3.1297 \(\int \frac{(c+d x)^7}{(a+b x)^{15}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.142819, 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} \[ -\frac{7 d^6 (b c-a d)}{8 b^8 (a+b x)^8}-\frac{7 d^5 (b c-a d)^2}{3 b^8 (a+b x)^9}-\frac{7 d^4 (b c-a d)^3}{2 b^8 (a+b x)^{10}}-\frac{35 d^3 (b c-a d)^4}{11 b^8 (a+b x)^{11}}-\frac{7 d^2 (b c-a d)^5}{4 b^8 (a+b x)^{12}}-\frac{7 d (b c-a d)^6}{13 b^8 (a+b x)^{13}}-\frac{(b c-a d)^7}{14 b^8 (a+b x)^{14}}-\frac{d^7}{7 b^8 (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^15,x]

[Out]

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

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

Mathematica [A]  time = 0.120119, size = 371, normalized size = 1.86 \[ -\frac{7 a^2 b^5 d^2 \left (1092 c^3 d^2 x^2+1456 c^2 d^3 x^3+420 c^4 d x+66 c^5+1001 c d^4 x^4+286 d^5 x^5\right )+7 a^3 b^4 d^3 \left (364 c^2 d^2 x^2+168 c^3 d x+30 c^4+364 c d^3 x^3+143 d^4 x^4\right )+7 a^4 b^3 d^4 \left (56 c^2 d x+12 c^3+91 c d^2 x^2+52 d^3 x^3\right )+7 a^5 b^2 d^5 \left (4 c^2+14 c d x+13 d^2 x^2\right )+7 a^6 b d^6 (c+2 d x)+a^7 d^7+7 a b^6 d \left (2730 c^4 d^2 x^2+4368 c^3 d^3 x^3+4004 c^2 d^4 x^4+924 c^5 d x+132 c^6+2002 c d^5 x^5+429 d^6 x^6\right )+b^7 \left (42042 c^5 d^2 x^2+76440 c^4 d^3 x^3+84084 c^3 d^4 x^4+56056 c^2 d^5 x^5+12936 c^6 d x+1716 c^7+21021 c d^6 x^6+3432 d^7 x^7\right )}{24024 b^8 (a+b x)^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^15,x]

[Out]

-(a^7*d^7 + 7*a^6*b*d^6*(c + 2*d*x) + 7*a^5*b^2*d^5*(4*c^2 + 14*c*d*x + 13*d^2*x^2) + 7*a^4*b^3*d^4*(12*c^3 +
56*c^2*d*x + 91*c*d^2*x^2 + 52*d^3*x^3) + 7*a^3*b^4*d^3*(30*c^4 + 168*c^3*d*x + 364*c^2*d^2*x^2 + 364*c*d^3*x^
3 + 143*d^4*x^4) + 7*a^2*b^5*d^2*(66*c^5 + 420*c^4*d*x + 1092*c^3*d^2*x^2 + 1456*c^2*d^3*x^3 + 1001*c*d^4*x^4
+ 286*d^5*x^5) + 7*a*b^6*d*(132*c^6 + 924*c^5*d*x + 2730*c^4*d^2*x^2 + 4368*c^3*d^3*x^3 + 4004*c^2*d^4*x^4 + 2
002*c*d^5*x^5 + 429*d^6*x^6) + b^7*(1716*c^7 + 12936*c^6*d*x + 42042*c^5*d^2*x^2 + 76440*c^4*d^3*x^3 + 84084*c
^3*d^4*x^4 + 56056*c^2*d^5*x^5 + 21021*c*d^6*x^6 + 3432*d^7*x^7))/(24024*b^8*(a + b*x)^14)

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Maple [B]  time = 0.008, size = 464, normalized size = 2.3 \begin{align*}{\frac{7\,{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 ) }{2\,{b}^{8} \left ( bx+a \right ) ^{10}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\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}}{14\,{b}^{8} \left ( bx+a \right ) ^{14}}}-{\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 ) }{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}-{\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 ) }{13\,{b}^{8} \left ( bx+a \right ) ^{13}}}+{\frac{7\,{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 ) }{4\,{b}^{8} \left ( bx+a \right ) ^{12}}}-{\frac{7\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{3\,{b}^{8} \left ( bx+a \right ) ^{9}}}-{\frac{{d}^{7}}{7\,{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)^15,x)

[Out]

7/2*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)^10+7/8*d^6*(a*d-b*c)/b^8/(b*x+a)^8-1/14*(-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)^14-35/11*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)^
11-7/13*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)^13+7/4*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)^12-7/3*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^8/(b*x+a)^9-1/7*d^7/b^8/(b*x+a)^7

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Maxima [B]  time = 1.13914, size = 814, normalized size = 4.07 \begin{align*} -\frac{3432 \, b^{7} d^{7} x^{7} + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7} + 3003 \,{\left (7 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 2002 \,{\left (28 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1001 \,{\left (84 \, b^{7} c^{3} d^{4} + 28 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 364 \,{\left (210 \, b^{7} c^{4} d^{3} + 84 \, a b^{6} c^{3} d^{4} + 28 \, a^{2} b^{5} c^{2} d^{5} + 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 91 \,{\left (462 \, b^{7} c^{5} d^{2} + 210 \, a b^{6} c^{4} d^{3} + 84 \, a^{2} b^{5} c^{3} d^{4} + 28 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 14 \,{\left (924 \, b^{7} c^{6} d + 462 \, a b^{6} c^{5} d^{2} + 210 \, a^{2} b^{5} c^{4} d^{3} + 84 \, a^{3} b^{4} c^{3} d^{4} + 28 \, a^{4} b^{3} c^{2} d^{5} + 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{24024 \,{\left (b^{22} x^{14} + 14 \, a b^{21} x^{13} + 91 \, a^{2} b^{20} x^{12} + 364 \, a^{3} b^{19} x^{11} + 1001 \, a^{4} b^{18} x^{10} + 2002 \, a^{5} b^{17} x^{9} + 3003 \, a^{6} b^{16} x^{8} + 3432 \, a^{7} b^{15} x^{7} + 3003 \, a^{8} b^{14} x^{6} + 2002 \, a^{9} b^{13} x^{5} + 1001 \, a^{10} b^{12} x^{4} + 364 \, a^{11} b^{11} x^{3} + 91 \, a^{12} b^{10} x^{2} + 14 \, a^{13} b^{9} x + a^{14} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="maxima")

[Out]

-1/24024*(3432*b^7*d^7*x^7 + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a
^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7 + 3003*(7*b^7*c*d^6 + a*b^6*d^7)*x^6 + 2002*(28*
b^7*c^2*d^5 + 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1001*(84*b^7*c^3*d^4 + 28*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + a
^3*b^4*d^7)*x^4 + 364*(210*b^7*c^4*d^3 + 84*a*b^6*c^3*d^4 + 28*a^2*b^5*c^2*d^5 + 7*a^3*b^4*c*d^6 + a^4*b^3*d^7
)*x^3 + 91*(462*b^7*c^5*d^2 + 210*a*b^6*c^4*d^3 + 84*a^2*b^5*c^3*d^4 + 28*a^3*b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 +
a^5*b^2*d^7)*x^2 + 14*(924*b^7*c^6*d + 462*a*b^6*c^5*d^2 + 210*a^2*b^5*c^4*d^3 + 84*a^3*b^4*c^3*d^4 + 28*a^4*b
^3*c^2*d^5 + 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^22*x^14 + 14*a*b^21*x^13 + 91*a^2*b^20*x^12 + 364*a^3*b^19*x^1
1 + 1001*a^4*b^18*x^10 + 2002*a^5*b^17*x^9 + 3003*a^6*b^16*x^8 + 3432*a^7*b^15*x^7 + 3003*a^8*b^14*x^6 + 2002*
a^9*b^13*x^5 + 1001*a^10*b^12*x^4 + 364*a^11*b^11*x^3 + 91*a^12*b^10*x^2 + 14*a^13*b^9*x + a^14*b^8)

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Fricas [B]  time = 1.85291, size = 1326, normalized size = 6.63 \begin{align*} -\frac{3432 \, b^{7} d^{7} x^{7} + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7} + 3003 \,{\left (7 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 2002 \,{\left (28 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1001 \,{\left (84 \, b^{7} c^{3} d^{4} + 28 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 364 \,{\left (210 \, b^{7} c^{4} d^{3} + 84 \, a b^{6} c^{3} d^{4} + 28 \, a^{2} b^{5} c^{2} d^{5} + 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 91 \,{\left (462 \, b^{7} c^{5} d^{2} + 210 \, a b^{6} c^{4} d^{3} + 84 \, a^{2} b^{5} c^{3} d^{4} + 28 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 14 \,{\left (924 \, b^{7} c^{6} d + 462 \, a b^{6} c^{5} d^{2} + 210 \, a^{2} b^{5} c^{4} d^{3} + 84 \, a^{3} b^{4} c^{3} d^{4} + 28 \, a^{4} b^{3} c^{2} d^{5} + 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{24024 \,{\left (b^{22} x^{14} + 14 \, a b^{21} x^{13} + 91 \, a^{2} b^{20} x^{12} + 364 \, a^{3} b^{19} x^{11} + 1001 \, a^{4} b^{18} x^{10} + 2002 \, a^{5} b^{17} x^{9} + 3003 \, a^{6} b^{16} x^{8} + 3432 \, a^{7} b^{15} x^{7} + 3003 \, a^{8} b^{14} x^{6} + 2002 \, a^{9} b^{13} x^{5} + 1001 \, a^{10} b^{12} x^{4} + 364 \, a^{11} b^{11} x^{3} + 91 \, a^{12} b^{10} x^{2} + 14 \, a^{13} b^{9} x + a^{14} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="fricas")

[Out]

-1/24024*(3432*b^7*d^7*x^7 + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a
^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7 + 3003*(7*b^7*c*d^6 + a*b^6*d^7)*x^6 + 2002*(28*
b^7*c^2*d^5 + 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1001*(84*b^7*c^3*d^4 + 28*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + a
^3*b^4*d^7)*x^4 + 364*(210*b^7*c^4*d^3 + 84*a*b^6*c^3*d^4 + 28*a^2*b^5*c^2*d^5 + 7*a^3*b^4*c*d^6 + a^4*b^3*d^7
)*x^3 + 91*(462*b^7*c^5*d^2 + 210*a*b^6*c^4*d^3 + 84*a^2*b^5*c^3*d^4 + 28*a^3*b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 +
a^5*b^2*d^7)*x^2 + 14*(924*b^7*c^6*d + 462*a*b^6*c^5*d^2 + 210*a^2*b^5*c^4*d^3 + 84*a^3*b^4*c^3*d^4 + 28*a^4*b
^3*c^2*d^5 + 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^22*x^14 + 14*a*b^21*x^13 + 91*a^2*b^20*x^12 + 364*a^3*b^19*x^1
1 + 1001*a^4*b^18*x^10 + 2002*a^5*b^17*x^9 + 3003*a^6*b^16*x^8 + 3432*a^7*b^15*x^7 + 3003*a^8*b^14*x^6 + 2002*
a^9*b^13*x^5 + 1001*a^10*b^12*x^4 + 364*a^11*b^11*x^3 + 91*a^12*b^10*x^2 + 14*a^13*b^9*x + a^14*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)**15,x)

[Out]

Timed out

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Giac [B]  time = 1.06709, size = 670, normalized size = 3.35 \begin{align*} -\frac{3432 \, b^{7} d^{7} x^{7} + 21021 \, b^{7} c d^{6} x^{6} + 3003 \, a b^{6} d^{7} x^{6} + 56056 \, b^{7} c^{2} d^{5} x^{5} + 14014 \, a b^{6} c d^{6} x^{5} + 2002 \, a^{2} b^{5} d^{7} x^{5} + 84084 \, b^{7} c^{3} d^{4} x^{4} + 28028 \, a b^{6} c^{2} d^{5} x^{4} + 7007 \, a^{2} b^{5} c d^{6} x^{4} + 1001 \, a^{3} b^{4} d^{7} x^{4} + 76440 \, b^{7} c^{4} d^{3} x^{3} + 30576 \, a b^{6} c^{3} d^{4} x^{3} + 10192 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 2548 \, a^{3} b^{4} c d^{6} x^{3} + 364 \, a^{4} b^{3} d^{7} x^{3} + 42042 \, b^{7} c^{5} d^{2} x^{2} + 19110 \, a b^{6} c^{4} d^{3} x^{2} + 7644 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 2548 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 637 \, a^{4} b^{3} c d^{6} x^{2} + 91 \, a^{5} b^{2} d^{7} x^{2} + 12936 \, b^{7} c^{6} d x + 6468 \, a b^{6} c^{5} d^{2} x + 2940 \, a^{2} b^{5} c^{4} d^{3} x + 1176 \, a^{3} b^{4} c^{3} d^{4} x + 392 \, a^{4} b^{3} c^{2} d^{5} x + 98 \, a^{5} b^{2} c d^{6} x + 14 \, a^{6} b d^{7} x + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7}}{24024 \,{\left (b x + a\right )}^{14} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="giac")

[Out]

-1/24024*(3432*b^7*d^7*x^7 + 21021*b^7*c*d^6*x^6 + 3003*a*b^6*d^7*x^6 + 56056*b^7*c^2*d^5*x^5 + 14014*a*b^6*c*
d^6*x^5 + 2002*a^2*b^5*d^7*x^5 + 84084*b^7*c^3*d^4*x^4 + 28028*a*b^6*c^2*d^5*x^4 + 7007*a^2*b^5*c*d^6*x^4 + 10
01*a^3*b^4*d^7*x^4 + 76440*b^7*c^4*d^3*x^3 + 30576*a*b^6*c^3*d^4*x^3 + 10192*a^2*b^5*c^2*d^5*x^3 + 2548*a^3*b^
4*c*d^6*x^3 + 364*a^4*b^3*d^7*x^3 + 42042*b^7*c^5*d^2*x^2 + 19110*a*b^6*c^4*d^3*x^2 + 7644*a^2*b^5*c^3*d^4*x^2
+ 2548*a^3*b^4*c^2*d^5*x^2 + 637*a^4*b^3*c*d^6*x^2 + 91*a^5*b^2*d^7*x^2 + 12936*b^7*c^6*d*x + 6468*a*b^6*c^5*
d^2*x + 2940*a^2*b^5*c^4*d^3*x + 1176*a^3*b^4*c^3*d^4*x + 392*a^4*b^3*c^2*d^5*x + 98*a^5*b^2*c*d^6*x + 14*a^6*
b*d^7*x + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a^4*b^3*c^3*d^4 + 28
*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^14*b^8)