3.2.65 \(\int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx\) [165]

Optimal. Leaf size=188 \[ -\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}+\frac {b^2 (6 A b-17 a B) (a+b x)^{11}}{2856 a^4 x^{14}}-\frac {b^3 (6 A b-17 a B) (a+b x)^{11}}{12376 a^5 x^{13}}+\frac {b^4 (6 A b-17 a B) (a+b x)^{11}}{74256 a^6 x^{12}}-\frac {b^5 (6 A b-17 a B) (a+b x)^{11}}{816816 a^7 x^{11}} \]

[Out]

-1/17*A*(b*x+a)^11/a/x^17+1/272*(6*A*b-17*B*a)*(b*x+a)^11/a^2/x^16-1/816*b*(6*A*b-17*B*a)*(b*x+a)^11/a^3/x^15+
1/2856*b^2*(6*A*b-17*B*a)*(b*x+a)^11/a^4/x^14-1/12376*b^3*(6*A*b-17*B*a)*(b*x+a)^11/a^5/x^13+1/74256*b^4*(6*A*
b-17*B*a)*(b*x+a)^11/a^6/x^12-1/816816*b^5*(6*A*b-17*B*a)*(b*x+a)^11/a^7/x^11

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {79, 47, 37} \begin {gather*} -\frac {b^5 (a+b x)^{11} (6 A b-17 a B)}{816816 a^7 x^{11}}+\frac {b^4 (a+b x)^{11} (6 A b-17 a B)}{74256 a^6 x^{12}}-\frac {b^3 (a+b x)^{11} (6 A b-17 a B)}{12376 a^5 x^{13}}+\frac {b^2 (a+b x)^{11} (6 A b-17 a B)}{2856 a^4 x^{14}}-\frac {b (a+b x)^{11} (6 A b-17 a B)}{816 a^3 x^{15}}+\frac {(a+b x)^{11} (6 A b-17 a B)}{272 a^2 x^{16}}-\frac {A (a+b x)^{11}}{17 a x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^18,x]

[Out]

-1/17*(A*(a + b*x)^11)/(a*x^17) + ((6*A*b - 17*a*B)*(a + b*x)^11)/(272*a^2*x^16) - (b*(6*A*b - 17*a*B)*(a + b*
x)^11)/(816*a^3*x^15) + (b^2*(6*A*b - 17*a*B)*(a + b*x)^11)/(2856*a^4*x^14) - (b^3*(6*A*b - 17*a*B)*(a + b*x)^
11)/(12376*a^5*x^13) + (b^4*(6*A*b - 17*a*B)*(a + b*x)^11)/(74256*a^6*x^12) - (b^5*(6*A*b - 17*a*B)*(a + b*x)^
11)/(816816*a^7*x^11)

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]

Rule 47

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(-6 A b+17 a B) \int \frac {(a+b x)^{10}}{x^{17}} \, dx}{17 a}\\ &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}+\frac {(5 b (6 A b-17 a B)) \int \frac {(a+b x)^{10}}{x^{16}} \, dx}{272 a^2}\\ &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}-\frac {\left (b^2 (6 A b-17 a B)\right ) \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{204 a^3}\\ &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}+\frac {b^2 (6 A b-17 a B) (a+b x)^{11}}{2856 a^4 x^{14}}+\frac {\left (b^3 (6 A b-17 a B)\right ) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{952 a^4}\\ &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}+\frac {b^2 (6 A b-17 a B) (a+b x)^{11}}{2856 a^4 x^{14}}-\frac {b^3 (6 A b-17 a B) (a+b x)^{11}}{12376 a^5 x^{13}}-\frac {\left (b^4 (6 A b-17 a B)\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{6188 a^5}\\ &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}+\frac {b^2 (6 A b-17 a B) (a+b x)^{11}}{2856 a^4 x^{14}}-\frac {b^3 (6 A b-17 a B) (a+b x)^{11}}{12376 a^5 x^{13}}+\frac {b^4 (6 A b-17 a B) (a+b x)^{11}}{74256 a^6 x^{12}}+\frac {\left (b^5 (6 A b-17 a B)\right ) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{74256 a^6}\\ &=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}+\frac {b^2 (6 A b-17 a B) (a+b x)^{11}}{2856 a^4 x^{14}}-\frac {b^3 (6 A b-17 a B) (a+b x)^{11}}{12376 a^5 x^{13}}+\frac {b^4 (6 A b-17 a B) (a+b x)^{11}}{74256 a^6 x^{12}}-\frac {b^5 (6 A b-17 a B) (a+b x)^{11}}{816816 a^7 x^{11}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 222, normalized size = 1.18 \begin {gather*} -\frac {b^{10} (6 A+7 B x)}{42 x^7}-\frac {5 a b^9 (7 A+8 B x)}{28 x^8}-\frac {5 a^2 b^8 (8 A+9 B x)}{8 x^9}-\frac {4 a^3 b^7 (9 A+10 B x)}{3 x^{10}}-\frac {21 a^4 b^6 (10 A+11 B x)}{11 x^{11}}-\frac {21 a^5 b^5 (11 A+12 B x)}{11 x^{12}}-\frac {35 a^6 b^4 (12 A+13 B x)}{26 x^{13}}-\frac {60 a^7 b^3 (13 A+14 B x)}{91 x^{14}}-\frac {3 a^8 b^2 (14 A+15 B x)}{14 x^{15}}-\frac {a^9 b (15 A+16 B x)}{24 x^{16}}-\frac {a^{10} (16 A+17 B x)}{272 x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^18,x]

[Out]

-1/42*(b^10*(6*A + 7*B*x))/x^7 - (5*a*b^9*(7*A + 8*B*x))/(28*x^8) - (5*a^2*b^8*(8*A + 9*B*x))/(8*x^9) - (4*a^3
*b^7*(9*A + 10*B*x))/(3*x^10) - (21*a^4*b^6*(10*A + 11*B*x))/(11*x^11) - (21*a^5*b^5*(11*A + 12*B*x))/(11*x^12
) - (35*a^6*b^4*(12*A + 13*B*x))/(26*x^13) - (60*a^7*b^3*(13*A + 14*B*x))/(91*x^14) - (3*a^8*b^2*(14*A + 15*B*
x))/(14*x^15) - (a^9*b*(15*A + 16*B*x))/(24*x^16) - (a^10*(16*A + 17*B*x))/(272*x^17)

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 208, normalized size = 1.11

method	result	size
default	\(-\frac {a^{8} b \left (9 A b +2 B a \right )}{3 x^{15}}-\frac {5 a^{2} b^{7} \left (3 A b +8 B a \right )}{3 x^{9}}-\frac {42 a^{4} b^{5} \left (5 A b +6 B a \right )}{11 x^{11}}-\frac {7 a^{5} b^{4} \left (6 A b +5 B a \right )}{2 x^{12}}-\frac {a^{10} A}{17 x^{17}}-\frac {b^{10} B}{6 x^{6}}-\frac {a^{9} \left (10 A b +B a \right )}{16 x^{16}}-\frac {3 a^{3} b^{6} \left (4 A b +7 B a \right )}{x^{10}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{14 x^{14}}-\frac {30 a^{6} b^{3} \left (7 A b +4 B a \right )}{13 x^{13}}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{8 x^{8}}-\frac {b^{9} \left (A b +10 B a \right )}{7 x^{7}}\)	\(208\)
norman	\(\frac {-\frac {a^{10} A}{17}+\left (-\frac {5}{8} a^{9} b A -\frac {1}{16} a^{10} B \right ) x +\left (-3 a^{8} b^{2} A -\frac {2}{3} a^{9} b B \right ) x^{2}+\left (-\frac {60}{7} a^{7} b^{3} A -\frac {45}{14} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {210}{13} a^{6} b^{4} A -\frac {120}{13} a^{7} b^{3} B \right ) x^{4}+\left (-21 a^{5} b^{5} A -\frac {35}{2} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {210}{11} a^{4} b^{6} A -\frac {252}{11} a^{5} b^{5} B \right ) x^{6}+\left (-12 a^{3} b^{7} A -21 a^{4} b^{6} B \right ) x^{7}+\left (-5 a^{2} b^{8} A -\frac {40}{3} a^{3} b^{7} B \right ) x^{8}+\left (-\frac {5}{4} a \,b^{9} A -\frac {45}{8} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{7} b^{10} A -\frac {10}{7} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{6}}{x^{17}}\)	\(235\)
risch	\(\frac {-\frac {a^{10} A}{17}+\left (-\frac {5}{8} a^{9} b A -\frac {1}{16} a^{10} B \right ) x +\left (-3 a^{8} b^{2} A -\frac {2}{3} a^{9} b B \right ) x^{2}+\left (-\frac {60}{7} a^{7} b^{3} A -\frac {45}{14} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {210}{13} a^{6} b^{4} A -\frac {120}{13} a^{7} b^{3} B \right ) x^{4}+\left (-21 a^{5} b^{5} A -\frac {35}{2} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {210}{11} a^{4} b^{6} A -\frac {252}{11} a^{5} b^{5} B \right ) x^{6}+\left (-12 a^{3} b^{7} A -21 a^{4} b^{6} B \right ) x^{7}+\left (-5 a^{2} b^{8} A -\frac {40}{3} a^{3} b^{7} B \right ) x^{8}+\left (-\frac {5}{4} a \,b^{9} A -\frac {45}{8} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{7} b^{10} A -\frac {10}{7} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{6}}{x^{17}}\)	\(235\)
gosper	\(-\frac {136136 b^{10} B \,x^{11}+116688 A \,b^{10} x^{10}+1166880 B a \,b^{9} x^{10}+1021020 a A \,b^{9} x^{9}+4594590 B \,a^{2} b^{8} x^{9}+4084080 a^{2} A \,b^{8} x^{8}[phone] B \,a^{3} b^{7} x^{8}+9801792 a^{3} A \,b^{7} x^{7}[phone] B \,a^{4} b^{6} x^{7}[phone] a^{4} A \,b^{6} x^{6}[phone] B \,a^{5} b^{5} x^{6}[phone] a^{5} A \,b^{5} x^{5}[phone] B \,a^{6} b^{4} x^{5}[phone] a^{6} A \,b^{4} x^{4}+7539840 B \,a^{7} b^{3} x^{4}+7001280 a^{7} A \,b^{3} x^{3}+2625480 B \,a^{8} b^{2} x^{3}+2450448 a^{8} A \,b^{2} x^{2}+544544 B \,a^{9} b \,x^{2}+510510 a^{9} A b x +51051 a^{10} B x +48048 a^{10} A}{816816 x^{17}}\)	\(244\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^18,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^8*b*(9*A*b+2*B*a)/x^15-5/3*a^2*b^7*(3*A*b+8*B*a)/x^9-42/11*a^4*b^5*(5*A*b+6*B*a)/x^11-7/2*a^5*b^4*(6*A*
b+5*B*a)/x^12-1/17*a^10*A/x^17-1/6*b^10*B/x^6-1/16*a^9*(10*A*b+B*a)/x^16-3*a^3*b^6*(4*A*b+7*B*a)/x^10-15/14*a^
7*b^2*(8*A*b+3*B*a)/x^14-30/13*a^6*b^3*(7*A*b+4*B*a)/x^13-5/8*a*b^8*(2*A*b+9*B*a)/x^8-1/7*b^9*(A*b+10*B*a)/x^7

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 243, normalized size = 1.29 \begin {gather*} -\frac {136136 \, B b^{10} x^{11} + 48048 \, A a^{10} + 116688 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 510510 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1361360 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 2450448 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3118752 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2858856 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1884960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 875160 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 272272 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 51051 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{816816 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^18,x, algorithm="maxima")

[Out]

-1/816816*(136136*B*b^10*x^11 + 48048*A*a^10 + 116688*(10*B*a*b^9 + A*b^10)*x^10 + 510510*(9*B*a^2*b^8 + 2*A*a
*b^9)*x^9 + 1361360*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 2450448*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 3118752*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*x^6 + 2858856*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 1884960*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4
+ 875160*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 272272*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 51051*(B*a^10 + 10*A*a^9*b)*
x)/x^17

________________________________________________________________________________________

Fricas [A]
time = 0.69, size = 243, normalized size = 1.29 \begin {gather*} -\frac {136136 \, B b^{10} x^{11} + 48048 \, A a^{10} + 116688 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 510510 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1361360 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 2450448 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3118752 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2858856 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1884960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 875160 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 272272 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 51051 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{816816 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^18,x, algorithm="fricas")

[Out]

-1/816816*(136136*B*b^10*x^11 + 48048*A*a^10 + 116688*(10*B*a*b^9 + A*b^10)*x^10 + 510510*(9*B*a^2*b^8 + 2*A*a
*b^9)*x^9 + 1361360*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 2450448*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 3118752*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*x^6 + 2858856*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 1884960*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4
+ 875160*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 272272*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 51051*(B*a^10 + 10*A*a^9*b)*
x)/x^17

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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)**10*(B*x+A)/x**18,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.98, size = 243, normalized size = 1.29 \begin {gather*} -\frac {136136 \, B b^{10} x^{11} + 1166880 \, B a b^{9} x^{10} + 116688 \, A b^{10} x^{10} + 4594590 \, B a^{2} b^{8} x^{9} + 1021020 \, A a b^{9} x^{9} + 10890880 \, B a^{3} b^{7} x^{8} + 4084080 \, A a^{2} b^{8} x^{8} + 17153136 \, B a^{4} b^{6} x^{7} + 9801792 \, A a^{3} b^{7} x^{7} + 18712512 \, B a^{5} b^{5} x^{6} + 15593760 \, A a^{4} b^{6} x^{6} + 14294280 \, B a^{6} b^{4} x^{5} + 17153136 \, A a^{5} b^{5} x^{5} + 7539840 \, B a^{7} b^{3} x^{4} + 13194720 \, A a^{6} b^{4} x^{4} + 2625480 \, B a^{8} b^{2} x^{3} + 7001280 \, A a^{7} b^{3} x^{3} + 544544 \, B a^{9} b x^{2} + 2450448 \, A a^{8} b^{2} x^{2} + 51051 \, B a^{10} x + 510510 \, A a^{9} b x + 48048 \, A a^{10}}{816816 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^18,x, algorithm="giac")

[Out]

-1/816816*(136136*B*b^10*x^11 + 1166880*B*a*b^9*x^10 + 116688*A*b^10*x^10 + 4594590*B*a^2*b^8*x^9 + 1021020*A*
a*b^9*x^9 + 10890880*B*a^3*b^7*x^8 + 4084080*A*a^2*b^8*x^8 + 17153136*B*a^4*b^6*x^7 + 9801792*A*a^3*b^7*x^7 +
18712512*B*a^5*b^5*x^6 + 15593760*A*a^4*b^6*x^6 + 14294280*B*a^6*b^4*x^5 + 17153136*A*a^5*b^5*x^5 + 7539840*B*
a^7*b^3*x^4 + 13194720*A*a^6*b^4*x^4 + 2625480*B*a^8*b^2*x^3 + 7001280*A*a^7*b^3*x^3 + 544544*B*a^9*b*x^2 + 24
50448*A*a^8*b^2*x^2 + 51051*B*a^10*x + 510510*A*a^9*b*x + 48048*A*a^10)/x^17

________________________________________________________________________________________

Mupad [B]
time = 0.14, size = 235, normalized size = 1.25 \begin {gather*} -\frac {x\,\left (\frac {B\,a^{10}}{16}+\frac {5\,A\,b\,a^9}{8}\right )+\frac {A\,a^{10}}{17}+x^2\,\left (\frac {2\,B\,a^9\,b}{3}+3\,A\,a^8\,b^2\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{8}+\frac {5\,A\,a\,b^9}{4}\right )+x^{10}\,\left (\frac {A\,b^{10}}{7}+\frac {10\,B\,a\,b^9}{7}\right )+x^7\,\left (21\,B\,a^4\,b^6+12\,A\,a^3\,b^7\right )+x^8\,\left (\frac {40\,B\,a^3\,b^7}{3}+5\,A\,a^2\,b^8\right )+x^5\,\left (\frac {35\,B\,a^6\,b^4}{2}+21\,A\,a^5\,b^5\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{14}+\frac {60\,A\,a^7\,b^3}{7}\right )+x^4\,\left (\frac {120\,B\,a^7\,b^3}{13}+\frac {210\,A\,a^6\,b^4}{13}\right )+x^6\,\left (\frac {252\,B\,a^5\,b^5}{11}+\frac {210\,A\,a^4\,b^6}{11}\right )+\frac {B\,b^{10}\,x^{11}}{6}}{x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^18,x)

[Out]

-(x*((B*a^10)/16 + (5*A*a^9*b)/8) + (A*a^10)/17 + x^2*(3*A*a^8*b^2 + (2*B*a^9*b)/3) + x^9*((45*B*a^2*b^8)/8 +
(5*A*a*b^9)/4) + x^10*((A*b^10)/7 + (10*B*a*b^9)/7) + x^7*(12*A*a^3*b^7 + 21*B*a^4*b^6) + x^8*(5*A*a^2*b^8 + (
40*B*a^3*b^7)/3) + x^5*(21*A*a^5*b^5 + (35*B*a^6*b^4)/2) + x^3*((60*A*a^7*b^3)/7 + (45*B*a^8*b^2)/14) + x^4*((
210*A*a^6*b^4)/13 + (120*B*a^7*b^3)/13) + x^6*((210*A*a^4*b^6)/11 + (252*B*a^5*b^5)/11) + (B*b^10*x^11)/6)/x^1
7

________________________________________________________________________________________