3.13.13 \(\int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx\)

Optimal. Leaf size=258 \[ \frac {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}} \]

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Rubi [A]  time = 0.36, antiderivative size = 258, 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 {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^8,x]

[Out]

(45*d^8*(b*c - a*d)^2*x)/b^10 - (b*c - a*d)^10/(7*b^11*(a + b*x)^7) - (5*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^6)
- (9*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^5) - (30*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^4) - (70*d^4*(b*c - a*d)^
6)/(b^11*(a + b*x)^3) - (126*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^2) - (210*d^6*(b*c - a*d)^4)/(b^11*(a + b*x))
+ (5*d^9*(b*c - a*d)*(a + b*x)^2)/b^11 + (d^10*(a + b*x)^3)/(3*b^11) + (120*d^7*(b*c - a*d)^3*Log[a + b*x])/b^
11

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)^{10}}{(a+b x)^8} \, dx &=\int \left (\frac {45 d^8 (b c-a d)^2}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^8}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^7}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^6}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^5}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^4}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^3}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^2}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)}+\frac {10 d^9 (b c-a d) (a+b x)}{b^{10}}+\frac {d^{10} (a+b x)^2}{b^{10}}\right ) \, dx\\ &=\frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 239, normalized size = 0.93 \begin {gather*} \frac {21 b d^8 x \left (36 a^2 d^2-80 a b c d+45 b^2 c^2\right )+21 b^2 d^9 x^2 (5 b c-4 a d)+2520 d^7 (b c-a d)^3 \log (a+b x)-\frac {4410 d^6 (b c-a d)^4}{a+b x}+\frac {2646 d^5 (a d-b c)^5}{(a+b x)^2}-\frac {1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac {630 d^3 (a d-b c)^7}{(a+b x)^4}-\frac {189 d^2 (b c-a d)^8}{(a+b x)^5}+\frac {35 d (a d-b c)^9}{(a+b x)^6}-\frac {3 (b c-a d)^{10}}{(a+b x)^7}+7 b^3 d^{10} x^3}{21 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^8,x]

[Out]

(21*b*d^8*(45*b^2*c^2 - 80*a*b*c*d + 36*a^2*d^2)*x + 21*b^2*d^9*(5*b*c - 4*a*d)*x^2 + 7*b^3*d^10*x^3 - (3*(b*c
- a*d)^10)/(a + b*x)^7 + (35*d*(-(b*c) + a*d)^9)/(a + b*x)^6 - (189*d^2*(b*c - a*d)^8)/(a + b*x)^5 + (630*d^3
*(-(b*c) + a*d)^7)/(a + b*x)^4 - (1470*d^4*(b*c - a*d)^6)/(a + b*x)^3 + (2646*d^5*(-(b*c) + a*d)^5)/(a + b*x)^
2 - (4410*d^6*(b*c - a*d)^4)/(a + b*x) + 2520*d^7*(b*c - a*d)^3*Log[a + b*x])/(21*b^11)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^8,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^8, x]

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fricas [B]  time = 1.40, size = 1362, normalized size = 5.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="fricas")

[Out]

1/21*(7*b^10*d^10*x^10 - 3*b^10*c^10 - 5*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 18*a^3*b^7*c^7*d^3 - 42*a^4*b^6*c^6
*d^4 - 126*a^5*b^5*c^5*d^5 - 630*a^6*b^4*c^4*d^6 + 6534*a^7*b^3*c^3*d^7 - 12987*a^8*b^2*c^2*d^8 + 10047*a^9*b*
c*d^9 - 2761*a^10*d^10 + 35*(3*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(3*b^10*c^2*d^8 - 3*a*b^9*c*d^9 + a^2*b^8*d^
10)*x^8 + 49*(135*a*b^9*c^2*d^8 - 195*a^2*b^8*c*d^9 + 77*a^3*b^7*d^10)*x^7 - 49*(90*b^10*c^4*d^6 - 360*a*b^9*c
^3*d^7 + 135*a^2*b^8*c^2*d^8 + 285*a^3*b^7*c*d^9 - 179*a^4*b^6*d^10)*x^6 - 147*(18*b^10*c^5*d^5 + 90*a*b^9*c^4
*d^6 - 540*a^2*b^8*c^3*d^7 + 675*a^3*b^7*c^2*d^8 - 255*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 - 245*(6*b^10*c^6*d^4
+ 18*a*b^9*c^5*d^5 + 90*a^2*b^8*c^4*d^6 - 660*a^3*b^7*c^3*d^7 + 1035*a^4*b^6*c^2*d^8 - 615*a^5*b^5*c*d^9 + 12
1*a^6*b^4*d^10)*x^4 - 35*(18*b^10*c^7*d^3 + 42*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 630*a^3*b^7*c^4*d^6 - 525
0*a^4*b^6*c^3*d^7 + 9135*a^5*b^5*c^2*d^8 - 6195*a^6*b^4*c*d^9 + 1477*a^7*b^3*d^10)*x^3 - 21*(9*b^10*c^8*d^2 +
18*a*b^9*c^7*d^3 + 42*a^2*b^8*c^6*d^4 + 126*a^3*b^7*c^5*d^5 + 630*a^4*b^6*c^4*d^6 - 5754*a^5*b^5*c^3*d^7 + 106
47*a^6*b^4*c^2*d^8 - 7707*a^7*b^3*c*d^9 + 1981*a^8*b^2*d^10)*x^2 - 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*
b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 11907*a^
7*b^3*c^2*d^8 - 8967*a^8*b^2*c*d^9 + 2401*a^9*b*d^10)*x + 2520*(a^7*b^3*c^3*d^7 - 3*a^8*b^2*c^2*d^8 + 3*a^9*b*
c*d^9 - a^10*d^10 + (b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 7*(a*b^9*c^3*d^7 -
3*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 - a^4*b^6*d^10)*x^6 + 21*(a^2*b^8*c^3*d^7 - 3*a^3*b^7*c^2*d^8 + 3*a^4*b^6
*c*d^9 - a^5*b^5*d^10)*x^5 + 35*(a^3*b^7*c^3*d^7 - 3*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 3
5*(a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 21*(a^5*b^5*c^3*d^7 - 3*a^6*b^4
*c^2*d^8 + 3*a^7*b^3*c*d^9 - a^8*b^2*d^10)*x^2 + 7*(a^6*b^4*c^3*d^7 - 3*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 - a^
9*b*d^10)*x)*log(b*x + a))/(b^18*x^7 + 7*a*b^17*x^6 + 21*a^2*b^16*x^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21
*a^5*b^13*x^2 + 7*a^6*b^12*x + a^7*b^11)

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giac [B]  time = 1.26, size = 872, normalized size = 3.38 \begin {gather*} \frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b x + a\right )}^{7} b^{11}} + \frac {b^{16} d^{10} x^{3} + 15 \, b^{16} c d^{9} x^{2} - 12 \, a b^{15} d^{10} x^{2} + 135 \, b^{16} c^{2} d^{8} x - 240 \, a b^{15} c d^{9} x + 108 \, a^{2} b^{14} d^{10} x}{3 \, b^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="giac")

[Out]

120*(b^3*c^3*d^7 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(abs(b*x + a))/b^11 - 1/21*(3*b^10*c^10 + 5*
a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*
c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*d^10 + 4410*(b^10*c^4*d
^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(b^10*c^5*d^5 + 5*a*b^9*
c^4*d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*x^5 + 1470*(b^10*c^6*d^
4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 - 141*a^5*b^5*c*d^9 + 37*
a^6*b^4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 - 875*a^4
*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(3*b^10*c^8*d^2 + 6*a*b^
9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^5*c^3*d^7 + 3654*a^6*b^
4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*b^8*c^7*d^
3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 12042*a^7*b^3*c^2*
d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/((b*x + a)^7*b^11) + 1/3*(b^16*d^10*x^3 + 15*b^16*c*d^9*x^2 - 1
2*a*b^15*d^10*x^2 + 135*b^16*c^2*d^8*x - 240*a*b^15*c*d^9*x + 108*a^2*b^14*d^10*x)/b^24

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maple [B]  time = 0.02, size = 1241, normalized size = 4.81

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^8,x)

[Out]

-30/b^4*d^3/(b*x+a)^4*c^7-210/b^11*d^10/(b*x+a)*a^4-210/b^7*d^6/(b*x+a)*c^4+5/3/b^11*d^10/(b*x+a)^6*a^9-5/3/b^
2*d/(b*x+a)^6*c^9+120/b^8*d^7*ln(b*x+a)*c^3+30/b^11*d^10/(b*x+a)^4*a^7-9/b^11*d^10/(b*x+a)^5*a^8-9/b^3*d^2/(b*
x+a)^5*c^8+126/b^11*d^10/(b*x+a)^2*a^5-126/b^6*d^5/(b*x+a)^2*c^5-120/b^11*d^10*ln(b*x+a)*a^3-70/b^5*d^4/(b*x+a
)^3*c^6-1/7/b^11/(b*x+a)^7*a^10*d^10-4*d^10/b^9*x^2*a+5*d^9/b^8*x^2*c+36*d^10/b^10*a^2*x+45*d^8/b^8*c^2*x-70/b
^11*d^10/(b*x+a)^3*a^6+360/b^10*d^9*ln(b*x+a)*a^2*c-360/b^9*d^8*ln(b*x+a)*a*c^2-210/b^10*d^9/(b*x+a)^4*a^6*c+6
30/b^9*d^8/(b*x+a)^4*a^5*c^2-1050/b^8*d^7/(b*x+a)^4*a^4*c^3+1050/b^7*d^6/(b*x+a)^4*a^3*c^4-630/b^6*d^5/(b*x+a)
^4*a^2*c^5+420/b^6*d^5/(b*x+a)^3*a*c^5+10/7/b^10/(b*x+a)^7*a^9*c*d^9-45/7/b^9/(b*x+a)^7*a^8*c^2*d^8+120/7/b^8/
(b*x+a)^7*a^7*c^3*d^7-30/b^7/(b*x+a)^7*a^6*c^4*d^6+36/b^6/(b*x+a)^7*a^5*c^5*d^5-30/b^5/(b*x+a)^7*a^4*c^6*d^4+1
20/7/b^4/(b*x+a)^7*a^3*c^7*d^3-45/7/b^3/(b*x+a)^7*a^2*c^8*d^2+10/7/b^2/(b*x+a)^7*a*c^9*d+72/b^10*d^9/(b*x+a)^5
*a^7*c-252/b^9*d^8/(b*x+a)^5*a^6*c^2+504/b^8*d^7/(b*x+a)^5*a^5*c^3-630/b^7*d^6/(b*x+a)^5*a^4*c^4+504/b^6*d^5/(
b*x+a)^5*a^3*c^5-252/b^5*d^4/(b*x+a)^5*a^2*c^6+72/b^4*d^3/(b*x+a)^5*a*c^7-630/b^10*d^9/(b*x+a)^2*a^4*c+1260/b^
9*d^8/(b*x+a)^2*a^3*c^2-1260/b^8*d^7/(b*x+a)^2*a^2*c^3+630/b^7*d^6/(b*x+a)^2*a*c^4-80*d^9/b^9*a*c*x+140/b^5*d^
4/(b*x+a)^6*a^3*c^6-60/b^4*d^3/(b*x+a)^6*a^2*c^7+15/b^3*d^2/(b*x+a)^6*a*c^8-15/b^10*d^9/(b*x+a)^6*a^8*c+60/b^9
*d^8/(b*x+a)^6*a^7*c^2+420/b^10*d^9/(b*x+a)^3*a^5*c-1050/b^9*d^8/(b*x+a)^3*a^4*c^2+1400/b^8*d^7/(b*x+a)^3*a^3*
c^3-1050/b^7*d^6/(b*x+a)^3*a^2*c^4+210/b^7*d^6/(b*x+a)^6*a^5*c^4-210/b^6*d^5/(b*x+a)^6*a^4*c^5+210/b^5*d^4/(b*
x+a)^4*a*c^6+840/b^10*d^9/(b*x+a)*a^3*c-1260/b^9*d^8/(b*x+a)*a^2*c^2+840/b^8*d^7/(b*x+a)*a*c^3-140/b^8*d^7/(b*
x+a)^6*a^6*c^3+1/3*d^10/b^8*x^3-1/7/b/(b*x+a)^7*c^10

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maxima [B]  time = 2.44, size = 934, normalized size = 3.62 \begin {gather*} -\frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b^{18} x^{7} + 7 \, a b^{17} x^{6} + 21 \, a^{2} b^{16} x^{5} + 35 \, a^{3} b^{15} x^{4} + 35 \, a^{4} b^{14} x^{3} + 21 \, a^{5} b^{13} x^{2} + 7 \, a^{6} b^{12} x + a^{7} b^{11}\right )}} + \frac {b^{2} d^{10} x^{3} + 3 \, {\left (5 \, b^{2} c d^{9} - 4 \, a b d^{10}\right )} x^{2} + 3 \, {\left (45 \, b^{2} c^{2} d^{8} - 80 \, a b c d^{9} + 36 \, a^{2} d^{10}\right )} x}{3 \, b^{10}} + \frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/21*(3*b^10*c^10 + 5*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5
*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*
d^10 + 4410*(b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(
b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*
x^5 + 1470*(b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 -
141*a^5*b^5*c*d^9 + 37*a^6*b^4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^
3*b^7*c^4*d^6 - 875*a^4*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(
3*b^10*c^8*d^2 + 6*a*b^9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^
5*c^3*d^7 + 3654*a^6*b^4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*
d^2 + 18*a^2*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d
^7 + 12042*a^7*b^3*c^2*d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/(b^18*x^7 + 7*a*b^17*x^6 + 21*a^2*b^16*x
^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21*a^5*b^13*x^2 + 7*a^6*b^12*x + a^7*b^11) + 1/3*(b^2*d^10*x^3 + 3*(5
*b^2*c*d^9 - 4*a*b*d^10)*x^2 + 3*(45*b^2*c^2*d^8 - 80*a*b*c*d^9 + 36*a^2*d^10)*x)/b^10 + 120*(b^3*c^3*d^7 - 3*
a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.43, size = 950, normalized size = 3.68 \begin {gather*} x\,\left (\frac {8\,a\,\left (\frac {8\,a\,d^{10}}{b^9}-\frac {10\,c\,d^9}{b^8}\right )}{b}-\frac {28\,a^2\,d^{10}}{b^{10}}+\frac {45\,c^2\,d^8}{b^8}\right )-\frac {x^4\,\left (2590\,a^6\,b^3\,d^{10}-9870\,a^5\,b^4\,c\,d^9+13650\,a^4\,b^5\,c^2\,d^8-7700\,a^3\,b^6\,c^3\,d^7+1050\,a^2\,b^7\,c^4\,d^6+210\,a\,b^8\,c^5\,d^5+70\,b^9\,c^6\,d^4\right )+x^6\,\left (210\,a^4\,b^5\,d^{10}-840\,a^3\,b^6\,c\,d^9+1260\,a^2\,b^7\,c^2\,d^8-840\,a\,b^8\,c^3\,d^7+210\,b^9\,c^4\,d^6\right )+\frac {2761\,a^{10}\,d^{10}-10047\,a^9\,b\,c\,d^9+12987\,a^8\,b^2\,c^2\,d^8-6534\,a^7\,b^3\,c^3\,d^7+630\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+42\,a^4\,b^6\,c^6\,d^4+18\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+3\,b^{10}\,c^{10}}{21\,b}+x\,\left (\frac {2509\,a^9\,d^{10}}{3}-3069\,a^8\,b\,c\,d^9+4014\,a^7\,b^2\,c^2\,d^8-2058\,a^6\,b^3\,c^3\,d^7+210\,a^5\,b^4\,c^4\,d^6+42\,a^4\,b^5\,c^5\,d^5+14\,a^3\,b^6\,c^6\,d^4+6\,a^2\,b^7\,c^7\,d^3+3\,a\,b^8\,c^8\,d^2+\frac {5\,b^9\,c^9\,d}{3}\right )+x^3\,\left (3190\,a^7\,b^2\,d^{10}-11970\,a^6\,b^3\,c\,d^9+16170\,a^5\,b^4\,c^2\,d^8-8750\,a^4\,b^5\,c^3\,d^7+1050\,a^3\,b^6\,c^4\,d^6+210\,a^2\,b^7\,c^5\,d^5+70\,a\,b^8\,c^6\,d^4+30\,b^9\,c^7\,d^3\right )+x^2\,\left (2229\,a^8\,b\,d^{10}-8262\,a^7\,b^2\,c\,d^9+10962\,a^6\,b^3\,c^2\,d^8-5754\,a^5\,b^4\,c^3\,d^7+630\,a^4\,b^5\,c^4\,d^6+126\,a^3\,b^6\,c^5\,d^5+42\,a^2\,b^7\,c^6\,d^4+18\,a\,b^8\,c^7\,d^3+9\,b^9\,c^8\,d^2\right )+x^5\,\left (1134\,a^5\,b^4\,d^{10}-4410\,a^4\,b^5\,c\,d^9+6300\,a^3\,b^6\,c^2\,d^8-3780\,a^2\,b^7\,c^3\,d^7+630\,a\,b^8\,c^4\,d^6+126\,b^9\,c^5\,d^5\right )}{a^7\,b^{10}+7\,a^6\,b^{11}\,x+21\,a^5\,b^{12}\,x^2+35\,a^4\,b^{13}\,x^3+35\,a^3\,b^{14}\,x^4+21\,a^2\,b^{15}\,x^5+7\,a\,b^{16}\,x^6+b^{17}\,x^7}-x^2\,\left (\frac {4\,a\,d^{10}}{b^9}-\frac {5\,c\,d^9}{b^8}\right )-\frac {\ln \left (a+b\,x\right )\,\left (120\,a^3\,d^{10}-360\,a^2\,b\,c\,d^9+360\,a\,b^2\,c^2\,d^8-120\,b^3\,c^3\,d^7\right )}{b^{11}}+\frac {d^{10}\,x^3}{3\,b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^8,x)

[Out]

x*((8*a*((8*a*d^10)/b^9 - (10*c*d^9)/b^8))/b - (28*a^2*d^10)/b^10 + (45*c^2*d^8)/b^8) - (x^4*(2590*a^6*b^3*d^1
0 + 70*b^9*c^6*d^4 + 210*a*b^8*c^5*d^5 - 9870*a^5*b^4*c*d^9 + 1050*a^2*b^7*c^4*d^6 - 7700*a^3*b^6*c^3*d^7 + 13
650*a^4*b^5*c^2*d^8) + x^6*(210*a^4*b^5*d^10 + 210*b^9*c^4*d^6 - 840*a*b^8*c^3*d^7 - 840*a^3*b^6*c*d^9 + 1260*
a^2*b^7*c^2*d^8) + (2761*a^10*d^10 + 3*b^10*c^10 + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4
+ 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 + 5*a*b^9*c^9*d -
10047*a^9*b*c*d^9)/(21*b) + x*((2509*a^9*d^10)/3 + (5*b^9*c^9*d)/3 + 3*a*b^8*c^8*d^2 + 6*a^2*b^7*c^7*d^3 + 14*
a^3*b^6*c^6*d^4 + 42*a^4*b^5*c^5*d^5 + 210*a^5*b^4*c^4*d^6 - 2058*a^6*b^3*c^3*d^7 + 4014*a^7*b^2*c^2*d^8 - 306
9*a^8*b*c*d^9) + x^3*(3190*a^7*b^2*d^10 + 30*b^9*c^7*d^3 + 70*a*b^8*c^6*d^4 - 11970*a^6*b^3*c*d^9 + 210*a^2*b^
7*c^5*d^5 + 1050*a^3*b^6*c^4*d^6 - 8750*a^4*b^5*c^3*d^7 + 16170*a^5*b^4*c^2*d^8) + x^2*(2229*a^8*b*d^10 + 9*b^
9*c^8*d^2 + 18*a*b^8*c^7*d^3 - 8262*a^7*b^2*c*d^9 + 42*a^2*b^7*c^6*d^4 + 126*a^3*b^6*c^5*d^5 + 630*a^4*b^5*c^4
*d^6 - 5754*a^5*b^4*c^3*d^7 + 10962*a^6*b^3*c^2*d^8) + x^5*(1134*a^5*b^4*d^10 + 126*b^9*c^5*d^5 + 630*a*b^8*c^
4*d^6 - 4410*a^4*b^5*c*d^9 - 3780*a^2*b^7*c^3*d^7 + 6300*a^3*b^6*c^2*d^8))/(a^7*b^10 + b^17*x^7 + 7*a^6*b^11*x
+ 7*a*b^16*x^6 + 21*a^5*b^12*x^2 + 35*a^4*b^13*x^3 + 35*a^3*b^14*x^4 + 21*a^2*b^15*x^5) - x^2*((4*a*d^10)/b^9
- (5*c*d^9)/b^8) - (log(a + b*x)*(120*a^3*d^10 - 120*b^3*c^3*d^7 + 360*a*b^2*c^2*d^8 - 360*a^2*b*c*d^9))/b^11
+ (d^10*x^3)/(3*b^8)

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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)**10/(b*x+a)**8,x)

[Out]

Timed out

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