
*(b*c - a*d)^2)/(2*d^8*(c + d*x)^2) + (7*b^6*(b*c - a*d))/(d^8*(c + d*x)) + (b^7*Log[c + d*x])/d^8

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Rubi [A]  time = 0.209351, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{7 b^6 (b c-a d)}{d^8 (c+d x)}-\frac{21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac{35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac{35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac{21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac{7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac{(b c-a d)^7}{7 d^8 (c+d x)^7}+\frac{b^7 \log (c+d x)}{d^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/(c + d*x)^8,x]

[Out]

(b*c - a*d)^7/(7*d^8*(c + d*x)^7) - (7*b*(b*c - a*d)^6)/(6*d^8*(c + d*x)^6) + (21*b^2*(b*c - a*d)^5)/(5*d^8*(c
+ d*x)^5) - (35*b^3*(b*c - a*d)^4)/(4*d^8*(c + d*x)^4) + (35*b^4*(b*c - a*d)^3)/(3*d^8*(c + d*x)^3) - (21*b^5
*(b*c - a*d)^2)/(2*d^8*(c + d*x)^2) + (7*b^6*(b*c - a*d))/(d^8*(c + d*x)) + (b^7*Log[c + d*x])/d^8

Rule 43

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*} \int \frac{(a+b x)^7}{(c+d x)^8} \, dx &=\int \left (\frac{(-b c+a d)^7}{d^7 (c+d x)^8}+\frac{7 b (b c-a d)^6}{d^7 (c+d x)^7}-\frac{21 b^2 (b c-a d)^5}{d^7 (c+d x)^6}+\frac{35 b^3 (b c-a d)^4}{d^7 (c+d x)^5}-\frac{35 b^4 (b c-a d)^3}{d^7 (c+d x)^4}+\frac{21 b^5 (b c-a d)^2}{d^7 (c+d x)^3}-\frac{7 b^6 (b c-a d)}{d^7 (c+d x)^2}+\frac{b^7}{d^7 (c+d x)}\right ) \, dx\\ &=\frac{(b c-a d)^7}{7 d^8 (c+d x)^7}-\frac{7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac{21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac{35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac{35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac{21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac{7 b^6 (b c-a d)}{d^8 (c+d x)}+\frac{b^7 \log (c+d x)}{d^8}\\ \end{align*}

Mathematica [A]  time = 0.15939, size = 308, normalized size = 1.59 \[ \frac{(b c-a d) \left (a^2 b^4 d^2 \left (6909 c^2 d^2 x^2+2793 c^3 d x+459 c^4+8575 c d^3 x^3+4900 d^4 x^4\right )+a^3 b^3 d^3 \left (1813 c^2 d x+319 c^3+3969 c d^2 x^2+3675 d^3 x^3\right )+2 a^4 b^2 d^4 \left (107 c^2+539 c d x+882 d^2 x^2\right )+10 a^5 b d^5 (13 c+49 d x)+60 a^6 d^6+a b^5 d \left (11319 c^3 d^2 x^2+15925 c^2 d^3 x^3+4263 c^4 d x+669 c^5+12250 c d^4 x^4+4410 d^5 x^5\right )+b^6 \left (20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+7203 c^5 d x+1089 c^6+13230 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 d^8 (c+d x)^7}+\frac{b^7 \log (c+d x)}{d^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/(c + d*x)^8,x]

[Out]

((b*c - a*d)*(60*a^6*d^6 + 10*a^5*b*d^5*(13*c + 49*d*x) + 2*a^4*b^2*d^4*(107*c^2 + 539*c*d*x + 882*d^2*x^2) +
a^3*b^3*d^3*(319*c^3 + 1813*c^2*d*x + 3969*c*d^2*x^2 + 3675*d^3*x^3) + a^2*b^4*d^2*(459*c^4 + 2793*c^3*d*x + 6
909*c^2*d^2*x^2 + 8575*c*d^3*x^3 + 4900*d^4*x^4) + a*b^5*d*(669*c^5 + 4263*c^4*d*x + 11319*c^3*d^2*x^2 + 15925
*c^2*d^3*x^3 + 12250*c*d^4*x^4 + 4410*d^5*x^5) + b^6*(1089*c^6 + 7203*c^5*d*x + 20139*c^4*d^2*x^2 + 30625*c^3*
d^3*x^3 + 26950*c^2*d^4*x^4 + 13230*c*d^5*x^5 + 2940*d^6*x^6)))/(420*d^8*(c + d*x)^7) + (b^7*Log[c + d*x])/d^8

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Maple [B]  time = 0.008, size = 672, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/(d*x+c)^8,x)

[Out]

-1/7/d/(d*x+c)^7*a^7+70/3*b^4/d^5/(d*x+c)^6*a^3*c^3-35/2*b^5/d^6/(d*x+c)^6*a^2*c^4+7*b^2/d^3/(d*x+c)^6*a^5*c-3
5/2*b^3/d^4/(d*x+c)^6*a^4*c^2+21*b^6/d^7/(d*x+c)^2*a*c+35*b^6/d^7/(d*x+c)^4*a*c^3-105/2*b^5/d^6/(d*x+c)^4*a^2*
c^2-3/d^3/(d*x+c)^7*a^5*b^2*c^2+5/d^4/(d*x+c)^7*a^4*b^3*c^3-5/d^5/(d*x+c)^7*a^3*b^4*c^4+3/d^6/(d*x+c)^7*a^2*b^
5*c^5-1/d^7/(d*x+c)^7*a*b^6*c^6+35*b^4/d^5/(d*x+c)^4*a^3*c+1/d^2/(d*x+c)^7*a^6*b*c+b^7*ln(d*x+c)/d^8-35/4*b^7/
d^8/(d*x+c)^4*c^4-35/4*b^3/d^4/(d*x+c)^4*a^4+1/7/d^8/(d*x+c)^7*b^7*c^7-7*b^6/d^7/(d*x+c)*a+7*b^7/d^8/(d*x+c)*c
-35/3*b^4/d^5/(d*x+c)^3*a^3+35/3*b^7/d^8/(d*x+c)^3*c^3-21/5*b^2/d^3/(d*x+c)^5*a^5+21/5*b^7/d^8/(d*x+c)^5*c^5-7
/6*b/d^2/(d*x+c)^6*a^6-7/6*b^7/d^8/(d*x+c)^6*c^6-21/2*b^5/d^6/(d*x+c)^2*a^2-21/2*b^7/d^8/(d*x+c)^2*c^2+35*b^5/
d^6/(d*x+c)^3*a^2*c-35*b^6/d^7/(d*x+c)^3*a*c^2+7*b^6/d^7/(d*x+c)^6*a*c^5+21*b^3/d^4/(d*x+c)^5*a^4*c-42*b^4/d^5
/(d*x+c)^5*a^3*c^2+42*b^5/d^6/(d*x+c)^5*a^2*c^3-21*b^6/d^7/(d*x+c)^5*a*c^4

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Maxima [B]  time = 1.04702, size = 722, normalized size = 3.72 \begin{align*} \frac{1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \,{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \,{\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \,{\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \,{\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \,{\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \,{\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x}{420 \,{\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} + \frac{b^{7} \log \left (d x + c\right )}{d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(d*x+c)^8,x, algorithm="maxima")

[Out]

1/420*(1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4*c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a
^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(3*b^7*c^2*d^5 - 2*a*b^
6*c*d^6 - a^2*b^5*d^7)*x^5 + 2450*(11*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 - 3*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 1
225*(25*b^7*c^4*d^3 - 12*a*b^6*c^3*d^4 - 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 147*(137*b
^7*c^5*d^2 - 60*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 - 20*a^3*b^4*c^2*d^5 - 15*a^4*b^3*c*d^6 - 12*a^5*b^2*d^7)*x
^2 + 49*(147*b^7*c^6*d - 60*a*b^6*c^5*d^2 - 30*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*d^4 - 15*a^4*b^3*c^2*d^5 - 12*
a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x)/(d^15*x^7 + 7*c*d^14*x^6 + 21*c^2*d^13*x^5 + 35*c^3*d^12*x^4 + 35*c^4*d^11*x^
3 + 21*c^5*d^10*x^2 + 7*c^6*d^9*x + c^7*d^8) + b^7*log(d*x + c)/d^8

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Fricas [B]  time = 1.82887, size = 1319, normalized size = 6.8 \begin{align*} \frac{1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \,{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \,{\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \,{\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \,{\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \,{\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \,{\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x + 420 \,{\left (b^{7} d^{7} x^{7} + 7 \, b^{7} c d^{6} x^{6} + 21 \, b^{7} c^{2} d^{5} x^{5} + 35 \, b^{7} c^{3} d^{4} x^{4} + 35 \, b^{7} c^{4} d^{3} x^{3} + 21 \, b^{7} c^{5} d^{2} x^{2} + 7 \, b^{7} c^{6} d x + b^{7} c^{7}\right )} \log \left (d x + c\right )}{420 \,{\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(d*x+c)^8,x, algorithm="fricas")

[Out]

1/420*(1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4*c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a
^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(3*b^7*c^2*d^5 - 2*a*b^
6*c*d^6 - a^2*b^5*d^7)*x^5 + 2450*(11*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 - 3*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 1
225*(25*b^7*c^4*d^3 - 12*a*b^6*c^3*d^4 - 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 147*(137*b
^7*c^5*d^2 - 60*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 - 20*a^3*b^4*c^2*d^5 - 15*a^4*b^3*c*d^6 - 12*a^5*b^2*d^7)*x
^2 + 49*(147*b^7*c^6*d - 60*a*b^6*c^5*d^2 - 30*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*d^4 - 15*a^4*b^3*c^2*d^5 - 12*
a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x + 420*(b^7*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^2*d^5*x^5 + 35*b^7*c^3*d^4*x^4
+ 35*b^7*c^4*d^3*x^3 + 21*b^7*c^5*d^2*x^2 + 7*b^7*c^6*d*x + b^7*c^7)*log(d*x + c))/(d^15*x^7 + 7*c*d^14*x^6 +
21*c^2*d^13*x^5 + 35*c^3*d^12*x^4 + 35*c^4*d^11*x^3 + 21*c^5*d^10*x^2 + 7*c^6*d^9*x + c^7*d^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/(d*x+c)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.07333, size = 630, normalized size = 3.25 \begin{align*} \frac{b^{7} \log \left ({\left | d x + c \right |}\right )}{d^{8}} + \frac{2940 \,{\left (b^{7} c d^{5} - a b^{6} d^{6}\right )} x^{6} + 4410 \,{\left (3 \, b^{7} c^{2} d^{4} - 2 \, a b^{6} c d^{5} - a^{2} b^{5} d^{6}\right )} x^{5} + 2450 \,{\left (11 \, b^{7} c^{3} d^{3} - 6 \, a b^{6} c^{2} d^{4} - 3 \, a^{2} b^{5} c d^{5} - 2 \, a^{3} b^{4} d^{6}\right )} x^{4} + 1225 \,{\left (25 \, b^{7} c^{4} d^{2} - 12 \, a b^{6} c^{3} d^{3} - 6 \, a^{2} b^{5} c^{2} d^{4} - 4 \, a^{3} b^{4} c d^{5} - 3 \, a^{4} b^{3} d^{6}\right )} x^{3} + 147 \,{\left (137 \, b^{7} c^{5} d - 60 \, a b^{6} c^{4} d^{2} - 30 \, a^{2} b^{5} c^{3} d^{3} - 20 \, a^{3} b^{4} c^{2} d^{4} - 15 \, a^{4} b^{3} c d^{5} - 12 \, a^{5} b^{2} d^{6}\right )} x^{2} + 49 \,{\left (147 \, b^{7} c^{6} - 60 \, a b^{6} c^{5} d - 30 \, a^{2} b^{5} c^{4} d^{2} - 20 \, a^{3} b^{4} c^{3} d^{3} - 15 \, a^{4} b^{3} c^{2} d^{4} - 12 \, a^{5} b^{2} c d^{5} - 10 \, a^{6} b d^{6}\right )} x + \frac{1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7}}{d}}{420 \,{\left (d x + c\right )}^{7} d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(d*x+c)^8,x, algorithm="giac")

[Out]

b^7*log(abs(d*x + c))/d^8 + 1/420*(2940*(b^7*c*d^5 - a*b^6*d^6)*x^6 + 4410*(3*b^7*c^2*d^4 - 2*a*b^6*c*d^5 - a^
2*b^5*d^6)*x^5 + 2450*(11*b^7*c^3*d^3 - 6*a*b^6*c^2*d^4 - 3*a^2*b^5*c*d^5 - 2*a^3*b^4*d^6)*x^4 + 1225*(25*b^7*
c^4*d^2 - 12*a*b^6*c^3*d^3 - 6*a^2*b^5*c^2*d^4 - 4*a^3*b^4*c*d^5 - 3*a^4*b^3*d^6)*x^3 + 147*(137*b^7*c^5*d - 6
0*a*b^6*c^4*d^2 - 30*a^2*b^5*c^3*d^3 - 20*a^3*b^4*c^2*d^4 - 15*a^4*b^3*c*d^5 - 12*a^5*b^2*d^6)*x^2 + 49*(147*b
^7*c^6 - 60*a*b^6*c^5*d - 30*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 - 15*a^4*b^3*c^2*d^4 - 12*a^5*b^2*c*d^5 - 10
*a^6*b*d^6)*x + (1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4*c^4*d^3 - 105*a^4*b^3*c^3*
d^4 - 84*a^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a^7*d^7)/d)/((d*x + c)^7*d^7)