3.165 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{18}} \, dx\)

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

[Out]

-(A*(a + b*x)^11)/(17*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)

________________________________________________________________________________________

Rubi [A]  time = 0.0767778, antiderivative size = 188, normalized size of antiderivative = 1., 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 = {78, 45, 37} \[ -\frac{b^5 (a+b x)^{11} (6 A b-17 a B)}{816816 a^7 x^{11}}+\frac{b^4 (a+b x)^{11} (6 A b-17 a B)}{74256 a^6 x^{12}}-\frac{b^3 (a+b x)^{11} (6 A b-17 a B)}{12376 a^5 x^{13}}+\frac{b^2 (a+b x)^{11} (6 A b-17 a B)}{2856 a^4 x^{14}}-\frac{b (a+b x)^{11} (6 A b-17 a B)}{816 a^3 x^{15}}+\frac{(a+b x)^{11} (6 A b-17 a B)}{272 a^2 x^{16}}-\frac{A (a+b x)^{11}}{17 a x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^18,x]

[Out]

-(A*(a + b*x)^11)/(17*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 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]))))

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 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]

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 = 0.0668163, size = 222, normalized size = 1.18 \[ -\frac{3 a^8 b^2 (14 A+15 B x)}{14 x^{15}}-\frac{60 a^7 b^3 (13 A+14 B x)}{91 x^{14}}-\frac{35 a^6 b^4 (12 A+13 B x)}{26 x^{13}}-\frac{21 a^5 b^5 (11 A+12 B x)}{11 x^{12}}-\frac{21 a^4 b^6 (10 A+11 B x)}{11 x^{11}}-\frac{4 a^3 b^7 (9 A+10 B x)}{3 x^{10}}-\frac{5 a^2 b^8 (8 A+9 B x)}{8 x^9}-\frac{a^9 b (15 A+16 B x)}{24 x^{16}}-\frac{a^{10} (16 A+17 B x)}{272 x^{17}}-\frac{5 a b^9 (7 A+8 B x)}{28 x^8}-\frac{b^{10} (6 A+7 B x)}{42 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^10*(6*A + 7*B*x))/(42*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 = 0.009, size = 208, normalized size = 1.1 \begin{align*} -{\frac{7\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{2\,{x}^{12}}}-{\frac{{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{3\,{x}^{15}}}-{\frac{42\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{11\,{x}^{11}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{14\,{x}^{14}}}-{\frac{A{a}^{10}}{17\,{x}^{17}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{16\,{x}^{16}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{8\,{x}^{8}}}-{\frac{30\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{13\,{x}^{13}}}-{\frac{B{b}^{10}}{6\,{x}^{6}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{7\,{x}^{7}}}-{\frac{5\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{3\,{x}^{9}}}-3\,{\frac{{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02133, size = 328, normalized size = 1.74 \begin{align*} -\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{align*}

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

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Fricas [A]  time = 1.43826, size = 606, normalized size = 3.22 \begin{align*} -\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{align*}

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 = 1.16488, size = 328, normalized size = 1.74 \begin{align*} -\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{align*}

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