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]

-(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)

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Rubi [A]  time = 0.0584872, antiderivative size = 159, normalized size of antiderivative = 1., 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 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^{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*}

Mathematica [A]  time = 0.0660365, size = 222, normalized size = 1.4 \[ -\frac{45 a^8 b^2 (13 A+14 B x)}{182 x^{14}}-\frac{10 a^7 b^3 (12 A+13 B x)}{13 x^{13}}-\frac{35 a^6 b^4 (11 A+12 B x)}{22 x^{12}}-\frac{126 a^5 b^5 (10 A+11 B x)}{55 x^{11}}-\frac{7 a^4 b^6 (9 A+10 B x)}{3 x^{10}}-\frac{5 a^3 b^7 (8 A+9 B x)}{3 x^9}-\frac{45 a^2 b^8 (7 A+8 B x)}{56 x^8}-\frac{a^9 b (14 A+15 B x)}{21 x^{15}}-\frac{a^{10} (15 A+16 B x)}{240 x^{16}}-\frac{5 a b^9 (6 A+7 B x)}{21 x^7}-\frac{b^{10} (5 A+6 B x)}{30 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^10*(5*A + 6*B*x))/(30*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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Maple [A]  time = 0.007, size = 208, normalized size = 1.3 \begin{align*} -{\frac{B{b}^{10}}{5\,{x}^{5}}}-{\frac{A{a}^{10}}{16\,{x}^{16}}}-{\frac{5\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{2\,{x}^{12}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{15\,{x}^{15}}}-{\frac{42\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{14\,{x}^{14}}}-{\frac{15\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{8\,{x}^{8}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{13\,{x}^{13}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{6\,{x}^{6}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{7\,{x}^{7}}}-{\frac{10\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{3\,{x}^{9}}}-{\frac{21\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{5\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02026, size = 328, normalized size = 2.06 \begin{align*} -\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}} \end{align*}

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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Fricas [A]  time = 1.41839, size = 597, normalized size = 3.75 \begin{align*} -\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}} \end{align*}

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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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**17,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.24773, size = 328, normalized size = 2.06 \begin{align*} -\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}} \end{align*}

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