3.163 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{16}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac{b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac{b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac{(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac{A (a+b x)^{11}}{15 a x^{15}} \]

[Out]

-(A*(a + b*x)^11)/(15*a*x^15) + ((4*A*b - 15*a*B)*(a + b*x)^11)/(210*a^2*x^14) - (b*(4*A*b - 15*a*B)*(a + b*x)
^11)/(910*a^3*x^13) + (b^2*(4*A*b - 15*a*B)*(a + b*x)^11)/(5460*a^4*x^12) - (b^3*(4*A*b - 15*a*B)*(a + b*x)^11
)/(60060*a^5*x^11)

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Rubi [A]  time = 0.0470627, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ -\frac{b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac{b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac{b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac{(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac{A (a+b x)^{11}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^16,x]

[Out]

-(A*(a + b*x)^11)/(15*a*x^15) + ((4*A*b - 15*a*B)*(a + b*x)^11)/(210*a^2*x^14) - (b*(4*A*b - 15*a*B)*(a + b*x)
^11)/(910*a^3*x^13) + (b^2*(4*A*b - 15*a*B)*(a + b*x)^11)/(5460*a^4*x^12) - (b^3*(4*A*b - 15*a*B)*(a + b*x)^11
)/(60060*a^5*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^{16}} \, dx &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(-4 A b+15 a B) \int \frac{(a+b x)^{10}}{x^{15}} \, dx}{15 a}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}+\frac{(b (4 A b-15 a B)) \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{70 a^2}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac{b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}-\frac{\left (b^2 (4 A b-15 a B)\right ) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{455 a^3}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac{b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac{b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}+\frac{\left (b^3 (4 A b-15 a B)\right ) \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{5460 a^4}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac{b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac{b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}-\frac{b^3 (4 A b-15 a B) (a+b x)^{11}}{60060 a^5 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.052496, size = 202, normalized size = 1.55 \[ -\frac{17325 a^8 b^2 x^2 (12 A+13 B x)+54600 a^7 b^3 x^3 (11 A+12 B x)+114660 a^6 b^4 x^4 (10 A+11 B x)+168168 a^5 b^5 x^5 (9 A+10 B x)+175175 a^4 b^6 x^6 (8 A+9 B x)+128700 a^3 b^7 x^7 (7 A+8 B x)+64350 a^2 b^8 x^8 (6 A+7 B x)+3300 a^9 b x (13 A+14 B x)+286 a^{10} (14 A+15 B x)+20020 a b^9 x^9 (5 A+6 B x)+3003 b^{10} x^{10} (4 A+5 B x)}{60060 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^16,x]

[Out]

-(3003*b^10*x^10*(4*A + 5*B*x) + 20020*a*b^9*x^9*(5*A + 6*B*x) + 64350*a^2*b^8*x^8*(6*A + 7*B*x) + 128700*a^3*
b^7*x^7*(7*A + 8*B*x) + 175175*a^4*b^6*x^6*(8*A + 9*B*x) + 168168*a^5*b^5*x^5*(9*A + 10*B*x) + 114660*a^6*b^4*
x^4*(10*A + 11*B*x) + 54600*a^7*b^3*x^3*(11*A + 12*B*x) + 17325*a^8*b^2*x^2*(12*A + 13*B*x) + 3300*a^9*b*x*(13
*A + 14*B*x) + 286*a^10*(14*A + 15*B*x))/(60060*x^15)

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Maple [A]  time = 0.006, size = 208, normalized size = 1.6 \begin{align*} -{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{5\,{x}^{5}}}-{\frac{5\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{4\,{x}^{12}}}-{\frac{A{a}^{10}}{15\,{x}^{15}}}-{\frac{30\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{11\,{x}^{11}}}-{\frac{B{b}^{10}}{4\,{x}^{4}}}-{\frac{15\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{4\,{x}^{8}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{6\,{x}^{6}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{15\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{7\,{x}^{7}}}-{\frac{14\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{3\,{x}^{9}}}-{\frac{21\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,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^16,x)

[Out]

-1/5*b^9*(A*b+10*B*a)/x^5-5/4*a^7*b^2*(8*A*b+3*B*a)/x^12-1/15*A*a^10/x^15-30/11*a^6*b^3*(7*A*b+4*B*a)/x^11-1/4
*B*b^10/x^4-15/4*a^3*b^6*(4*A*b+7*B*a)/x^8-5/13*a^8*b*(9*A*b+2*B*a)/x^13-5/6*a*b^8*(2*A*b+9*B*a)/x^6-1/14*a^9*
(10*A*b+B*a)/x^14-15/7*a^2*b^7*(3*A*b+8*B*a)/x^7-14/3*a^4*b^5*(5*A*b+6*B*a)/x^9-21/5*a^5*b^4*(6*A*b+5*B*a)/x^1
0

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Maxima [B]  time = 1.02173, size = 328, normalized size = 2.52 \begin{align*} -\frac{15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^16,x, algorithm="maxima")

[Out]

-1/60060*(15015*B*b^10*x^11 + 4004*A*a^10 + 12012*(10*B*a*b^9 + A*b^10)*x^10 + 50050*(9*B*a^2*b^8 + 2*A*a*b^9)
*x^9 + 128700*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 225225*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 280280*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 252252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 163800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 75075*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 23100*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4290*(B*a^10 + 10*A*a^9*b)*x)/x^15

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Fricas [B]  time = 1.46514, size = 589, normalized size = 4.53 \begin{align*} -\frac{15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^16,x, algorithm="fricas")

[Out]

-1/60060*(15015*B*b^10*x^11 + 4004*A*a^10 + 12012*(10*B*a*b^9 + A*b^10)*x^10 + 50050*(9*B*a^2*b^8 + 2*A*a*b^9)
*x^9 + 128700*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 225225*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 280280*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 252252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 163800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 75075*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 23100*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4290*(B*a^10 + 10*A*a^9*b)*x)/x^15

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Sympy [A]  time = 146.726, size = 245, normalized size = 1.88 \begin{align*} - \frac{4004 A a^{10} + 15015 B b^{10} x^{11} + x^{10} \left (12012 A b^{10} + 120120 B a b^{9}\right ) + x^{9} \left (100100 A a b^{9} + 450450 B a^{2} b^{8}\right ) + x^{8} \left (386100 A a^{2} b^{8} + 1029600 B a^{3} b^{7}\right ) + x^{7} \left (900900 A a^{3} b^{7} + 1576575 B a^{4} b^{6}\right ) + x^{6} \left (1401400 A a^{4} b^{6} + 1681680 B a^{5} b^{5}\right ) + x^{5} \left (1513512 A a^{5} b^{5} + 1261260 B a^{6} b^{4}\right ) + x^{4} \left (1146600 A a^{6} b^{4} + 655200 B a^{7} b^{3}\right ) + x^{3} \left (600600 A a^{7} b^{3} + 225225 B a^{8} b^{2}\right ) + x^{2} \left (207900 A a^{8} b^{2} + 46200 B a^{9} b\right ) + x \left (42900 A a^{9} b + 4290 B a^{10}\right )}{60060 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**16,x)

[Out]

-(4004*A*a**10 + 15015*B*b**10*x**11 + x**10*(12012*A*b**10 + 120120*B*a*b**9) + x**9*(100100*A*a*b**9 + 45045
0*B*a**2*b**8) + x**8*(386100*A*a**2*b**8 + 1029600*B*a**3*b**7) + x**7*(900900*A*a**3*b**7 + 1576575*B*a**4*b
**6) + x**6*(1401400*A*a**4*b**6 + 1681680*B*a**5*b**5) + x**5*(1513512*A*a**5*b**5 + 1261260*B*a**6*b**4) + x
**4*(1146600*A*a**6*b**4 + 655200*B*a**7*b**3) + x**3*(600600*A*a**7*b**3 + 225225*B*a**8*b**2) + x**2*(207900
*A*a**8*b**2 + 46200*B*a**9*b) + x*(42900*A*a**9*b + 4290*B*a**10))/(60060*x**15)

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Giac [B]  time = 1.21491, size = 328, normalized size = 2.52 \begin{align*} -\frac{15015 \, B b^{10} x^{11} + 120120 \, B a b^{9} x^{10} + 12012 \, A b^{10} x^{10} + 450450 \, B a^{2} b^{8} x^{9} + 100100 \, A a b^{9} x^{9} + 1029600 \, B a^{3} b^{7} x^{8} + 386100 \, A a^{2} b^{8} x^{8} + 1576575 \, B a^{4} b^{6} x^{7} + 900900 \, A a^{3} b^{7} x^{7} + 1681680 \, B a^{5} b^{5} x^{6} + 1401400 \, A a^{4} b^{6} x^{6} + 1261260 \, B a^{6} b^{4} x^{5} + 1513512 \, A a^{5} b^{5} x^{5} + 655200 \, B a^{7} b^{3} x^{4} + 1146600 \, A a^{6} b^{4} x^{4} + 225225 \, B a^{8} b^{2} x^{3} + 600600 \, A a^{7} b^{3} x^{3} + 46200 \, B a^{9} b x^{2} + 207900 \, A a^{8} b^{2} x^{2} + 4290 \, B a^{10} x + 42900 \, A a^{9} b x + 4004 \, A a^{10}}{60060 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^16,x, algorithm="giac")

[Out]

-1/60060*(15015*B*b^10*x^11 + 120120*B*a*b^9*x^10 + 12012*A*b^10*x^10 + 450450*B*a^2*b^8*x^9 + 100100*A*a*b^9*
x^9 + 1029600*B*a^3*b^7*x^8 + 386100*A*a^2*b^8*x^8 + 1576575*B*a^4*b^6*x^7 + 900900*A*a^3*b^7*x^7 + 1681680*B*
a^5*b^5*x^6 + 1401400*A*a^4*b^6*x^6 + 1261260*B*a^6*b^4*x^5 + 1513512*A*a^5*b^5*x^5 + 655200*B*a^7*b^3*x^4 + 1
146600*A*a^6*b^4*x^4 + 225225*B*a^8*b^2*x^3 + 600600*A*a^7*b^3*x^3 + 46200*B*a^9*b*x^2 + 207900*A*a^8*b^2*x^2
+ 4290*B*a^10*x + 42900*A*a^9*b*x + 4004*A*a^10)/x^15