3.1298 \(\int \frac{(c+d x)^7}{(a+b x)^{16}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac{21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac{35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac{35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac{21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac{d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac{(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac{d^7}{8 b^8 (a+b x)^8} \]

[Out]

-(b*c - a*d)^7/(15*b^8*(a + b*x)^15) - (d*(b*c - a*d)^6)/(2*b^8*(a + b*x)^14) - (21*d^2*(b*c - a*d)^5)/(13*b^8
*(a + b*x)^13) - (35*d^3*(b*c - a*d)^4)/(12*b^8*(a + b*x)^12) - (35*d^4*(b*c - a*d)^3)/(11*b^8*(a + b*x)^11) -
(21*d^5*(b*c - a*d)^2)/(10*b^8*(a + b*x)^10) - (7*d^6*(b*c - a*d))/(9*b^8*(a + b*x)^9) - d^7/(8*b^8*(a + b*x)
^8)

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Rubi [A]  time = 0.139636, antiderivative size = 200, 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} \[ -\frac{7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac{21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac{35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac{35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac{21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac{d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac{(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac{d^7}{8 b^8 (a+b x)^8} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^16,x]

[Out]

-(b*c - a*d)^7/(15*b^8*(a + b*x)^15) - (d*(b*c - a*d)^6)/(2*b^8*(a + b*x)^14) - (21*d^2*(b*c - a*d)^5)/(13*b^8
*(a + b*x)^13) - (35*d^3*(b*c - a*d)^4)/(12*b^8*(a + b*x)^12) - (35*d^4*(b*c - a*d)^3)/(11*b^8*(a + b*x)^11) -
(21*d^5*(b*c - a*d)^2)/(10*b^8*(a + b*x)^10) - (7*d^6*(b*c - a*d))/(9*b^8*(a + b*x)^9) - d^7/(8*b^8*(a + b*x)
^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{(c+d x)^7}{(a+b x)^{16}} \, dx &=\int \left (\frac{(b c-a d)^7}{b^7 (a+b x)^{16}}+\frac{7 d (b c-a d)^6}{b^7 (a+b x)^{15}}+\frac{21 d^2 (b c-a d)^5}{b^7 (a+b x)^{14}}+\frac{35 d^3 (b c-a d)^4}{b^7 (a+b x)^{13}}+\frac{35 d^4 (b c-a d)^3}{b^7 (a+b x)^{12}}+\frac{21 d^5 (b c-a d)^2}{b^7 (a+b x)^{11}}+\frac{7 d^6 (b c-a d)}{b^7 (a+b x)^{10}}+\frac{d^7}{b^7 (a+b x)^9}\right ) \, dx\\ &=-\frac{(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac{d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac{21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac{35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac{35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac{21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac{7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac{d^7}{8 b^8 (a+b x)^8}\\ \end{align*}

Mathematica [A]  time = 0.120596, size = 371, normalized size = 1.86 \[ -\frac{3 a^2 b^5 d^2 \left (4200 c^3 d^2 x^2+5460 c^2 d^3 x^3+1650 c^4 d x+264 c^5+3640 c d^4 x^4+1001 d^5 x^5\right )+5 a^3 b^4 d^3 \left (756 c^2 d^2 x^2+360 c^3 d x+66 c^4+728 c d^3 x^3+273 d^4 x^4\right )+5 a^4 b^3 d^4 \left (108 c^2 d x+24 c^3+168 c d^2 x^2+91 d^3 x^3\right )+3 a^5 b^2 d^5 \left (12 c^2+40 c d x+35 d^2 x^2\right )+a^6 b d^6 (8 c+15 d x)+a^7 d^7+a b^6 d \left (34650 c^4 d^2 x^2+54600 c^3 d^3 x^3+49140 c^2 d^4 x^4+11880 c^5 d x+1716 c^6+24024 c d^5 x^5+5005 d^6 x^6\right )+b^7 \left (83160 c^5 d^2 x^2+150150 c^4 d^3 x^3+163800 c^3 d^4 x^4+108108 c^2 d^5 x^5+25740 c^6 d x+3432 c^7+40040 c d^6 x^6+6435 d^7 x^7\right )}{51480 b^8 (a+b x)^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^16,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(8*c + 15*d*x) + 3*a^5*b^2*d^5*(12*c^2 + 40*c*d*x + 35*d^2*x^2) + 5*a^4*b^3*d^4*(24*c^3
+ 108*c^2*d*x + 168*c*d^2*x^2 + 91*d^3*x^3) + 5*a^3*b^4*d^3*(66*c^4 + 360*c^3*d*x + 756*c^2*d^2*x^2 + 728*c*d^
3*x^3 + 273*d^4*x^4) + 3*a^2*b^5*d^2*(264*c^5 + 1650*c^4*d*x + 4200*c^3*d^2*x^2 + 5460*c^2*d^3*x^3 + 3640*c*d^
4*x^4 + 1001*d^5*x^5) + a*b^6*d*(1716*c^6 + 11880*c^5*d*x + 34650*c^4*d^2*x^2 + 54600*c^3*d^3*x^3 + 49140*c^2*
d^4*x^4 + 24024*c*d^5*x^5 + 5005*d^6*x^6) + b^7*(3432*c^7 + 25740*c^6*d*x + 83160*c^5*d^2*x^2 + 150150*c^4*d^3
*x^3 + 163800*c^3*d^4*x^4 + 108108*c^2*d^5*x^5 + 40040*c*d^6*x^6 + 6435*d^7*x^7))/(51480*b^8*(a + b*x)^15)

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Maple [B]  time = 0.008, size = 464, normalized size = 2.3 \begin{align*} -{\frac{21\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{10\,{b}^{8} \left ( bx+a \right ) ^{10}}}-{\frac{{d}^{7}}{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{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 ) }{2\,{b}^{8} \left ( bx+a \right ) ^{14}}}+{\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 ) }{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}+{\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 ) }{13\,{b}^{8} \left ( bx+a \right ) ^{13}}}-{\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}}{15\,{b}^{8} \left ( bx+a \right ) ^{15}}}-{\frac{35\,{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 ) }{12\,{b}^{8} \left ( bx+a \right ) ^{12}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{9\,{b}^{8} \left ( bx+a \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^16,x)

[Out]

-21/10*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^8/(b*x+a)^10-1/8*d^7/b^8/(b*x+a)^8-1/2*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)^14+35/11*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)^11+21/13*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)^13-1/15*(-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)^15-35/12*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)^12+7/9*d^6*(a*d-b*c)/b^8/(b*x+a)^9

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Maxima [B]  time = 1.13135, size = 829, normalized size = 4.14 \begin{align*} -\frac{6435 \, b^{7} d^{7} x^{7} + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7} + 5005 \,{\left (8 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 3003 \,{\left (36 \, b^{7} c^{2} d^{5} + 8 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1365 \,{\left (120 \, b^{7} c^{3} d^{4} + 36 \, a b^{6} c^{2} d^{5} + 8 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 455 \,{\left (330 \, b^{7} c^{4} d^{3} + 120 \, a b^{6} c^{3} d^{4} + 36 \, a^{2} b^{5} c^{2} d^{5} + 8 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 105 \,{\left (792 \, b^{7} c^{5} d^{2} + 330 \, a b^{6} c^{4} d^{3} + 120 \, a^{2} b^{5} c^{3} d^{4} + 36 \, a^{3} b^{4} c^{2} d^{5} + 8 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 15 \,{\left (1716 \, b^{7} c^{6} d + 792 \, a b^{6} c^{5} d^{2} + 330 \, a^{2} b^{5} c^{4} d^{3} + 120 \, a^{3} b^{4} c^{3} d^{4} + 36 \, a^{4} b^{3} c^{2} d^{5} + 8 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{51480 \,{\left (b^{23} x^{15} + 15 \, a b^{22} x^{14} + 105 \, a^{2} b^{21} x^{13} + 455 \, a^{3} b^{20} x^{12} + 1365 \, a^{4} b^{19} x^{11} + 3003 \, a^{5} b^{18} x^{10} + 5005 \, a^{6} b^{17} x^{9} + 6435 \, a^{7} b^{16} x^{8} + 6435 \, a^{8} b^{15} x^{7} + 5005 \, a^{9} b^{14} x^{6} + 3003 \, a^{10} b^{13} x^{5} + 1365 \, a^{11} b^{12} x^{4} + 455 \, a^{12} b^{11} x^{3} + 105 \, a^{13} b^{10} x^{2} + 15 \, a^{14} b^{9} x + a^{15} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^16,x, algorithm="maxima")

[Out]

-1/51480*(6435*b^7*d^7*x^7 + 3432*b^7*c^7 + 1716*a*b^6*c^6*d + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120
*a^4*b^3*c^3*d^4 + 36*a^5*b^2*c^2*d^5 + 8*a^6*b*c*d^6 + a^7*d^7 + 5005*(8*b^7*c*d^6 + a*b^6*d^7)*x^6 + 3003*(3
6*b^7*c^2*d^5 + 8*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1365*(120*b^7*c^3*d^4 + 36*a*b^6*c^2*d^5 + 8*a^2*b^5*c*d^6
+ a^3*b^4*d^7)*x^4 + 455*(330*b^7*c^4*d^3 + 120*a*b^6*c^3*d^4 + 36*a^2*b^5*c^2*d^5 + 8*a^3*b^4*c*d^6 + a^4*b^3
*d^7)*x^3 + 105*(792*b^7*c^5*d^2 + 330*a*b^6*c^4*d^3 + 120*a^2*b^5*c^3*d^4 + 36*a^3*b^4*c^2*d^5 + 8*a^4*b^3*c*
d^6 + a^5*b^2*d^7)*x^2 + 15*(1716*b^7*c^6*d + 792*a*b^6*c^5*d^2 + 330*a^2*b^5*c^4*d^3 + 120*a^3*b^4*c^3*d^4 +
36*a^4*b^3*c^2*d^5 + 8*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^23*x^15 + 15*a*b^22*x^14 + 105*a^2*b^21*x^13 + 455*a^3
*b^20*x^12 + 1365*a^4*b^19*x^11 + 3003*a^5*b^18*x^10 + 5005*a^6*b^17*x^9 + 6435*a^7*b^16*x^8 + 6435*a^8*b^15*x
^7 + 5005*a^9*b^14*x^6 + 3003*a^10*b^13*x^5 + 1365*a^11*b^12*x^4 + 455*a^12*b^11*x^3 + 105*a^13*b^10*x^2 + 15*
a^14*b^9*x + a^15*b^8)

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Fricas [B]  time = 1.84946, size = 1369, normalized size = 6.84 \begin{align*} -\frac{6435 \, b^{7} d^{7} x^{7} + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7} + 5005 \,{\left (8 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 3003 \,{\left (36 \, b^{7} c^{2} d^{5} + 8 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1365 \,{\left (120 \, b^{7} c^{3} d^{4} + 36 \, a b^{6} c^{2} d^{5} + 8 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 455 \,{\left (330 \, b^{7} c^{4} d^{3} + 120 \, a b^{6} c^{3} d^{4} + 36 \, a^{2} b^{5} c^{2} d^{5} + 8 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 105 \,{\left (792 \, b^{7} c^{5} d^{2} + 330 \, a b^{6} c^{4} d^{3} + 120 \, a^{2} b^{5} c^{3} d^{4} + 36 \, a^{3} b^{4} c^{2} d^{5} + 8 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 15 \,{\left (1716 \, b^{7} c^{6} d + 792 \, a b^{6} c^{5} d^{2} + 330 \, a^{2} b^{5} c^{4} d^{3} + 120 \, a^{3} b^{4} c^{3} d^{4} + 36 \, a^{4} b^{3} c^{2} d^{5} + 8 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{51480 \,{\left (b^{23} x^{15} + 15 \, a b^{22} x^{14} + 105 \, a^{2} b^{21} x^{13} + 455 \, a^{3} b^{20} x^{12} + 1365 \, a^{4} b^{19} x^{11} + 3003 \, a^{5} b^{18} x^{10} + 5005 \, a^{6} b^{17} x^{9} + 6435 \, a^{7} b^{16} x^{8} + 6435 \, a^{8} b^{15} x^{7} + 5005 \, a^{9} b^{14} x^{6} + 3003 \, a^{10} b^{13} x^{5} + 1365 \, a^{11} b^{12} x^{4} + 455 \, a^{12} b^{11} x^{3} + 105 \, a^{13} b^{10} x^{2} + 15 \, a^{14} b^{9} x + a^{15} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^16,x, algorithm="fricas")

[Out]

-1/51480*(6435*b^7*d^7*x^7 + 3432*b^7*c^7 + 1716*a*b^6*c^6*d + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120
*a^4*b^3*c^3*d^4 + 36*a^5*b^2*c^2*d^5 + 8*a^6*b*c*d^6 + a^7*d^7 + 5005*(8*b^7*c*d^6 + a*b^6*d^7)*x^6 + 3003*(3
6*b^7*c^2*d^5 + 8*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1365*(120*b^7*c^3*d^4 + 36*a*b^6*c^2*d^5 + 8*a^2*b^5*c*d^6
+ a^3*b^4*d^7)*x^4 + 455*(330*b^7*c^4*d^3 + 120*a*b^6*c^3*d^4 + 36*a^2*b^5*c^2*d^5 + 8*a^3*b^4*c*d^6 + a^4*b^3
*d^7)*x^3 + 105*(792*b^7*c^5*d^2 + 330*a*b^6*c^4*d^3 + 120*a^2*b^5*c^3*d^4 + 36*a^3*b^4*c^2*d^5 + 8*a^4*b^3*c*
d^6 + a^5*b^2*d^7)*x^2 + 15*(1716*b^7*c^6*d + 792*a*b^6*c^5*d^2 + 330*a^2*b^5*c^4*d^3 + 120*a^3*b^4*c^3*d^4 +
36*a^4*b^3*c^2*d^5 + 8*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^23*x^15 + 15*a*b^22*x^14 + 105*a^2*b^21*x^13 + 455*a^3
*b^20*x^12 + 1365*a^4*b^19*x^11 + 3003*a^5*b^18*x^10 + 5005*a^6*b^17*x^9 + 6435*a^7*b^16*x^8 + 6435*a^8*b^15*x
^7 + 5005*a^9*b^14*x^6 + 3003*a^10*b^13*x^5 + 1365*a^11*b^12*x^4 + 455*a^12*b^11*x^3 + 105*a^13*b^10*x^2 + 15*
a^14*b^9*x + a^15*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)**16,x)

[Out]

Timed out

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Giac [B]  time = 1.06347, size = 670, normalized size = 3.35 \begin{align*} -\frac{6435 \, b^{7} d^{7} x^{7} + 40040 \, b^{7} c d^{6} x^{6} + 5005 \, a b^{6} d^{7} x^{6} + 108108 \, b^{7} c^{2} d^{5} x^{5} + 24024 \, a b^{6} c d^{6} x^{5} + 3003 \, a^{2} b^{5} d^{7} x^{5} + 163800 \, b^{7} c^{3} d^{4} x^{4} + 49140 \, a b^{6} c^{2} d^{5} x^{4} + 10920 \, a^{2} b^{5} c d^{6} x^{4} + 1365 \, a^{3} b^{4} d^{7} x^{4} + 150150 \, b^{7} c^{4} d^{3} x^{3} + 54600 \, a b^{6} c^{3} d^{4} x^{3} + 16380 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 3640 \, a^{3} b^{4} c d^{6} x^{3} + 455 \, a^{4} b^{3} d^{7} x^{3} + 83160 \, b^{7} c^{5} d^{2} x^{2} + 34650 \, a b^{6} c^{4} d^{3} x^{2} + 12600 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 3780 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 840 \, a^{4} b^{3} c d^{6} x^{2} + 105 \, a^{5} b^{2} d^{7} x^{2} + 25740 \, b^{7} c^{6} d x + 11880 \, a b^{6} c^{5} d^{2} x + 4950 \, a^{2} b^{5} c^{4} d^{3} x + 1800 \, a^{3} b^{4} c^{3} d^{4} x + 540 \, a^{4} b^{3} c^{2} d^{5} x + 120 \, a^{5} b^{2} c d^{6} x + 15 \, a^{6} b d^{7} x + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7}}{51480 \,{\left (b x + a\right )}^{15} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^16,x, algorithm="giac")

[Out]

-1/51480*(6435*b^7*d^7*x^7 + 40040*b^7*c*d^6*x^6 + 5005*a*b^6*d^7*x^6 + 108108*b^7*c^2*d^5*x^5 + 24024*a*b^6*c
*d^6*x^5 + 3003*a^2*b^5*d^7*x^5 + 163800*b^7*c^3*d^4*x^4 + 49140*a*b^6*c^2*d^5*x^4 + 10920*a^2*b^5*c*d^6*x^4 +
1365*a^3*b^4*d^7*x^4 + 150150*b^7*c^4*d^3*x^3 + 54600*a*b^6*c^3*d^4*x^3 + 16380*a^2*b^5*c^2*d^5*x^3 + 3640*a^
3*b^4*c*d^6*x^3 + 455*a^4*b^3*d^7*x^3 + 83160*b^7*c^5*d^2*x^2 + 34650*a*b^6*c^4*d^3*x^2 + 12600*a^2*b^5*c^3*d^
4*x^2 + 3780*a^3*b^4*c^2*d^5*x^2 + 840*a^4*b^3*c*d^6*x^2 + 105*a^5*b^2*d^7*x^2 + 25740*b^7*c^6*d*x + 11880*a*b
^6*c^5*d^2*x + 4950*a^2*b^5*c^4*d^3*x + 1800*a^3*b^4*c^3*d^4*x + 540*a^4*b^3*c^2*d^5*x + 120*a^5*b^2*c*d^6*x +
15*a^6*b*d^7*x + 3432*b^7*c^7 + 1716*a*b^6*c^6*d + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120*a^4*b^3*c^
3*d^4 + 36*a^5*b^2*c^2*d^5 + 8*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^15*b^8)