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

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

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Rubi [A]  time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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^{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*}

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Mathematica [A]  time = 0.07, size = 202, normalized size = 2.00 \begin {gather*} -\frac {66 a^{10} (13 A+14 B x)+770 a^9 b x (12 A+13 B x)+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)+6006 a b^9 x^9 (4 A+5 B x)+1001 b^{10} x^{10} (3 A+4 B x)}{12012 x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12012*(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) + 3432
0*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))/x^14

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.08, size = 243, normalized size = 2.41 \begin {gather*} -\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 {gather*}

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

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giac [B]  time = 0.78, size = 243, normalized size = 2.41 \begin {gather*} -\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 {gather*}

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

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maple [B]  time = 0.01, size = 208, normalized size = 2.06 \begin {gather*} -\frac {B \,b^{10}}{3 x^{3}}-\frac {\left (A b +10 B a \right ) b^{9}}{4 x^{4}}-\frac {\left (2 A b +9 B a \right ) a \,b^{8}}{x^{5}}-\frac {5 \left (3 A b +8 B a \right ) a^{2} b^{7}}{2 x^{6}}-\frac {30 \left (4 A b +7 B a \right ) a^{3} b^{6}}{7 x^{7}}-\frac {21 \left (5 A b +6 B a \right ) a^{4} b^{5}}{4 x^{8}}-\frac {14 \left (6 A b +5 B a \right ) a^{5} b^{4}}{3 x^{9}}-\frac {3 \left (7 A b +4 B a \right ) a^{6} b^{3}}{x^{10}}-\frac {15 \left (8 A b +3 B a \right ) a^{7} b^{2}}{11 x^{11}}-\frac {A \,a^{10}}{14 x^{14}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{12 x^{12}}-\frac {\left (10 A b +B a \right ) a^{9}}{13 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.12, size = 243, normalized size = 2.41 \begin {gather*} -\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 {gather*}

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

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mupad [B]  time = 0.14, size = 235, normalized size = 2.33 \begin {gather*} -\frac {x\,\left (\frac {B\,a^{10}}{13}+\frac {10\,A\,b\,a^9}{13}\right )+\frac {A\,a^{10}}{14}+x^9\,\left (9\,B\,a^2\,b^8+2\,A\,a\,b^9\right )+x^2\,\left (\frac {5\,B\,a^9\,b}{6}+\frac {15\,A\,a^8\,b^2}{4}\right )+x^{10}\,\left (\frac {A\,b^{10}}{4}+\frac {5\,B\,a\,b^9}{2}\right )+x^4\,\left (12\,B\,a^7\,b^3+21\,A\,a^6\,b^4\right )+x^8\,\left (20\,B\,a^3\,b^7+\frac {15\,A\,a^2\,b^8}{2}\right )+x^5\,\left (\frac {70\,B\,a^6\,b^4}{3}+28\,A\,a^5\,b^5\right )+x^7\,\left (30\,B\,a^4\,b^6+\frac {120\,A\,a^3\,b^7}{7}\right )+x^6\,\left (\frac {63\,B\,a^5\,b^5}{2}+\frac {105\,A\,a^4\,b^6}{4}\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{11}+\frac {120\,A\,a^7\,b^3}{11}\right )+\frac {B\,b^{10}\,x^{11}}{3}}{x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^15,x)

[Out]

-(x*((B*a^10)/13 + (10*A*a^9*b)/13) + (A*a^10)/14 + x^9*(9*B*a^2*b^8 + 2*A*a*b^9) + x^2*((15*A*a^8*b^2)/4 + (5
*B*a^9*b)/6) + x^10*((A*b^10)/4 + (5*B*a*b^9)/2) + x^4*(21*A*a^6*b^4 + 12*B*a^7*b^3) + x^8*((15*A*a^2*b^8)/2 +
20*B*a^3*b^7) + x^5*(28*A*a^5*b^5 + (70*B*a^6*b^4)/3) + x^7*((120*A*a^3*b^7)/7 + 30*B*a^4*b^6) + x^6*((105*A*
a^4*b^6)/4 + (63*B*a^5*b^5)/2) + x^3*((120*A*a^7*b^3)/11 + (45*B*a^8*b^2)/11) + (B*b^10*x^11)/3)/x^14

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sympy [B]  time = 52.24, size = 260, normalized size = 2.57 \begin {gather*} \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 {gather*}

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)

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