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

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

[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)

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Rubi [A]  time = 0.0224224, antiderivative size = 72, normalized size of antiderivative = 1., 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} \[ -\frac{b (a+b x)^{11} (2 A b-13 a B)}{1716 a^3 x^{11}}+\frac{(a+b x)^{11} (2 A b-13 a B)}{156 a^2 x^{12}}-\frac{A (a+b x)^{11}}{13 a x^{13}} \]

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 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^{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*}

Mathematica [B]  time = 0.0525338, size = 202, normalized size = 2.81 \[ -\frac{702 a^8 b^2 x^2 (10 A+11 B x)+2288 a^7 b^3 x^3 (9 A+10 B x)+5005 a^6 b^4 x^4 (8 A+9 B x)+7722 a^5 b^5 x^5 (7 A+8 B x)+8580 a^4 b^6 x^6 (6 A+7 B x)+6864 a^3 b^7 x^7 (5 A+6 B x)+3861 a^2 b^8 x^8 (4 A+5 B x)+130 a^9 b x (11 A+12 B x)+11 a^{10} (12 A+13 B x)+1430 a b^9 x^9 (3 A+4 B x)+286 b^{10} x^{10} (2 A+3 B x)}{1716 x^{13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/(1716*x^13)

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Maple [B]  time = 0.007, size = 208, normalized size = 2.9 \begin{align*} -{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{3\,{x}^{3}}}-3\,{\frac{{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{{x}^{5}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{12\,{x}^{12}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{4\,{x}^{4}}}-{\frac{21\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{4\,{x}^{8}}}-{\frac{B{b}^{10}}{2\,{x}^{2}}}-{\frac{A{a}^{10}}{13\,{x}^{13}}}-5\,{\frac{{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{{x}^{6}}}-6\,{\frac{{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{{x}^{7}}}-{\frac{10\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{3\,{x}^{9}}}-{\frac{3\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{2\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04916, size = 328, normalized size = 4.56 \begin{align*} -\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{align*}

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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Fricas [B]  time = 1.39208, size = 562, normalized size = 7.81 \begin{align*} -\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{align*}

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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Sympy [B]  time = 73.6814, size = 245, normalized size = 3.4 \begin{align*} - \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{align*}

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*(514
80*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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Giac [B]  time = 1.17201, size = 328, normalized size = 4.56 \begin{align*} -\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{align*}

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