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

Optimal. Leaf size=76 \[ \frac{b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac{b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac{(a+b x)^{11}}{14 a x^{14}} \]

[Out]

-(a + b*x)^11/(14*a*x^14) + (3*b*(a + b*x)^11)/(182*a^2*x^13) - (b^2*(a + b*x)^11)/(364*a^3*x^12) + (b^3*(a +
b*x)^11)/(4004*a^4*x^11)

________________________________________________________________________________________

Rubi [A]  time = 0.0170879, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac{b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac{(a+b x)^{11}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^15,x]

[Out]

-(a + b*x)^11/(14*a*x^14) + (3*b*(a + b*x)^11)/(182*a^2*x^13) - (b^2*(a + b*x)^11)/(364*a^3*x^12) + (b^3*(a +
b*x)^11)/(4004*a^4*x^11)

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}}{x^{15}} \, dx &=-\frac{(a+b x)^{11}}{14 a x^{14}}-\frac{(3 b) \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{14 a}\\ &=-\frac{(a+b x)^{11}}{14 a x^{14}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}+\frac{\left (3 b^2\right ) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{91 a^2}\\ &=-\frac{(a+b x)^{11}}{14 a x^{14}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac{b^2 (a+b x)^{11}}{364 a^3 x^{12}}-\frac{b^3 \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{364 a^3}\\ &=-\frac{(a+b x)^{11}}{14 a x^{14}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac{b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac{b^3 (a+b x)^{11}}{4004 a^4 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0112619, size = 128, normalized size = 1.68 \[ -\frac{15 a^8 b^2}{4 x^{12}}-\frac{120 a^7 b^3}{11 x^{11}}-\frac{21 a^6 b^4}{x^{10}}-\frac{28 a^5 b^5}{x^9}-\frac{105 a^4 b^6}{4 x^8}-\frac{120 a^3 b^7}{7 x^7}-\frac{15 a^2 b^8}{2 x^6}-\frac{10 a^9 b}{13 x^{13}}-\frac{a^{10}}{14 x^{14}}-\frac{2 a b^9}{x^5}-\frac{b^{10}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^15,x]

[Out]

-a^10/(14*x^14) - (10*a^9*b)/(13*x^13) - (15*a^8*b^2)/(4*x^12) - (120*a^7*b^3)/(11*x^11) - (21*a^6*b^4)/x^10 -
(28*a^5*b^5)/x^9 - (105*a^4*b^6)/(4*x^8) - (120*a^3*b^7)/(7*x^7) - (15*a^2*b^8)/(2*x^6) - (2*a*b^9)/x^5 - b^1
0/(4*x^4)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 113, normalized size = 1.5 \begin{align*} -21\,{\frac{{a}^{6}{b}^{4}}{{x}^{10}}}-{\frac{15\,{a}^{8}{b}^{2}}{4\,{x}^{12}}}-2\,{\frac{a{b}^{9}}{{x}^{5}}}-{\frac{120\,{a}^{7}{b}^{3}}{11\,{x}^{11}}}-{\frac{{b}^{10}}{4\,{x}^{4}}}-{\frac{15\,{a}^{2}{b}^{8}}{2\,{x}^{6}}}-{\frac{105\,{a}^{4}{b}^{6}}{4\,{x}^{8}}}-{\frac{10\,{a}^{9}b}{13\,{x}^{13}}}-{\frac{120\,{a}^{3}{b}^{7}}{7\,{x}^{7}}}-28\,{\frac{{a}^{5}{b}^{5}}{{x}^{9}}}-{\frac{{a}^{10}}{14\,{x}^{14}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-21*a^6*b^4/x^10-15/4*a^8*b^2/x^12-2*a*b^9/x^5-120/11*a^7*b^3/x^11-1/4*b^10/x^4-15/2*a^2*b^8/x^6-105/4*a^4*b^6
/x^8-10/13*a^9*b/x^13-120/7*a^3*b^7/x^7-28*a^5*b^5/x^9-1/14*a^10/x^14

________________________________________________________________________________________

Maxima [A]  time = 1.00864, size = 151, normalized size = 1.99 \begin{align*} -\frac{1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7 + 105105*a^4*b^6*x^6 + 112112
*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

________________________________________________________________________________________

Fricas [A]  time = 1.40059, size = 292, normalized size = 3.84 \begin{align*} -\frac{1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7 + 105105*a^4*b^6*x^6 + 112112
*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

________________________________________________________________________________________

Sympy [A]  time = 1.87377, size = 121, normalized size = 1.59 \begin{align*} - \frac{286 a^{10} + 3080 a^{9} b x + 15015 a^{8} b^{2} x^{2} + 43680 a^{7} b^{3} x^{3} + 84084 a^{6} b^{4} x^{4} + 112112 a^{5} b^{5} x^{5} + 105105 a^{4} b^{6} x^{6} + 68640 a^{3} b^{7} x^{7} + 30030 a^{2} b^{8} x^{8} + 8008 a b^{9} x^{9} + 1001 b^{10} x^{10}}{4004 x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(286*a**10 + 3080*a**9*b*x + 15015*a**8*b**2*x**2 + 43680*a**7*b**3*x**3 + 84084*a**6*b**4*x**4 + 112112*a**5
*b**5*x**5 + 105105*a**4*b**6*x**6 + 68640*a**3*b**7*x**7 + 30030*a**2*b**8*x**8 + 8008*a*b**9*x**9 + 1001*b**
10*x**10)/(4004*x**14)

________________________________________________________________________________________

Giac [A]  time = 1.14972, size = 151, normalized size = 1.99 \begin{align*} -\frac{1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7 + 105105*a^4*b^6*x^6 + 112112
*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14