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

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

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Rubi [A]  time = 0.16, antiderivative size = 194, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^8,x]

[Out]

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

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Mathematica [A]  time = 0.16, size = 308, normalized size = 1.59 \begin {gather*} \frac {d^7 \log (a+b x)}{b^8}-\frac {(b c-a d) \left (1089 a^6 d^6+3 a^5 b d^5 (223 c+2401 d x)+3 a^4 b^2 d^4 \left (153 c^2+1421 c d x+6713 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+2793 c^2 d x+11319 c d^2 x^2+30625 d^3 x^3\right )+a^2 b^4 d^2 \left (214 c^4+1813 c^3 d x+6909 c^2 d^2 x^2+15925 c d^3 x^3+26950 d^4 x^4\right )+a b^5 d \left (130 c^5+1078 c^4 d x+3969 c^3 d^2 x^2+8575 c^2 d^3 x^3+12250 c d^4 x^4+13230 d^5 x^5\right )+b^6 \left (60 c^6+490 c^5 d x+1764 c^4 d^2 x^2+3675 c^3 d^3 x^3+4900 c^2 d^4 x^4+4410 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 b^8 (a+b x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^8,x]

[Out]

-1/420*((b*c - a*d)*(1089*a^6*d^6 + 3*a^5*b*d^5*(223*c + 2401*d*x) + 3*a^4*b^2*d^4*(153*c^2 + 1421*c*d*x + 671
3*d^2*x^2) + a^3*b^3*d^3*(319*c^3 + 2793*c^2*d*x + 11319*c*d^2*x^2 + 30625*d^3*x^3) + a^2*b^4*d^2*(214*c^4 + 1
813*c^3*d*x + 6909*c^2*d^2*x^2 + 15925*c*d^3*x^3 + 26950*d^4*x^4) + a*b^5*d*(130*c^5 + 1078*c^4*d*x + 3969*c^3
*d^2*x^2 + 8575*c^2*d^3*x^3 + 12250*c*d^4*x^4 + 13230*d^5*x^5) + b^6*(60*c^6 + 490*c^5*d*x + 1764*c^4*d^2*x^2
+ 3675*c^3*d^3*x^3 + 4900*c^2*d^4*x^4 + 4410*c*d^5*x^5 + 2940*d^6*x^6)))/(b^8*(a + b*x)^7) + (d^7*Log[a + b*x]
)/b^8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^7}{(a+b x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^8,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^8, x]

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fricas [B]  time = 1.42, size = 624, normalized size = 3.22 \begin {gather*} -\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x - 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, a b^{6} d^{7} x^{6} + 21 \, a^{2} b^{5} d^{7} x^{5} + 35 \, a^{3} b^{4} d^{7} x^{4} + 35 \, a^{4} b^{3} d^{7} x^{3} + 21 \, a^{5} b^{2} d^{7} x^{2} + 7 \, a^{6} b d^{7} x + a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 + 210*a^5
*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(b^7*c^2*d^5 + 2*a*b^6
*c*d^6 - 3*a^2*b^5*d^7)*x^5 + 2450*(2*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 - 11*a^3*b^4*d^7)*x^4 +
1225*(3*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 12*a^3*b^4*c*d^6 - 25*a^4*b^3*d^7)*x^3 + 147*(12*b
^7*c^5*d^2 + 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 + 60*a^4*b^3*c*d^6 - 137*a^5*b^2*d^7)*
x^2 + 49*(10*b^7*c^6*d + 12*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 + 60*
a^5*b^2*c*d^6 - 147*a^6*b*d^7)*x - 420*(b^7*d^7*x^7 + 7*a*b^6*d^7*x^6 + 21*a^2*b^5*d^7*x^5 + 35*a^3*b^4*d^7*x^
4 + 35*a^4*b^3*d^7*x^3 + 21*a^5*b^2*d^7*x^2 + 7*a^6*b*d^7*x + a^7*d^7)*log(b*x + a))/(b^15*x^7 + 7*a*b^14*x^6
+ 21*a^2*b^13*x^5 + 35*a^3*b^12*x^4 + 35*a^4*b^11*x^3 + 21*a^5*b^10*x^2 + 7*a^6*b^9*x + a^7*b^8)

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giac [B]  time = 1.28, size = 466, normalized size = 2.40 \begin {gather*} \frac {d^{7} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {2940 \, {\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 4410 \, {\left (b^{6} c^{2} d^{5} + 2 \, a b^{5} c d^{6} - 3 \, a^{2} b^{4} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{6} c^{3} d^{4} + 3 \, a b^{5} c^{2} d^{5} + 6 \, a^{2} b^{4} c d^{6} - 11 \, a^{3} b^{3} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{6} c^{4} d^{3} + 4 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} + 12 \, a^{3} b^{3} c d^{6} - 25 \, a^{4} b^{2} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{6} c^{5} d^{2} + 15 \, a b^{5} c^{4} d^{3} + 20 \, a^{2} b^{4} c^{3} d^{4} + 30 \, a^{3} b^{3} c^{2} d^{5} + 60 \, a^{4} b^{2} c d^{6} - 137 \, a^{5} b d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{6} c^{6} d + 12 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} + 20 \, a^{3} b^{3} c^{3} d^{4} + 30 \, a^{4} b^{2} c^{2} d^{5} + 60 \, a^{5} b c d^{6} - 147 \, a^{6} d^{7}\right )} x + \frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7}}{b}}{420 \, {\left (b x + a\right )}^{7} b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 672, normalized size = 3.46 \begin {gather*} \frac {a^{7} d^{7}}{7 \left (b x +a \right )^{7} b^{8}}-\frac {a^{6} c \,d^{6}}{\left (b x +a \right )^{7} b^{7}}+\frac {3 a^{5} c^{2} d^{5}}{\left (b x +a \right )^{7} b^{6}}-\frac {5 a^{4} c^{3} d^{4}}{\left (b x +a \right )^{7} b^{5}}+\frac {5 a^{3} c^{4} d^{3}}{\left (b x +a \right )^{7} b^{4}}-\frac {3 a^{2} c^{5} d^{2}}{\left (b x +a \right )^{7} b^{3}}+\frac {a \,c^{6} d}{\left (b x +a \right )^{7} b^{2}}-\frac {c^{7}}{7 \left (b x +a \right )^{7} b}-\frac {7 a^{6} d^{7}}{6 \left (b x +a \right )^{6} b^{8}}+\frac {7 a^{5} c \,d^{6}}{\left (b x +a \right )^{6} b^{7}}-\frac {35 a^{4} c^{2} d^{5}}{2 \left (b x +a \right )^{6} b^{6}}+\frac {70 a^{3} c^{3} d^{4}}{3 \left (b x +a \right )^{6} b^{5}}-\frac {35 a^{2} c^{4} d^{3}}{2 \left (b x +a \right )^{6} b^{4}}+\frac {7 a \,c^{5} d^{2}}{\left (b x +a \right )^{6} b^{3}}-\frac {7 c^{6} d}{6 \left (b x +a \right )^{6} b^{2}}+\frac {21 a^{5} d^{7}}{5 \left (b x +a \right )^{5} b^{8}}-\frac {21 a^{4} c \,d^{6}}{\left (b x +a \right )^{5} b^{7}}+\frac {42 a^{3} c^{2} d^{5}}{\left (b x +a \right )^{5} b^{6}}-\frac {42 a^{2} c^{3} d^{4}}{\left (b x +a \right )^{5} b^{5}}+\frac {21 a \,c^{4} d^{3}}{\left (b x +a \right )^{5} b^{4}}-\frac {21 c^{5} d^{2}}{5 \left (b x +a \right )^{5} b^{3}}-\frac {35 a^{4} d^{7}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {35 a^{3} c \,d^{6}}{\left (b x +a \right )^{4} b^{7}}-\frac {105 a^{2} c^{2} d^{5}}{2 \left (b x +a \right )^{4} b^{6}}+\frac {35 a \,c^{3} d^{4}}{\left (b x +a \right )^{4} b^{5}}-\frac {35 c^{4} d^{3}}{4 \left (b x +a \right )^{4} b^{4}}+\frac {35 a^{3} d^{7}}{3 \left (b x +a \right )^{3} b^{8}}-\frac {35 a^{2} c \,d^{6}}{\left (b x +a \right )^{3} b^{7}}+\frac {35 a \,c^{2} d^{5}}{\left (b x +a \right )^{3} b^{6}}-\frac {35 c^{3} d^{4}}{3 \left (b x +a \right )^{3} b^{5}}-\frac {21 a^{2} d^{7}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {21 a c \,d^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {21 c^{2} d^{5}}{2 \left (b x +a \right )^{2} b^{6}}+\frac {7 a \,d^{7}}{\left (b x +a \right ) b^{8}}-\frac {7 c \,d^{6}}{\left (b x +a \right ) b^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.65, size = 534, normalized size = 2.75 \begin {gather*} -\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} + \frac {d^{7} \log \left (b x + a\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 + 210*a^5
*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(b^7*c^2*d^5 + 2*a*b^6
*c*d^6 - 3*a^2*b^5*d^7)*x^5 + 2450*(2*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 - 11*a^3*b^4*d^7)*x^4 +
1225*(3*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 12*a^3*b^4*c*d^6 - 25*a^4*b^3*d^7)*x^3 + 147*(12*b
^7*c^5*d^2 + 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 + 60*a^4*b^3*c*d^6 - 137*a^5*b^2*d^7)*
x^2 + 49*(10*b^7*c^6*d + 12*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 + 60*
a^5*b^2*c*d^6 - 147*a^6*b*d^7)*x)/(b^15*x^7 + 7*a*b^14*x^6 + 21*a^2*b^13*x^5 + 35*a^3*b^12*x^4 + 35*a^4*b^11*x
^3 + 21*a^5*b^10*x^2 + 7*a^6*b^9*x + a^7*b^8) + d^7*log(b*x + a)/b^8

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mupad [B]  time = 0.35, size = 461, normalized size = 2.38 \begin {gather*} \frac {d^7\,\ln \left (a+b\,x\right )}{b^8}-\frac {x\,\left (-\frac {343\,a^6\,b\,d^7}{20}+7\,a^5\,b^2\,c\,d^6+\frac {7\,a^4\,b^3\,c^2\,d^5}{2}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{4}+\frac {7\,a\,b^6\,c^5\,d^2}{5}+\frac {7\,b^7\,c^6\,d}{6}\right )-x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (-\frac {875\,a^4\,b^3\,d^7}{12}+35\,a^3\,b^4\,c\,d^6+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+\frac {35\,a\,b^6\,c^3\,d^4}{3}+\frac {35\,b^7\,c^4\,d^3}{4}\right )+x^5\,\left (-\frac {63\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6+\frac {21\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (-\frac {959\,a^5\,b^2\,d^7}{20}+21\,a^4\,b^3\,c\,d^6+\frac {21\,a^3\,b^4\,c^2\,d^5}{2}+7\,a^2\,b^5\,c^3\,d^4+\frac {21\,a\,b^6\,c^4\,d^3}{4}+\frac {21\,b^7\,c^5\,d^2}{5}\right )-\frac {363\,a^7\,d^7}{140}+\frac {b^7\,c^7}{7}+x^4\,\left (-\frac {385\,a^3\,b^4\,d^7}{6}+35\,a^2\,b^5\,c\,d^6+\frac {35\,a\,b^6\,c^2\,d^5}{2}+\frac {35\,b^7\,c^3\,d^4}{3}\right )+\frac {a^2\,b^5\,c^5\,d^2}{5}+\frac {a^3\,b^4\,c^4\,d^3}{4}+\frac {a^4\,b^3\,c^3\,d^4}{3}+\frac {a^5\,b^2\,c^2\,d^5}{2}+\frac {a\,b^6\,c^6\,d}{6}+a^6\,b\,c\,d^6}{b^8\,{\left (a+b\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^8,x)

[Out]

(d^7*log(a + b*x))/b^8 - (x*((7*b^7*c^6*d)/6 - (343*a^6*b*d^7)/20 + (7*a*b^6*c^5*d^2)/5 + 7*a^5*b^2*c*d^6 + (7
*a^2*b^5*c^4*d^3)/4 + (7*a^3*b^4*c^3*d^4)/3 + (7*a^4*b^3*c^2*d^5)/2) - x^6*(7*a*b^6*d^7 - 7*b^7*c*d^6) + x^3*(
(35*b^7*c^4*d^3)/4 - (875*a^4*b^3*d^7)/12 + (35*a*b^6*c^3*d^4)/3 + 35*a^3*b^4*c*d^6 + (35*a^2*b^5*c^2*d^5)/2)
+ x^5*((21*b^7*c^2*d^5)/2 - (63*a^2*b^5*d^7)/2 + 21*a*b^6*c*d^6) + x^2*((21*b^7*c^5*d^2)/5 - (959*a^5*b^2*d^7)
/20 + (21*a*b^6*c^4*d^3)/4 + 21*a^4*b^3*c*d^6 + 7*a^2*b^5*c^3*d^4 + (21*a^3*b^4*c^2*d^5)/2) - (363*a^7*d^7)/14
0 + (b^7*c^7)/7 + x^4*((35*b^7*c^3*d^4)/3 - (385*a^3*b^4*d^7)/6 + (35*a*b^6*c^2*d^5)/2 + 35*a^2*b^5*c*d^6) + (
a^2*b^5*c^5*d^2)/5 + (a^3*b^4*c^4*d^3)/4 + (a^4*b^3*c^3*d^4)/3 + (a^5*b^2*c^2*d^5)/2 + (a*b^6*c^6*d)/6 + a^6*b
*c*d^6)/(b^8*(a + b*x)^7)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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