3.164 \(\int \frac {(a+b x)^{10} (A+B x)}{x^{17}} \, dx\)

Optimal. Leaf size=159 \[ \frac {b^4 (a+b x)^{11} (5 A b-16 a B)}{240240 a^6 x^{11}}-\frac {b^3 (a+b x)^{11} (5 A b-16 a B)}{21840 a^5 x^{12}}+\frac {b^2 (a+b x)^{11} (5 A b-16 a B)}{3640 a^4 x^{13}}-\frac {b (a+b x)^{11} (5 A b-16 a B)}{840 a^3 x^{14}}+\frac {(a+b x)^{11} (5 A b-16 a B)}{240 a^2 x^{15}}-\frac {A (a+b x)^{11}}{16 a x^{16}} \]

[Out]

-1/16*A*(b*x+a)^11/a/x^16+1/240*(5*A*b-16*B*a)*(b*x+a)^11/a^2/x^15-1/840*b*(5*A*b-16*B*a)*(b*x+a)^11/a^3/x^14+
1/3640*b^2*(5*A*b-16*B*a)*(b*x+a)^11/a^4/x^13-1/21840*b^3*(5*A*b-16*B*a)*(b*x+a)^11/a^5/x^12+1/240240*b^4*(5*A
*b-16*B*a)*(b*x+a)^11/a^6/x^11

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Rubi [A]  time = 0.06, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ \frac {b^4 (a+b x)^{11} (5 A b-16 a B)}{240240 a^6 x^{11}}-\frac {b^3 (a+b x)^{11} (5 A b-16 a B)}{21840 a^5 x^{12}}+\frac {b^2 (a+b x)^{11} (5 A b-16 a B)}{3640 a^4 x^{13}}-\frac {b (a+b x)^{11} (5 A b-16 a B)}{840 a^3 x^{14}}+\frac {(a+b x)^{11} (5 A b-16 a B)}{240 a^2 x^{15}}-\frac {A (a+b x)^{11}}{16 a x^{16}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^17,x]

[Out]

-(A*(a + b*x)^11)/(16*a*x^16) + ((5*A*b - 16*a*B)*(a + b*x)^11)/(240*a^2*x^15) - (b*(5*A*b - 16*a*B)*(a + b*x)
^11)/(840*a^3*x^14) + (b^2*(5*A*b - 16*a*B)*(a + b*x)^11)/(3640*a^4*x^13) - (b^3*(5*A*b - 16*a*B)*(a + b*x)^11
)/(21840*a^5*x^12) + (b^4*(5*A*b - 16*a*B)*(a + b*x)^11)/(240240*a^6*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 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 78

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] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^{17}} \, dx &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(-5 A b+16 a B) \int \frac {(a+b x)^{10}}{x^{16}} \, dx}{16 a}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}+\frac {(b (5 A b-16 a B)) \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{60 a^2}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}-\frac {\left (b^2 (5 A b-16 a B)\right ) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{280 a^3}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}+\frac {b^2 (5 A b-16 a B) (a+b x)^{11}}{3640 a^4 x^{13}}+\frac {\left (b^3 (5 A b-16 a B)\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{1820 a^4}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}+\frac {b^2 (5 A b-16 a B) (a+b x)^{11}}{3640 a^4 x^{13}}-\frac {b^3 (5 A b-16 a B) (a+b x)^{11}}{21840 a^5 x^{12}}-\frac {\left (b^4 (5 A b-16 a B)\right ) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{21840 a^5}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}+\frac {b^2 (5 A b-16 a B) (a+b x)^{11}}{3640 a^4 x^{13}}-\frac {b^3 (5 A b-16 a B) (a+b x)^{11}}{21840 a^5 x^{12}}+\frac {b^4 (5 A b-16 a B) (a+b x)^{11}}{240240 a^6 x^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 222, normalized size = 1.40 \[ -\frac {a^{10} (15 A+16 B x)}{240 x^{16}}-\frac {a^9 b (14 A+15 B x)}{21 x^{15}}-\frac {45 a^8 b^2 (13 A+14 B x)}{182 x^{14}}-\frac {10 a^7 b^3 (12 A+13 B x)}{13 x^{13}}-\frac {35 a^6 b^4 (11 A+12 B x)}{22 x^{12}}-\frac {126 a^5 b^5 (10 A+11 B x)}{55 x^{11}}-\frac {7 a^4 b^6 (9 A+10 B x)}{3 x^{10}}-\frac {5 a^3 b^7 (8 A+9 B x)}{3 x^9}-\frac {45 a^2 b^8 (7 A+8 B x)}{56 x^8}-\frac {5 a b^9 (6 A+7 B x)}{21 x^7}-\frac {b^{10} (5 A+6 B x)}{30 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^17,x]

[Out]

-1/30*(b^10*(5*A + 6*B*x))/x^6 - (5*a*b^9*(6*A + 7*B*x))/(21*x^7) - (45*a^2*b^8*(7*A + 8*B*x))/(56*x^8) - (5*a
^3*b^7*(8*A + 9*B*x))/(3*x^9) - (7*a^4*b^6*(9*A + 10*B*x))/(3*x^10) - (126*a^5*b^5*(10*A + 11*B*x))/(55*x^11)
- (35*a^6*b^4*(11*A + 12*B*x))/(22*x^12) - (10*a^7*b^3*(12*A + 13*B*x))/(13*x^13) - (45*a^8*b^2*(13*A + 14*B*x
))/(182*x^14) - (a^9*b*(14*A + 15*B*x))/(21*x^15) - (a^10*(15*A + 16*B*x))/(240*x^16)

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fricas [A]  time = 0.93, size = 243, normalized size = 1.53 \[ -\frac {48048 \, B b^{10} x^{11} + 15015 \, A a^{10} + 40040 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 171600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450450 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 800800 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 1009008 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 917280 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 600600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 277200 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 85800 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 16016 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{240240 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^17,x, algorithm="fricas")

[Out]

-1/240240*(48048*B*b^10*x^11 + 15015*A*a^10 + 40040*(10*B*a*b^9 + A*b^10)*x^10 + 171600*(9*B*a^2*b^8 + 2*A*a*b
^9)*x^9 + 450450*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 800800*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 1009008*(6*B*a^5*b
^5 + 5*A*a^4*b^6)*x^6 + 917280*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 600600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2772
00*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 85800*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 16016*(B*a^10 + 10*A*a^9*b)*x)/x^16

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giac [A]  time = 0.94, size = 243, normalized size = 1.53 \[ -\frac {48048 \, B b^{10} x^{11} + 400400 \, B a b^{9} x^{10} + 40040 \, A b^{10} x^{10} + 1544400 \, B a^{2} b^{8} x^{9} + 343200 \, A a b^{9} x^{9} + 3603600 \, B a^{3} b^{7} x^{8} + 1351350 \, A a^{2} b^{8} x^{8} + 5605600 \, B a^{4} b^{6} x^{7} + 3203200 \, A a^{3} b^{7} x^{7} + 6054048 \, B a^{5} b^{5} x^{6} + 5045040 \, A a^{4} b^{6} x^{6} + 4586400 \, B a^{6} b^{4} x^{5} + 5503680 \, A a^{5} b^{5} x^{5} + 2402400 \, B a^{7} b^{3} x^{4} + 4204200 \, A a^{6} b^{4} x^{4} + 831600 \, B a^{8} b^{2} x^{3} + 2217600 \, A a^{7} b^{3} x^{3} + 171600 \, B a^{9} b x^{2} + 772200 \, A a^{8} b^{2} x^{2} + 16016 \, B a^{10} x + 160160 \, A a^{9} b x + 15015 \, A a^{10}}{240240 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^17,x, algorithm="giac")

[Out]

-1/240240*(48048*B*b^10*x^11 + 400400*B*a*b^9*x^10 + 40040*A*b^10*x^10 + 1544400*B*a^2*b^8*x^9 + 343200*A*a*b^
9*x^9 + 3603600*B*a^3*b^7*x^8 + 1351350*A*a^2*b^8*x^8 + 5605600*B*a^4*b^6*x^7 + 3203200*A*a^3*b^7*x^7 + 605404
8*B*a^5*b^5*x^6 + 5045040*A*a^4*b^6*x^6 + 4586400*B*a^6*b^4*x^5 + 5503680*A*a^5*b^5*x^5 + 2402400*B*a^7*b^3*x^
4 + 4204200*A*a^6*b^4*x^4 + 831600*B*a^8*b^2*x^3 + 2217600*A*a^7*b^3*x^3 + 171600*B*a^9*b*x^2 + 772200*A*a^8*b
^2*x^2 + 16016*B*a^10*x + 160160*A*a^9*b*x + 15015*A*a^10)/x^16

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maple [A]  time = 0.01, size = 208, normalized size = 1.31 \[ -\frac {B \,b^{10}}{5 x^{5}}-\frac {\left (A b +10 B a \right ) b^{9}}{6 x^{6}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{7 x^{7}}-\frac {15 \left (3 A b +8 B a \right ) a^{2} b^{7}}{8 x^{8}}-\frac {10 \left (4 A b +7 B a \right ) a^{3} b^{6}}{3 x^{9}}-\frac {21 \left (5 A b +6 B a \right ) a^{4} b^{5}}{5 x^{10}}-\frac {42 \left (6 A b +5 B a \right ) a^{5} b^{4}}{11 x^{11}}-\frac {5 \left (7 A b +4 B a \right ) a^{6} b^{3}}{2 x^{12}}-\frac {15 \left (8 A b +3 B a \right ) a^{7} b^{2}}{13 x^{13}}-\frac {A \,a^{10}}{16 x^{16}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{14 x^{14}}-\frac {\left (10 A b +B a \right ) a^{9}}{15 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^17,x)

[Out]

-1/6*b^9*(A*b+10*B*a)/x^6-1/5*B*b^10/x^5-5/7*a*b^8*(2*A*b+9*B*a)/x^7-5/14*a^8*b*(9*A*b+2*B*a)/x^14-15/8*a^2*b^
7*(3*A*b+8*B*a)/x^8-10/3*a^3*b^6*(4*A*b+7*B*a)/x^9-15/13*a^7*b^2*(8*A*b+3*B*a)/x^13-21/5*a^4*b^5*(5*A*b+6*B*a)
/x^10-42/11*a^5*b^4*(6*A*b+5*B*a)/x^11-1/16*A*a^10/x^16-5/2*a^6*b^3*(7*A*b+4*B*a)/x^12-1/15*a^9*(10*A*b+B*a)/x
^15

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maxima [A]  time = 1.10, size = 243, normalized size = 1.53 \[ -\frac {48048 \, B b^{10} x^{11} + 15015 \, A a^{10} + 40040 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 171600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450450 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 800800 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 1009008 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 917280 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 600600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 277200 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 85800 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 16016 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{240240 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^17,x, algorithm="maxima")

[Out]

-1/240240*(48048*B*b^10*x^11 + 15015*A*a^10 + 40040*(10*B*a*b^9 + A*b^10)*x^10 + 171600*(9*B*a^2*b^8 + 2*A*a*b
^9)*x^9 + 450450*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 800800*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 1009008*(6*B*a^5*b
^5 + 5*A*a^4*b^6)*x^6 + 917280*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 600600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2772
00*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 85800*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 16016*(B*a^10 + 10*A*a^9*b)*x)/x^16

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mupad [B]  time = 0.39, size = 235, normalized size = 1.48 \[ -\frac {x\,\left (\frac {B\,a^{10}}{15}+\frac {2\,A\,b\,a^9}{3}\right )+\frac {A\,a^{10}}{16}+x^2\,\left (\frac {5\,B\,a^9\,b}{7}+\frac {45\,A\,a^8\,b^2}{14}\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{7}+\frac {10\,A\,a\,b^9}{7}\right )+x^{10}\,\left (\frac {A\,b^{10}}{6}+\frac {5\,B\,a\,b^9}{3}\right )+x^4\,\left (10\,B\,a^7\,b^3+\frac {35\,A\,a^6\,b^4}{2}\right )+x^8\,\left (15\,B\,a^3\,b^7+\frac {45\,A\,a^2\,b^8}{8}\right )+x^7\,\left (\frac {70\,B\,a^4\,b^6}{3}+\frac {40\,A\,a^3\,b^7}{3}\right )+x^6\,\left (\frac {126\,B\,a^5\,b^5}{5}+21\,A\,a^4\,b^6\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{13}+\frac {120\,A\,a^7\,b^3}{13}\right )+x^5\,\left (\frac {210\,B\,a^6\,b^4}{11}+\frac {252\,A\,a^5\,b^5}{11}\right )+\frac {B\,b^{10}\,x^{11}}{5}}{x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^17,x)

[Out]

-(x*((B*a^10)/15 + (2*A*a^9*b)/3) + (A*a^10)/16 + x^2*((45*A*a^8*b^2)/14 + (5*B*a^9*b)/7) + x^9*((45*B*a^2*b^8
)/7 + (10*A*a*b^9)/7) + x^10*((A*b^10)/6 + (5*B*a*b^9)/3) + x^4*((35*A*a^6*b^4)/2 + 10*B*a^7*b^3) + x^8*((45*A
*a^2*b^8)/8 + 15*B*a^3*b^7) + x^7*((40*A*a^3*b^7)/3 + (70*B*a^4*b^6)/3) + x^6*(21*A*a^4*b^6 + (126*B*a^5*b^5)/
5) + x^3*((120*A*a^7*b^3)/13 + (45*B*a^8*b^2)/13) + x^5*((252*A*a^5*b^5)/11 + (210*B*a^6*b^4)/11) + (B*b^10*x^
11)/5)/x^16

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sympy [A]  time = 86.74, size = 260, normalized size = 1.64 \[ \frac {- 15015 A a^{10} - 48048 B b^{10} x^{11} + x^{10} \left (- 40040 A b^{10} - 400400 B a b^{9}\right ) + x^{9} \left (- 343200 A a b^{9} - 1544400 B a^{2} b^{8}\right ) + x^{8} \left (- 1351350 A a^{2} b^{8} - 3603600 B a^{3} b^{7}\right ) + x^{7} \left (- 3203200 A a^{3} b^{7} - 5605600 B a^{4} b^{6}\right ) + x^{6} \left (- 5045040 A a^{4} b^{6} - 6054048 B a^{5} b^{5}\right ) + x^{5} \left (- 5503680 A a^{5} b^{5} - 4586400 B a^{6} b^{4}\right ) + x^{4} \left (- 4204200 A a^{6} b^{4} - 2402400 B a^{7} b^{3}\right ) + x^{3} \left (- 2217600 A a^{7} b^{3} - 831600 B a^{8} b^{2}\right ) + x^{2} \left (- 772200 A a^{8} b^{2} - 171600 B a^{9} b\right ) + x \left (- 160160 A a^{9} b - 16016 B a^{10}\right )}{240240 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**17,x)

[Out]

(-15015*A*a**10 - 48048*B*b**10*x**11 + x**10*(-40040*A*b**10 - 400400*B*a*b**9) + x**9*(-343200*A*a*b**9 - 15
44400*B*a**2*b**8) + x**8*(-1351350*A*a**2*b**8 - 3603600*B*a**3*b**7) + x**7*(-3203200*A*a**3*b**7 - 5605600*
B*a**4*b**6) + x**6*(-5045040*A*a**4*b**6 - 6054048*B*a**5*b**5) + x**5*(-5503680*A*a**5*b**5 - 4586400*B*a**6
*b**4) + x**4*(-4204200*A*a**6*b**4 - 2402400*B*a**7*b**3) + x**3*(-2217600*A*a**7*b**3 - 831600*B*a**8*b**2)
+ x**2*(-772200*A*a**8*b**2 - 171600*B*a**9*b) + x*(-160160*A*a**9*b - 16016*B*a**10))/(240240*x**16)

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