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

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

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Rubi [A]  time = 0.21, 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 b^6 (b c-a d)}{d^8 (c+d x)}-\frac {21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac {35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac {35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac {21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac {7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac {(b c-a d)^7}{7 d^8 (c+d x)^7}+\frac {b^7 \log (c+d x)}{d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/(c + d*x)^8,x]

[Out]

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

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Mathematica [A]  time = 0.16, size = 308, normalized size = 1.59 \begin {gather*} \frac {(b c-a d) \left (60 a^6 d^6+10 a^5 b d^5 (13 c+49 d x)+2 a^4 b^2 d^4 \left (107 c^2+539 c d x+882 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+1813 c^2 d x+3969 c d^2 x^2+3675 d^3 x^3\right )+a^2 b^4 d^2 \left (459 c^4+2793 c^3 d x+6909 c^2 d^2 x^2+8575 c d^3 x^3+4900 d^4 x^4\right )+a b^5 d \left (669 c^5+4263 c^4 d x+11319 c^3 d^2 x^2+15925 c^2 d^3 x^3+12250 c d^4 x^4+4410 d^5 x^5\right )+b^6 \left (1089 c^6+7203 c^5 d x+20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+13230 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 d^8 (c+d x)^7}+\frac {b^7 \log (c+d x)}{d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/(c + d*x)^8,x]

[Out]

((b*c - a*d)*(60*a^6*d^6 + 10*a^5*b*d^5*(13*c + 49*d*x) + 2*a^4*b^2*d^4*(107*c^2 + 539*c*d*x + 882*d^2*x^2) +
a^3*b^3*d^3*(319*c^3 + 1813*c^2*d*x + 3969*c*d^2*x^2 + 3675*d^3*x^3) + a^2*b^4*d^2*(459*c^4 + 2793*c^3*d*x + 6
909*c^2*d^2*x^2 + 8575*c*d^3*x^3 + 4900*d^4*x^4) + a*b^5*d*(669*c^5 + 4263*c^4*d*x + 11319*c^3*d^2*x^2 + 15925
*c^2*d^3*x^3 + 12250*c*d^4*x^4 + 4410*d^5*x^5) + b^6*(1089*c^6 + 7203*c^5*d*x + 20139*c^4*d^2*x^2 + 30625*c^3*
d^3*x^3 + 26950*c^2*d^4*x^4 + 13230*c*d^5*x^5 + 2940*d^6*x^6)))/(420*d^8*(c + d*x)^7) + (b^7*Log[c + d*x])/d^8

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.12, size = 625, normalized size = 3.22 \begin {gather*} \frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x + 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, b^{7} c d^{6} x^{6} + 21 \, b^{7} c^{2} d^{5} x^{5} + 35 \, b^{7} c^{3} d^{4} x^{4} + 35 \, b^{7} c^{4} d^{3} x^{3} + 21 \, b^{7} c^{5} d^{2} x^{2} + 7 \, b^{7} c^{6} d x + b^{7} c^{7}\right )} \log \left (d x + c\right )}{420 \, {\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4*c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a
^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(3*b^7*c^2*d^5 - 2*a*b^
6*c*d^6 - a^2*b^5*d^7)*x^5 + 2450*(11*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 - 3*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 1
225*(25*b^7*c^4*d^3 - 12*a*b^6*c^3*d^4 - 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 147*(137*b
^7*c^5*d^2 - 60*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 - 20*a^3*b^4*c^2*d^5 - 15*a^4*b^3*c*d^6 - 12*a^5*b^2*d^7)*x
^2 + 49*(147*b^7*c^6*d - 60*a*b^6*c^5*d^2 - 30*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*d^4 - 15*a^4*b^3*c^2*d^5 - 12*
a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x + 420*(b^7*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^2*d^5*x^5 + 35*b^7*c^3*d^4*x^4
+ 35*b^7*c^4*d^3*x^3 + 21*b^7*c^5*d^2*x^2 + 7*b^7*c^6*d*x + b^7*c^7)*log(d*x + c))/(d^15*x^7 + 7*c*d^14*x^6 +
21*c^2*d^13*x^5 + 35*c^3*d^12*x^4 + 35*c^4*d^11*x^3 + 21*c^5*d^10*x^2 + 7*c^6*d^9*x + c^7*d^8)

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giac [B]  time = 1.30, size = 467, normalized size = 2.41 \begin {gather*} \frac {b^{7} \log \left ({\left | d x + c \right |}\right )}{d^{8}} + \frac {2940 \, {\left (b^{7} c d^{5} - a b^{6} d^{6}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{4} - 2 \, a b^{6} c d^{5} - a^{2} b^{5} d^{6}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{3} - 6 \, a b^{6} c^{2} d^{4} - 3 \, a^{2} b^{5} c d^{5} - 2 \, a^{3} b^{4} d^{6}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{2} - 12 \, a b^{6} c^{3} d^{3} - 6 \, a^{2} b^{5} c^{2} d^{4} - 4 \, a^{3} b^{4} c d^{5} - 3 \, a^{4} b^{3} d^{6}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d - 60 \, a b^{6} c^{4} d^{2} - 30 \, a^{2} b^{5} c^{3} d^{3} - 20 \, a^{3} b^{4} c^{2} d^{4} - 15 \, a^{4} b^{3} c d^{5} - 12 \, a^{5} b^{2} d^{6}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} - 60 \, a b^{6} c^{5} d - 30 \, a^{2} b^{5} c^{4} d^{2} - 20 \, a^{3} b^{4} c^{3} d^{3} - 15 \, a^{4} b^{3} c^{2} d^{4} - 12 \, a^{5} b^{2} c d^{5} - 10 \, a^{6} b d^{6}\right )} x + \frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7}}{d}}{420 \, {\left (d x + c\right )}^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.66, size = 535, normalized size = 2.76 \begin {gather*} \frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x}{420 \, {\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} + \frac {b^{7} \log \left (d x + c\right )}{d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4*c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a
^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(3*b^7*c^2*d^5 - 2*a*b^
6*c*d^6 - a^2*b^5*d^7)*x^5 + 2450*(11*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 - 3*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 1
225*(25*b^7*c^4*d^3 - 12*a*b^6*c^3*d^4 - 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 147*(137*b
^7*c^5*d^2 - 60*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 - 20*a^3*b^4*c^2*d^5 - 15*a^4*b^3*c*d^6 - 12*a^5*b^2*d^7)*x
^2 + 49*(147*b^7*c^6*d - 60*a*b^6*c^5*d^2 - 30*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*d^4 - 15*a^4*b^3*c^2*d^5 - 12*
a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x)/(d^15*x^7 + 7*c*d^14*x^6 + 21*c^2*d^13*x^5 + 35*c^3*d^12*x^4 + 35*c^4*d^11*x^
3 + 21*c^5*d^10*x^2 + 7*c^6*d^9*x + c^7*d^8) + b^7*log(d*x + c)/d^8

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mupad [B]  time = 0.38, size = 460, normalized size = 2.37 \begin {gather*} \frac {b^7\,\ln \left (c+d\,x\right )}{d^8}-\frac {x\,\left (\frac {7\,a^6\,b\,d^7}{6}+\frac {7\,a^5\,b^2\,c\,d^6}{5}+\frac {7\,a^4\,b^3\,c^2\,d^5}{4}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{2}+7\,a\,b^6\,c^5\,d^2-\frac {343\,b^7\,c^6\,d}{20}\right )+x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (\frac {35\,a^4\,b^3\,d^7}{4}+\frac {35\,a^3\,b^4\,c\,d^6}{3}+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+35\,a\,b^6\,c^3\,d^4-\frac {875\,b^7\,c^4\,d^3}{12}\right )+x^5\,\left (\frac {21\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6-\frac {63\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (\frac {21\,a^5\,b^2\,d^7}{5}+\frac {21\,a^4\,b^3\,c\,d^6}{4}+7\,a^3\,b^4\,c^2\,d^5+\frac {21\,a^2\,b^5\,c^3\,d^4}{2}+21\,a\,b^6\,c^4\,d^3-\frac {959\,b^7\,c^5\,d^2}{20}\right )+\frac {a^7\,d^7}{7}-\frac {363\,b^7\,c^7}{140}+x^4\,\left (\frac {35\,a^3\,b^4\,d^7}{3}+\frac {35\,a^2\,b^5\,c\,d^6}{2}+35\,a\,b^6\,c^2\,d^5-\frac {385\,b^7\,c^3\,d^4}{6}\right )+\frac {a^2\,b^5\,c^5\,d^2}{2}+\frac {a^3\,b^4\,c^4\,d^3}{3}+\frac {a^4\,b^3\,c^3\,d^4}{4}+\frac {a^5\,b^2\,c^2\,d^5}{5}+a\,b^6\,c^6\,d+\frac {a^6\,b\,c\,d^6}{6}}{d^8\,{\left (c+d\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/(c + d*x)^8,x)

[Out]

(b^7*log(c + d*x))/d^8 - (x*((7*a^6*b*d^7)/6 - (343*b^7*c^6*d)/20 + 7*a*b^6*c^5*d^2 + (7*a^5*b^2*c*d^6)/5 + (7
*a^2*b^5*c^4*d^3)/2 + (7*a^3*b^4*c^3*d^4)/3 + (7*a^4*b^3*c^2*d^5)/4) + x^6*(7*a*b^6*d^7 - 7*b^7*c*d^6) + x^3*(
(35*a^4*b^3*d^7)/4 - (875*b^7*c^4*d^3)/12 + 35*a*b^6*c^3*d^4 + (35*a^3*b^4*c*d^6)/3 + (35*a^2*b^5*c^2*d^5)/2)
+ x^5*((21*a^2*b^5*d^7)/2 - (63*b^7*c^2*d^5)/2 + 21*a*b^6*c*d^6) + x^2*((21*a^5*b^2*d^7)/5 - (959*b^7*c^5*d^2)
/20 + 21*a*b^6*c^4*d^3 + (21*a^4*b^3*c*d^6)/4 + (21*a^2*b^5*c^3*d^4)/2 + 7*a^3*b^4*c^2*d^5) + (a^7*d^7)/7 - (3
63*b^7*c^7)/140 + x^4*((35*a^3*b^4*d^7)/3 - (385*b^7*c^3*d^4)/6 + 35*a*b^6*c^2*d^5 + (35*a^2*b^5*c*d^6)/2) + (
a^2*b^5*c^5*d^2)/2 + (a^3*b^4*c^4*d^3)/3 + (a^4*b^3*c^3*d^4)/4 + (a^5*b^2*c^2*d^5)/5 + a*b^6*c^6*d + (a^6*b*c*
d^6)/6)/(d^8*(c + d*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((b*x+a)**7/(d*x+c)**8,x)

[Out]

Timed out

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