3.1296 \(\int \frac{(c+d x)^7}{(a+b x)^{14}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac{21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac{35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac{7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac{21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac{7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac{(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac{d^7}{6 b^8 (a+b x)^6} \]

[Out]

-(b*c - a*d)^7/(13*b^8*(a + b*x)^13) - (7*d*(b*c - a*d)^6)/(12*b^8*(a + b*x)^12) - (21*d^2*(b*c - a*d)^5)/(11*
b^8*(a + b*x)^11) - (7*d^3*(b*c - a*d)^4)/(2*b^8*(a + b*x)^10) - (35*d^4*(b*c - a*d)^3)/(9*b^8*(a + b*x)^9) -
(21*d^5*(b*c - a*d)^2)/(8*b^8*(a + b*x)^8) - (d^6*(b*c - a*d))/(b^8*(a + b*x)^7) - d^7/(6*b^8*(a + b*x)^6)

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Rubi [A]  time = 0.153446, antiderivative size = 198, 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} \[ -\frac{d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac{21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac{35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac{7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac{21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac{7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac{(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac{d^7}{6 b^8 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^14,x]

[Out]

-(b*c - a*d)^7/(13*b^8*(a + b*x)^13) - (7*d*(b*c - a*d)^6)/(12*b^8*(a + b*x)^12) - (21*d^2*(b*c - a*d)^5)/(11*
b^8*(a + b*x)^11) - (7*d^3*(b*c - a*d)^4)/(2*b^8*(a + b*x)^10) - (35*d^4*(b*c - a*d)^3)/(9*b^8*(a + b*x)^9) -
(21*d^5*(b*c - a*d)^2)/(8*b^8*(a + b*x)^8) - (d^6*(b*c - a*d))/(b^8*(a + b*x)^7) - d^7/(6*b^8*(a + b*x)^6)

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{(c+d x)^7}{(a+b x)^{14}} \, dx &=\int \left (\frac{(b c-a d)^7}{b^7 (a+b x)^{14}}+\frac{7 d (b c-a d)^6}{b^7 (a+b x)^{13}}+\frac{21 d^2 (b c-a d)^5}{b^7 (a+b x)^{12}}+\frac{35 d^3 (b c-a d)^4}{b^7 (a+b x)^{11}}+\frac{35 d^4 (b c-a d)^3}{b^7 (a+b x)^{10}}+\frac{21 d^5 (b c-a d)^2}{b^7 (a+b x)^9}+\frac{7 d^6 (b c-a d)}{b^7 (a+b x)^8}+\frac{d^7}{b^7 (a+b x)^7}\right ) \, dx\\ &=-\frac{(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac{7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac{21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac{7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac{35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac{21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac{d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac{d^7}{6 b^8 (a+b x)^6}\\ \end{align*}

Mathematica [A]  time = 0.121782, size = 369, normalized size = 1.86 \[ -\frac{3 a^2 b^5 d^2 \left (1456 c^3 d^2 x^2+2002 c^2 d^3 x^3+546 c^4 d x+84 c^5+1430 c d^4 x^4+429 d^5 x^5\right )+a^3 b^4 d^3 \left (1638 c^2 d^2 x^2+728 c^3 d x+126 c^4+1716 c d^3 x^3+715 d^4 x^4\right )+a^4 b^3 d^4 \left (273 c^2 d x+56 c^3+468 c d^2 x^2+286 d^3 x^3\right )+3 a^5 b^2 d^5 \left (7 c^2+26 c d x+26 d^2 x^2\right )+a^6 b d^6 (6 c+13 d x)+a^7 d^7+a b^6 d \left (9828 c^4 d^2 x^2+16016 c^3 d^3 x^3+15015 c^2 d^4 x^4+3276 c^5 d x+462 c^6+7722 c d^5 x^5+1716 d^6 x^6\right )+b^7 \left (19656 c^5 d^2 x^2+36036 c^4 d^3 x^3+40040 c^3 d^4 x^4+27027 c^2 d^5 x^5+6006 c^6 d x+792 c^7+10296 c d^6 x^6+1716 d^7 x^7\right )}{10296 b^8 (a+b x)^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^14,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(6*c + 13*d*x) + 3*a^5*b^2*d^5*(7*c^2 + 26*c*d*x + 26*d^2*x^2) + a^4*b^3*d^4*(56*c^3 + 2
73*c^2*d*x + 468*c*d^2*x^2 + 286*d^3*x^3) + a^3*b^4*d^3*(126*c^4 + 728*c^3*d*x + 1638*c^2*d^2*x^2 + 1716*c*d^3
*x^3 + 715*d^4*x^4) + 3*a^2*b^5*d^2*(84*c^5 + 546*c^4*d*x + 1456*c^3*d^2*x^2 + 2002*c^2*d^3*x^3 + 1430*c*d^4*x
^4 + 429*d^5*x^5) + a*b^6*d*(462*c^6 + 3276*c^5*d*x + 9828*c^4*d^2*x^2 + 16016*c^3*d^3*x^3 + 15015*c^2*d^4*x^4
+ 7722*c*d^5*x^5 + 1716*d^6*x^6) + b^7*(792*c^7 + 6006*c^6*d*x + 19656*c^5*d^2*x^2 + 36036*c^4*d^3*x^3 + 4004
0*c^3*d^4*x^4 + 27027*c^2*d^5*x^5 + 10296*c*d^6*x^6 + 1716*d^7*x^7))/(10296*b^8*(a + b*x)^13)

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Maple [B]  time = 0.008, size = 463, normalized size = 2.3 \begin{align*} -{\frac{7\,{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{2\,{b}^{8} \left ( bx+a \right ) ^{10}}}-{\frac{21\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{{d}^{7}}{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}+{\frac{21\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}-{\frac{-{a}^{7}{d}^{7}+7\,{a}^{6}c{d}^{6}b-21\,{a}^{5}{b}^{2}{c}^{2}{d}^{5}+35\,{c}^{3}{d}^{4}{a}^{4}{b}^{3}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{c}^{6}d{b}^{6}+{b}^{7}{c}^{7}}{13\,{b}^{8} \left ( bx+a \right ) ^{13}}}-{\frac{7\,d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{12\,{b}^{8} \left ( bx+a \right ) ^{12}}}+{\frac{35\,{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{9\,{b}^{8} \left ( bx+a \right ) ^{9}}}+{\frac{{d}^{6} \left ( ad-bc \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^14,x)

[Out]

-7/2*d^3*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/b^8/(b*x+a)^10-21/8*d^5*(a^2*d^2-2*a*
b*c*d+b^2*c^2)/b^8/(b*x+a)^8-1/6*d^7/b^8/(b*x+a)^6+21/11*d^2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*
b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/b^8/(b*x+a)^11-1/13*(-a^7*d^7+7*a^6*b*c*d^6-21*a^5*b^2*c^2*d^5+35*a^4*b^3*c
^3*d^4-35*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2-7*a*b^6*c^6*d+b^7*c^7)/b^8/(b*x+a)^13-7/12*d*(a^6*d^6-6*a^5*b*c*d
^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/b^8/(b*x+a)^12+35/9*d^4*(a^
3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^8/(b*x+a)^9+d^6*(a*d-b*c)/b^8/(b*x+a)^7

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Maxima [B]  time = 1.11617, size = 799, normalized size = 4.04 \begin{align*} -\frac{1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \,{\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \,{\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \,{\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \,{\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \,{\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \,{\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \,{\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="maxima")

[Out]

-1/10296*(1716*b^7*d^7*x^7 + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^
4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7 + 1716*(6*b^7*c*d^6 + a*b^6*d^7)*x^6 + 1287*(21*b
^7*c^2*d^5 + 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 715*(56*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 + a^3
*b^4*d^7)*x^4 + 286*(126*b^7*c^4*d^3 + 56*a*b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 + 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*
x^3 + 78*(252*b^7*c^5*d^2 + 126*a*b^6*c^4*d^3 + 56*a^2*b^5*c^3*d^4 + 21*a^3*b^4*c^2*d^5 + 6*a^4*b^3*c*d^6 + a^
5*b^2*d^7)*x^2 + 13*(462*b^7*c^6*d + 252*a*b^6*c^5*d^2 + 126*a^2*b^5*c^4*d^3 + 56*a^3*b^4*c^3*d^4 + 21*a^4*b^3
*c^2*d^5 + 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^21*x^13 + 13*a*b^20*x^12 + 78*a^2*b^19*x^11 + 286*a^3*b^18*x^10
+ 715*a^4*b^17*x^9 + 1287*a^5*b^16*x^8 + 1716*a^6*b^15*x^7 + 1716*a^7*b^14*x^6 + 1287*a^8*b^13*x^5 + 715*a^9*b
^12*x^4 + 286*a^10*b^11*x^3 + 78*a^11*b^10*x^2 + 13*a^12*b^9*x + a^13*b^8)

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Fricas [B]  time = 1.76347, size = 1291, normalized size = 6.52 \begin{align*} -\frac{1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \,{\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \,{\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \,{\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \,{\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \,{\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \,{\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \,{\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="fricas")

[Out]

-1/10296*(1716*b^7*d^7*x^7 + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^
4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7 + 1716*(6*b^7*c*d^6 + a*b^6*d^7)*x^6 + 1287*(21*b
^7*c^2*d^5 + 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 715*(56*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 + a^3
*b^4*d^7)*x^4 + 286*(126*b^7*c^4*d^3 + 56*a*b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 + 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*
x^3 + 78*(252*b^7*c^5*d^2 + 126*a*b^6*c^4*d^3 + 56*a^2*b^5*c^3*d^4 + 21*a^3*b^4*c^2*d^5 + 6*a^4*b^3*c*d^6 + a^
5*b^2*d^7)*x^2 + 13*(462*b^7*c^6*d + 252*a*b^6*c^5*d^2 + 126*a^2*b^5*c^4*d^3 + 56*a^3*b^4*c^3*d^4 + 21*a^4*b^3
*c^2*d^5 + 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^21*x^13 + 13*a*b^20*x^12 + 78*a^2*b^19*x^11 + 286*a^3*b^18*x^10
+ 715*a^4*b^17*x^9 + 1287*a^5*b^16*x^8 + 1716*a^6*b^15*x^7 + 1716*a^7*b^14*x^6 + 1287*a^8*b^13*x^5 + 715*a^9*b
^12*x^4 + 286*a^10*b^11*x^3 + 78*a^11*b^10*x^2 + 13*a^12*b^9*x + a^13*b^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((d*x+c)**7/(b*x+a)**14,x)

[Out]

Timed out

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Giac [B]  time = 1.06428, size = 670, normalized size = 3.38 \begin{align*} -\frac{1716 \, b^{7} d^{7} x^{7} + 10296 \, b^{7} c d^{6} x^{6} + 1716 \, a b^{6} d^{7} x^{6} + 27027 \, b^{7} c^{2} d^{5} x^{5} + 7722 \, a b^{6} c d^{6} x^{5} + 1287 \, a^{2} b^{5} d^{7} x^{5} + 40040 \, b^{7} c^{3} d^{4} x^{4} + 15015 \, a b^{6} c^{2} d^{5} x^{4} + 4290 \, a^{2} b^{5} c d^{6} x^{4} + 715 \, a^{3} b^{4} d^{7} x^{4} + 36036 \, b^{7} c^{4} d^{3} x^{3} + 16016 \, a b^{6} c^{3} d^{4} x^{3} + 6006 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1716 \, a^{3} b^{4} c d^{6} x^{3} + 286 \, a^{4} b^{3} d^{7} x^{3} + 19656 \, b^{7} c^{5} d^{2} x^{2} + 9828 \, a b^{6} c^{4} d^{3} x^{2} + 4368 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 1638 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 468 \, a^{4} b^{3} c d^{6} x^{2} + 78 \, a^{5} b^{2} d^{7} x^{2} + 6006 \, b^{7} c^{6} d x + 3276 \, a b^{6} c^{5} d^{2} x + 1638 \, a^{2} b^{5} c^{4} d^{3} x + 728 \, a^{3} b^{4} c^{3} d^{4} x + 273 \, a^{4} b^{3} c^{2} d^{5} x + 78 \, a^{5} b^{2} c d^{6} x + 13 \, a^{6} b d^{7} x + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7}}{10296 \,{\left (b x + a\right )}^{13} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="giac")

[Out]

-1/10296*(1716*b^7*d^7*x^7 + 10296*b^7*c*d^6*x^6 + 1716*a*b^6*d^7*x^6 + 27027*b^7*c^2*d^5*x^5 + 7722*a*b^6*c*d
^6*x^5 + 1287*a^2*b^5*d^7*x^5 + 40040*b^7*c^3*d^4*x^4 + 15015*a*b^6*c^2*d^5*x^4 + 4290*a^2*b^5*c*d^6*x^4 + 715
*a^3*b^4*d^7*x^4 + 36036*b^7*c^4*d^3*x^3 + 16016*a*b^6*c^3*d^4*x^3 + 6006*a^2*b^5*c^2*d^5*x^3 + 1716*a^3*b^4*c
*d^6*x^3 + 286*a^4*b^3*d^7*x^3 + 19656*b^7*c^5*d^2*x^2 + 9828*a*b^6*c^4*d^3*x^2 + 4368*a^2*b^5*c^3*d^4*x^2 + 1
638*a^3*b^4*c^2*d^5*x^2 + 468*a^4*b^3*c*d^6*x^2 + 78*a^5*b^2*d^7*x^2 + 6006*b^7*c^6*d*x + 3276*a*b^6*c^5*d^2*x
+ 1638*a^2*b^5*c^4*d^3*x + 728*a^3*b^4*c^3*d^4*x + 273*a^4*b^3*c^2*d^5*x + 78*a^5*b^2*c*d^6*x + 13*a^6*b*d^7*
x + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^4*b^3*c^3*d^4 + 21*a^5*b^
2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^13*b^8)