3.162 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{15}} \, dx\)

Optimal. Leaf size=101 \[ \frac{b^2 (a+b x)^{11} (3 A b-14 a B)}{12012 a^4 x^{11}}-\frac{b (a+b x)^{11} (3 A b-14 a B)}{1092 a^3 x^{12}}+\frac{(a+b x)^{11} (3 A b-14 a B)}{182 a^2 x^{13}}-\frac{A (a+b x)^{11}}{14 a x^{14}} \]

[Out]

-(A*(a + b*x)^11)/(14*a*x^14) + ((3*A*b - 14*a*B)*(a + b*x)^11)/(182*a^2*x^13) - (b*(3*A*b - 14*a*B)*(a + b*x)
^11)/(1092*a^3*x^12) + (b^2*(3*A*b - 14*a*B)*(a + b*x)^11)/(12012*a^4*x^11)

________________________________________________________________________________________

Rubi [A]  time = 0.0344756, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ \frac{b^2 (a+b x)^{11} (3 A b-14 a B)}{12012 a^4 x^{11}}-\frac{b (a+b x)^{11} (3 A b-14 a B)}{1092 a^3 x^{12}}+\frac{(a+b x)^{11} (3 A b-14 a B)}{182 a^2 x^{13}}-\frac{A (a+b x)^{11}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^15,x]

[Out]

-(A*(a + b*x)^11)/(14*a*x^14) + ((3*A*b - 14*a*B)*(a + b*x)^11)/(182*a^2*x^13) - (b*(3*A*b - 14*a*B)*(a + b*x)
^11)/(1092*a^3*x^12) + (b^2*(3*A*b - 14*a*B)*(a + b*x)^11)/(12012*a^4*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^{15}} \, dx &=-\frac{A (a+b x)^{11}}{14 a x^{14}}+\frac{(-3 A b+14 a B) \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{14 a}\\ &=-\frac{A (a+b x)^{11}}{14 a x^{14}}+\frac{(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}+\frac{(b (3 A b-14 a B)) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{91 a^2}\\ &=-\frac{A (a+b x)^{11}}{14 a x^{14}}+\frac{(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}-\frac{b (3 A b-14 a B) (a+b x)^{11}}{1092 a^3 x^{12}}-\frac{\left (b^2 (3 A b-14 a B)\right ) \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{1092 a^3}\\ &=-\frac{A (a+b x)^{11}}{14 a x^{14}}+\frac{(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}-\frac{b (3 A b-14 a B) (a+b x)^{11}}{1092 a^3 x^{12}}+\frac{b^2 (3 A b-14 a B) (a+b x)^{11}}{12012 a^4 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0500587, size = 202, normalized size = 2. \[ -\frac{4095 a^8 b^2 x^2 (11 A+12 B x)+13104 a^7 b^3 x^3 (10 A+11 B x)+28028 a^6 b^4 x^4 (9 A+10 B x)+42042 a^5 b^5 x^5 (8 A+9 B x)+45045 a^4 b^6 x^6 (7 A+8 B x)+34320 a^3 b^7 x^7 (6 A+7 B x)+18018 a^2 b^8 x^8 (5 A+6 B x)+770 a^9 b x (12 A+13 B x)+66 a^{10} (13 A+14 B x)+6006 a b^9 x^9 (4 A+5 B x)+1001 b^{10} x^{10} (3 A+4 B x)}{12012 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^15,x]

[Out]

-(1001*b^10*x^10*(3*A + 4*B*x) + 6006*a*b^9*x^9*(4*A + 5*B*x) + 18018*a^2*b^8*x^8*(5*A + 6*B*x) + 34320*a^3*b^
7*x^7*(6*A + 7*B*x) + 45045*a^4*b^6*x^6*(7*A + 8*B*x) + 42042*a^5*b^5*x^5*(8*A + 9*B*x) + 28028*a^6*b^4*x^4*(9
*A + 10*B*x) + 13104*a^7*b^3*x^3*(10*A + 11*B*x) + 4095*a^8*b^2*x^2*(11*A + 12*B*x) + 770*a^9*b*x*(12*A + 13*B
*x) + 66*a^10*(13*A + 14*B*x))/(12012*x^14)

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 208, normalized size = 2.1 \begin{align*} -{\frac{B{b}^{10}}{3\,{x}^{3}}}-{\frac{a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{{x}^{5}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{12\,{x}^{12}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{11\,{x}^{11}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{4\,{x}^{4}}}-{\frac{21\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{4\,{x}^{8}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{2\,{x}^{6}}}-{\frac{30\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{7\,{x}^{7}}}-{\frac{14\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{3\,{x}^{9}}}-{\frac{A{a}^{10}}{14\,{x}^{14}}}-3\,{\frac{{a}^{6}{b}^{3} \left ( 7\,Ab+4\,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^15,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.03108, size = 328, normalized size = 3.25 \begin{align*} -\frac{4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^15,x, algorithm="maxima")

[Out]

-1/12012*(4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 12012*(9*B*a^2*b^8 + 2*A*a*b^9)*x^
9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 51480*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6*B*a^5*b^5 + 5*A*a
^4*b^6)*x^6 + 56056*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 16380*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^10 + 10*A*a^9*b)*x)/x^14

________________________________________________________________________________________

Fricas [B]  time = 1.5121, size = 575, normalized size = 5.69 \begin{align*} -\frac{4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^15,x, algorithm="fricas")

[Out]

-1/12012*(4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 12012*(9*B*a^2*b^8 + 2*A*a*b^9)*x^
9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 51480*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6*B*a^5*b^5 + 5*A*a
^4*b^6)*x^6 + 56056*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 16380*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^10 + 10*A*a^9*b)*x)/x^14

________________________________________________________________________________________

Sympy [B]  time = 110.354, size = 245, normalized size = 2.43 \begin{align*} - \frac{858 A a^{10} + 4004 B b^{10} x^{11} + x^{10} \left (3003 A b^{10} + 30030 B a b^{9}\right ) + x^{9} \left (24024 A a b^{9} + 108108 B a^{2} b^{8}\right ) + x^{8} \left (90090 A a^{2} b^{8} + 240240 B a^{3} b^{7}\right ) + x^{7} \left (205920 A a^{3} b^{7} + 360360 B a^{4} b^{6}\right ) + x^{6} \left (315315 A a^{4} b^{6} + 378378 B a^{5} b^{5}\right ) + x^{5} \left (336336 A a^{5} b^{5} + 280280 B a^{6} b^{4}\right ) + x^{4} \left (252252 A a^{6} b^{4} + 144144 B a^{7} b^{3}\right ) + x^{3} \left (131040 A a^{7} b^{3} + 49140 B a^{8} b^{2}\right ) + x^{2} \left (45045 A a^{8} b^{2} + 10010 B a^{9} b\right ) + x \left (9240 A a^{9} b + 924 B a^{10}\right )}{12012 x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**15,x)

[Out]

-(858*A*a**10 + 4004*B*b**10*x**11 + x**10*(3003*A*b**10 + 30030*B*a*b**9) + x**9*(24024*A*a*b**9 + 108108*B*a
**2*b**8) + x**8*(90090*A*a**2*b**8 + 240240*B*a**3*b**7) + x**7*(205920*A*a**3*b**7 + 360360*B*a**4*b**6) + x
**6*(315315*A*a**4*b**6 + 378378*B*a**5*b**5) + x**5*(336336*A*a**5*b**5 + 280280*B*a**6*b**4) + x**4*(252252*
A*a**6*b**4 + 144144*B*a**7*b**3) + x**3*(131040*A*a**7*b**3 + 49140*B*a**8*b**2) + x**2*(45045*A*a**8*b**2 +
10010*B*a**9*b) + x*(9240*A*a**9*b + 924*B*a**10))/(12012*x**14)

________________________________________________________________________________________

Giac [B]  time = 1.16046, size = 328, normalized size = 3.25 \begin{align*} -\frac{4004 \, B b^{10} x^{11} + 30030 \, B a b^{9} x^{10} + 3003 \, A b^{10} x^{10} + 108108 \, B a^{2} b^{8} x^{9} + 24024 \, A a b^{9} x^{9} + 240240 \, B a^{3} b^{7} x^{8} + 90090 \, A a^{2} b^{8} x^{8} + 360360 \, B a^{4} b^{6} x^{7} + 205920 \, A a^{3} b^{7} x^{7} + 378378 \, B a^{5} b^{5} x^{6} + 315315 \, A a^{4} b^{6} x^{6} + 280280 \, B a^{6} b^{4} x^{5} + 336336 \, A a^{5} b^{5} x^{5} + 144144 \, B a^{7} b^{3} x^{4} + 252252 \, A a^{6} b^{4} x^{4} + 49140 \, B a^{8} b^{2} x^{3} + 131040 \, A a^{7} b^{3} x^{3} + 10010 \, B a^{9} b x^{2} + 45045 \, A a^{8} b^{2} x^{2} + 924 \, B a^{10} x + 9240 \, A a^{9} b x + 858 \, A a^{10}}{12012 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^15,x, algorithm="giac")

[Out]

-1/12012*(4004*B*b^10*x^11 + 30030*B*a*b^9*x^10 + 3003*A*b^10*x^10 + 108108*B*a^2*b^8*x^9 + 24024*A*a*b^9*x^9
+ 240240*B*a^3*b^7*x^8 + 90090*A*a^2*b^8*x^8 + 360360*B*a^4*b^6*x^7 + 205920*A*a^3*b^7*x^7 + 378378*B*a^5*b^5*
x^6 + 315315*A*a^4*b^6*x^6 + 280280*B*a^6*b^4*x^5 + 336336*A*a^5*b^5*x^5 + 144144*B*a^7*b^3*x^4 + 252252*A*a^6
*b^4*x^4 + 49140*B*a^8*b^2*x^3 + 131040*A*a^7*b^3*x^3 + 10010*B*a^9*b*x^2 + 45045*A*a^8*b^2*x^2 + 924*B*a^10*x
+ 9240*A*a^9*b*x + 858*A*a^10)/x^14