3.1295 \(\int \frac{(c+d x)^7}{(a+b x)^{13}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac{d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac{d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac{(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \]

[Out]

-(c + d*x)^8/(12*(b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^8)/(33*(b*c - a*d)^2*(a + b*x)^11) - (d^2*(c + d*x)^
8)/(110*(b*c - a*d)^3*(a + b*x)^10) + (d^3*(c + d*x)^8)/(495*(b*c - a*d)^4*(a + b*x)^9) - (d^4*(c + d*x)^8)/(3
960*(b*c - a*d)^5*(a + b*x)^8)

________________________________________________________________________________________

Rubi [A]  time = 0.0466408, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac{d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac{d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac{(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^13,x]

[Out]

-(c + d*x)^8/(12*(b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^8)/(33*(b*c - a*d)^2*(a + b*x)^11) - (d^2*(c + d*x)^
8)/(110*(b*c - a*d)^3*(a + b*x)^10) + (d^3*(c + d*x)^8)/(495*(b*c - a*d)^4*(a + b*x)^9) - (d^4*(c + d*x)^8)/(3
960*(b*c - a*d)^5*(a + b*x)^8)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^{13}} \, dx &=-\frac{(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}-\frac{d \int \frac{(c+d x)^7}{(a+b x)^{12}} \, dx}{3 (b c-a d)}\\ &=-\frac{(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac{d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}+\frac{d^2 \int \frac{(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)^2}\\ &=-\frac{(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac{d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac{d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}-\frac{d^3 \int \frac{(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^3}\\ &=-\frac{(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac{d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac{d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac{d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}+\frac{d^4 \int \frac{(c+d x)^7}{(a+b x)^9} \, dx}{495 (b c-a d)^4}\\ &=-\frac{(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac{d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac{d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac{d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}-\frac{d^4 (c+d x)^8}{3960 (b c-a d)^5 (a+b x)^8}\\ \end{align*}

Mathematica [B]  time = 0.120301, size = 371, normalized size = 2.46 \[ -\frac{3 a^2 b^5 d^2 \left (770 c^3 d^2 x^2+1100 c^2 d^3 x^3+280 c^4 d x+42 c^5+825 c d^4 x^4+264 d^5 x^5\right )+5 a^3 b^4 d^3 \left (198 c^2 d^2 x^2+84 c^3 d x+14 c^4+220 c d^3 x^3+99 d^4 x^4\right )+5 a^4 b^3 d^4 \left (36 c^2 d x+7 c^3+66 c d^2 x^2+44 d^3 x^3\right )+3 a^5 b^2 d^5 \left (5 c^2+20 c d x+22 d^2 x^2\right )+a^6 b d^6 (5 c+12 d x)+a^7 d^7+a b^6 d \left (4620 c^4 d^2 x^2+7700 c^3 d^3 x^3+7425 c^2 d^4 x^4+1512 c^5 d x+210 c^6+3960 c d^5 x^5+924 d^6 x^6\right )+b^7 \left (8316 c^5 d^2 x^2+15400 c^4 d^3 x^3+17325 c^3 d^4 x^4+11880 c^2 d^5 x^5+2520 c^6 d x+330 c^7+4620 c d^6 x^6+792 d^7 x^7\right )}{3960 b^8 (a+b x)^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^13,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(5*c + 12*d*x) + 3*a^5*b^2*d^5*(5*c^2 + 20*c*d*x + 22*d^2*x^2) + 5*a^4*b^3*d^4*(7*c^3 +
36*c^2*d*x + 66*c*d^2*x^2 + 44*d^3*x^3) + 5*a^3*b^4*d^3*(14*c^4 + 84*c^3*d*x + 198*c^2*d^2*x^2 + 220*c*d^3*x^3
+ 99*d^4*x^4) + 3*a^2*b^5*d^2*(42*c^5 + 280*c^4*d*x + 770*c^3*d^2*x^2 + 1100*c^2*d^3*x^3 + 825*c*d^4*x^4 + 26
4*d^5*x^5) + a*b^6*d*(210*c^6 + 1512*c^5*d*x + 4620*c^4*d^2*x^2 + 7700*c^3*d^3*x^3 + 7425*c^2*d^4*x^4 + 3960*c
*d^5*x^5 + 924*d^6*x^6) + b^7*(330*c^7 + 2520*c^6*d*x + 8316*c^5*d^2*x^2 + 15400*c^4*d^3*x^3 + 17325*c^3*d^4*x
^4 + 11880*c^2*d^5*x^5 + 4620*c*d^6*x^6 + 792*d^7*x^7))/(3960*b^8*(a + b*x)^12)

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 464, normalized size = 3.1 \begin{align*}{\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 ) }{10\,{b}^{8} \left ( bx+a \right ) ^{10}}}+{\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 ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\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 ) }{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}-{\frac{{d}^{7}}{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}-{\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}}{12\,{b}^{8} \left ( bx+a \right ) ^{12}}}-{\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 ) }{9\,{b}^{8} \left ( bx+a \right ) ^{9}}}-3\,{\frac{{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \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)^13,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.11723, size = 784, normalized size = 5.19 \begin{align*} -\frac{792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \,{\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \,{\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \,{\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \,{\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \,{\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \,{\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \,{\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="maxima")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b
^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^
2*d^5 + 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6 + a^3*b^4*
d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 + 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 +
66*(126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d
^7)*x^2 + 12*(210*b^7*c^6*d + 126*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5
+ 5*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^11 + 66*a^2*b^18*x^10 + 220*a^3*b^17*x^9 + 495*a^4
*b^16*x^8 + 792*a^5*b^15*x^7 + 924*a^6*b^14*x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*
a^10*b^10*x^2 + 12*a^11*b^9*x + a^12*b^8)

________________________________________________________________________________________

Fricas [B]  time = 1.84802, size = 1246, normalized size = 8.25 \begin{align*} -\frac{792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \,{\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \,{\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \,{\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \,{\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \,{\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \,{\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \,{\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="fricas")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b
^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^
2*d^5 + 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6 + a^3*b^4*
d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 + 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 +
66*(126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d
^7)*x^2 + 12*(210*b^7*c^6*d + 126*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5
+ 5*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^11 + 66*a^2*b^18*x^10 + 220*a^3*b^17*x^9 + 495*a^4
*b^16*x^8 + 792*a^5*b^15*x^7 + 924*a^6*b^14*x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*
a^10*b^10*x^2 + 12*a^11*b^9*x + a^12*b^8)

________________________________________________________________________________________

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)**13,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.08083, size = 670, normalized size = 4.44 \begin{align*} -\frac{792 \, b^{7} d^{7} x^{7} + 4620 \, b^{7} c d^{6} x^{6} + 924 \, a b^{6} d^{7} x^{6} + 11880 \, b^{7} c^{2} d^{5} x^{5} + 3960 \, a b^{6} c d^{6} x^{5} + 792 \, a^{2} b^{5} d^{7} x^{5} + 17325 \, b^{7} c^{3} d^{4} x^{4} + 7425 \, a b^{6} c^{2} d^{5} x^{4} + 2475 \, a^{2} b^{5} c d^{6} x^{4} + 495 \, a^{3} b^{4} d^{7} x^{4} + 15400 \, b^{7} c^{4} d^{3} x^{3} + 7700 \, a b^{6} c^{3} d^{4} x^{3} + 3300 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1100 \, a^{3} b^{4} c d^{6} x^{3} + 220 \, a^{4} b^{3} d^{7} x^{3} + 8316 \, b^{7} c^{5} d^{2} x^{2} + 4620 \, a b^{6} c^{4} d^{3} x^{2} + 2310 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 990 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 330 \, a^{4} b^{3} c d^{6} x^{2} + 66 \, a^{5} b^{2} d^{7} x^{2} + 2520 \, b^{7} c^{6} d x + 1512 \, a b^{6} c^{5} d^{2} x + 840 \, a^{2} b^{5} c^{4} d^{3} x + 420 \, a^{3} b^{4} c^{3} d^{4} x + 180 \, a^{4} b^{3} c^{2} d^{5} x + 60 \, a^{5} b^{2} c d^{6} x + 12 \, a^{6} b d^{7} x + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7}}{3960 \,{\left (b x + a\right )}^{12} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="giac")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 4620*b^7*c*d^6*x^6 + 924*a*b^6*d^7*x^6 + 11880*b^7*c^2*d^5*x^5 + 3960*a*b^6*c*d^6*x
^5 + 792*a^2*b^5*d^7*x^5 + 17325*b^7*c^3*d^4*x^4 + 7425*a*b^6*c^2*d^5*x^4 + 2475*a^2*b^5*c*d^6*x^4 + 495*a^3*b
^4*d^7*x^4 + 15400*b^7*c^4*d^3*x^3 + 7700*a*b^6*c^3*d^4*x^3 + 3300*a^2*b^5*c^2*d^5*x^3 + 1100*a^3*b^4*c*d^6*x^
3 + 220*a^4*b^3*d^7*x^3 + 8316*b^7*c^5*d^2*x^2 + 4620*a*b^6*c^4*d^3*x^2 + 2310*a^2*b^5*c^3*d^4*x^2 + 990*a^3*b
^4*c^2*d^5*x^2 + 330*a^4*b^3*c*d^6*x^2 + 66*a^5*b^2*d^7*x^2 + 2520*b^7*c^6*d*x + 1512*a*b^6*c^5*d^2*x + 840*a^
2*b^5*c^4*d^3*x + 420*a^3*b^4*c^3*d^4*x + 180*a^4*b^3*c^2*d^5*x + 60*a^5*b^2*c*d^6*x + 12*a^6*b*d^7*x + 330*b^
7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 +
5*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^12*b^8)