3.2.47 \(\int \frac {(a+b x)^{10} (A+B x)}{x^{14}} \, dx\)

Optimal. Leaf size=72 \[ -\frac {b (a+b x)^{11} (2 A b-13 a B)}{1716 a^3 x^{11}}+\frac {(a+b x)^{11} (2 A b-13 a B)}{156 a^2 x^{12}}-\frac {A (a+b x)^{11}}{13 a x^{13}} \]

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, 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 (a+b x)^{11} (2 A b-13 a B)}{1716 a^3 x^{11}}+\frac {(a+b x)^{11} (2 A b-13 a B)}{156 a^2 x^{12}}-\frac {A (a+b x)^{11}}{13 a x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^14,x]

[Out]

-(A*(a + b*x)^11)/(13*a*x^13) + ((2*A*b - 13*a*B)*(a + b*x)^11)/(156*a^2*x^12) - (b*(2*A*b - 13*a*B)*(a + b*x)
^11)/(1716*a^3*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^{14}} \, dx &=-\frac {A (a+b x)^{11}}{13 a x^{13}}+\frac {(-2 A b+13 a B) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{13 a}\\ &=-\frac {A (a+b x)^{11}}{13 a x^{13}}+\frac {(2 A b-13 a B) (a+b x)^{11}}{156 a^2 x^{12}}+\frac {(b (2 A b-13 a B)) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{156 a^2}\\ &=-\frac {A (a+b x)^{11}}{13 a x^{13}}+\frac {(2 A b-13 a B) (a+b x)^{11}}{156 a^2 x^{12}}-\frac {b (2 A b-13 a B) (a+b x)^{11}}{1716 a^3 x^{11}}\\ \end {align*}

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Mathematica [B]  time = 0.08, size = 202, normalized size = 2.81 \begin {gather*} -\frac {11 a^{10} (12 A+13 B x)+130 a^9 b x (11 A+12 B x)+702 a^8 b^2 x^2 (10 A+11 B x)+2288 a^7 b^3 x^3 (9 A+10 B x)+5005 a^6 b^4 x^4 (8 A+9 B x)+7722 a^5 b^5 x^5 (7 A+8 B x)+8580 a^4 b^6 x^6 (6 A+7 B x)+6864 a^3 b^7 x^7 (5 A+6 B x)+3861 a^2 b^8 x^8 (4 A+5 B x)+1430 a b^9 x^9 (3 A+4 B x)+286 b^{10} x^{10} (2 A+3 B x)}{1716 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^14,x]

[Out]

-1/1716*(286*b^10*x^10*(2*A + 3*B*x) + 1430*a*b^9*x^9*(3*A + 4*B*x) + 3861*a^2*b^8*x^8*(4*A + 5*B*x) + 6864*a^
3*b^7*x^7*(5*A + 6*B*x) + 8580*a^4*b^6*x^6*(6*A + 7*B*x) + 7722*a^5*b^5*x^5*(7*A + 8*B*x) + 5005*a^6*b^4*x^4*(
8*A + 9*B*x) + 2288*a^7*b^3*x^3*(9*A + 10*B*x) + 702*a^8*b^2*x^2*(10*A + 11*B*x) + 130*a^9*b*x*(11*A + 12*B*x)
+ 11*a^10*(12*A + 13*B*x))/x^13

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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^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/x^14,x]

[Out]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/x^14, x]

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fricas [B]  time = 0.95, size = 243, normalized size = 3.38 \begin {gather*} -\frac {858 \, B b^{10} x^{11} + 132 \, A a^{10} + 572 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2145 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 5148 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 8580 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 10296 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 9009 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5720 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2574 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 780 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 143 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{1716 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^14,x, algorithm="fricas")

[Out]

-1/1716*(858*B*b^10*x^11 + 132*A*a^10 + 572*(10*B*a*b^9 + A*b^10)*x^10 + 2145*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 +
5148*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 8580*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 10296*(6*B*a^5*b^5 + 5*A*a^4*b^6
)*x^6 + 9009*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5720*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2574*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*x^3 + 780*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 143*(B*a^10 + 10*A*a^9*b)*x)/x^13

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giac [B]  time = 1.17, size = 243, normalized size = 3.38 \begin {gather*} -\frac {858 \, B b^{10} x^{11} + 5720 \, B a b^{9} x^{10} + 572 \, A b^{10} x^{10} + 19305 \, B a^{2} b^{8} x^{9} + 4290 \, A a b^{9} x^{9} + 41184 \, B a^{3} b^{7} x^{8} + 15444 \, A a^{2} b^{8} x^{8} + 60060 \, B a^{4} b^{6} x^{7} + 34320 \, A a^{3} b^{7} x^{7} + 61776 \, B a^{5} b^{5} x^{6} + 51480 \, A a^{4} b^{6} x^{6} + 45045 \, B a^{6} b^{4} x^{5} + 54054 \, A a^{5} b^{5} x^{5} + 22880 \, B a^{7} b^{3} x^{4} + 40040 \, A a^{6} b^{4} x^{4} + 7722 \, B a^{8} b^{2} x^{3} + 20592 \, A a^{7} b^{3} x^{3} + 1560 \, B a^{9} b x^{2} + 7020 \, A a^{8} b^{2} x^{2} + 143 \, B a^{10} x + 1430 \, A a^{9} b x + 132 \, A a^{10}}{1716 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^14,x, algorithm="giac")

[Out]

-1/1716*(858*B*b^10*x^11 + 5720*B*a*b^9*x^10 + 572*A*b^10*x^10 + 19305*B*a^2*b^8*x^9 + 4290*A*a*b^9*x^9 + 4118
4*B*a^3*b^7*x^8 + 15444*A*a^2*b^8*x^8 + 60060*B*a^4*b^6*x^7 + 34320*A*a^3*b^7*x^7 + 61776*B*a^5*b^5*x^6 + 5148
0*A*a^4*b^6*x^6 + 45045*B*a^6*b^4*x^5 + 54054*A*a^5*b^5*x^5 + 22880*B*a^7*b^3*x^4 + 40040*A*a^6*b^4*x^4 + 7722
*B*a^8*b^2*x^3 + 20592*A*a^7*b^3*x^3 + 1560*B*a^9*b*x^2 + 7020*A*a^8*b^2*x^2 + 143*B*a^10*x + 1430*A*a^9*b*x +
132*A*a^10)/x^13

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maple [B]  time = 0.01, size = 208, normalized size = 2.89 \begin {gather*} -\frac {B \,b^{10}}{2 x^{2}}-\frac {\left (A b +10 B a \right ) b^{9}}{3 x^{3}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{4 x^{4}}-\frac {3 \left (3 A b +8 B a \right ) a^{2} b^{7}}{x^{5}}-\frac {5 \left (4 A b +7 B a \right ) a^{3} b^{6}}{x^{6}}-\frac {6 \left (5 A b +6 B a \right ) a^{4} b^{5}}{x^{7}}-\frac {21 \left (6 A b +5 B a \right ) a^{5} b^{4}}{4 x^{8}}-\frac {10 \left (7 A b +4 B a \right ) a^{6} b^{3}}{3 x^{9}}-\frac {3 \left (8 A b +3 B a \right ) a^{7} b^{2}}{2 x^{10}}-\frac {A \,a^{10}}{13 x^{13}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{11 x^{11}}-\frac {\left (10 A b +B a \right ) a^{9}}{12 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^14,x)

[Out]

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

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maxima [B]  time = 1.05, size = 243, normalized size = 3.38 \begin {gather*} -\frac {858 \, B b^{10} x^{11} + 132 \, A a^{10} + 572 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2145 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 5148 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 8580 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 10296 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 9009 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5720 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2574 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 780 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 143 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{1716 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^14,x, algorithm="maxima")

[Out]

-1/1716*(858*B*b^10*x^11 + 132*A*a^10 + 572*(10*B*a*b^9 + A*b^10)*x^10 + 2145*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 +
5148*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 8580*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 10296*(6*B*a^5*b^5 + 5*A*a^4*b^6
)*x^6 + 9009*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5720*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2574*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*x^3 + 780*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 143*(B*a^10 + 10*A*a^9*b)*x)/x^13

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mupad [B]  time = 0.41, size = 235, normalized size = 3.26 \begin {gather*} -\frac {x\,\left (\frac {B\,a^{10}}{12}+\frac {5\,A\,b\,a^9}{6}\right )+\frac {A\,a^{10}}{13}+x^9\,\left (\frac {45\,B\,a^2\,b^8}{4}+\frac {5\,A\,a\,b^9}{2}\right )+x^2\,\left (\frac {10\,B\,a^9\,b}{11}+\frac {45\,A\,a^8\,b^2}{11}\right )+x^{10}\,\left (\frac {A\,b^{10}}{3}+\frac {10\,B\,a\,b^9}{3}\right )+x^3\,\left (\frac {9\,B\,a^8\,b^2}{2}+12\,A\,a^7\,b^3\right )+x^8\,\left (24\,B\,a^3\,b^7+9\,A\,a^2\,b^8\right )+x^7\,\left (35\,B\,a^4\,b^6+20\,A\,a^3\,b^7\right )+x^6\,\left (36\,B\,a^5\,b^5+30\,A\,a^4\,b^6\right )+x^4\,\left (\frac {40\,B\,a^7\,b^3}{3}+\frac {70\,A\,a^6\,b^4}{3}\right )+x^5\,\left (\frac {105\,B\,a^6\,b^4}{4}+\frac {63\,A\,a^5\,b^5}{2}\right )+\frac {B\,b^{10}\,x^{11}}{2}}{x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^14,x)

[Out]

-(x*((B*a^10)/12 + (5*A*a^9*b)/6) + (A*a^10)/13 + x^9*((45*B*a^2*b^8)/4 + (5*A*a*b^9)/2) + x^2*((45*A*a^8*b^2)
/11 + (10*B*a^9*b)/11) + x^10*((A*b^10)/3 + (10*B*a*b^9)/3) + x^3*(12*A*a^7*b^3 + (9*B*a^8*b^2)/2) + x^8*(9*A*
a^2*b^8 + 24*B*a^3*b^7) + x^7*(20*A*a^3*b^7 + 35*B*a^4*b^6) + x^6*(30*A*a^4*b^6 + 36*B*a^5*b^5) + x^4*((70*A*a
^6*b^4)/3 + (40*B*a^7*b^3)/3) + x^5*((63*A*a^5*b^5)/2 + (105*B*a^6*b^4)/4) + (B*b^10*x^11)/2)/x^13

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sympy [B]  time = 37.03, size = 260, normalized size = 3.61 \begin {gather*} \frac {- 132 A a^{10} - 858 B b^{10} x^{11} + x^{10} \left (- 572 A b^{10} - 5720 B a b^{9}\right ) + x^{9} \left (- 4290 A a b^{9} - 19305 B a^{2} b^{8}\right ) + x^{8} \left (- 15444 A a^{2} b^{8} - 41184 B a^{3} b^{7}\right ) + x^{7} \left (- 34320 A a^{3} b^{7} - 60060 B a^{4} b^{6}\right ) + x^{6} \left (- 51480 A a^{4} b^{6} - 61776 B a^{5} b^{5}\right ) + x^{5} \left (- 54054 A a^{5} b^{5} - 45045 B a^{6} b^{4}\right ) + x^{4} \left (- 40040 A a^{6} b^{4} - 22880 B a^{7} b^{3}\right ) + x^{3} \left (- 20592 A a^{7} b^{3} - 7722 B a^{8} b^{2}\right ) + x^{2} \left (- 7020 A a^{8} b^{2} - 1560 B a^{9} b\right ) + x \left (- 1430 A a^{9} b - 143 B a^{10}\right )}{1716 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**14,x)

[Out]

(-132*A*a**10 - 858*B*b**10*x**11 + x**10*(-572*A*b**10 - 5720*B*a*b**9) + x**9*(-4290*A*a*b**9 - 19305*B*a**2
*b**8) + x**8*(-15444*A*a**2*b**8 - 41184*B*a**3*b**7) + x**7*(-34320*A*a**3*b**7 - 60060*B*a**4*b**6) + x**6*
(-51480*A*a**4*b**6 - 61776*B*a**5*b**5) + x**5*(-54054*A*a**5*b**5 - 45045*B*a**6*b**4) + x**4*(-40040*A*a**6
*b**4 - 22880*B*a**7*b**3) + x**3*(-20592*A*a**7*b**3 - 7722*B*a**8*b**2) + x**2*(-7020*A*a**8*b**2 - 1560*B*a
**9*b) + x*(-1430*A*a**9*b - 143*B*a**10))/(1716*x**13)

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