3.132 \(\int x^2 (a+b x)^{10} \, dx\)

Optimal. Leaf size=47 \[ \frac{a^2 (a+b x)^{11}}{11 b^3}+\frac{(a+b x)^{13}}{13 b^3}-\frac{a (a+b x)^{12}}{6 b^3} \]

[Out]

(a^2*(a + b*x)^11)/(11*b^3) - (a*(a + b*x)^12)/(6*b^3) + (a + b*x)^13/(13*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0296192, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{a^2 (a+b x)^{11}}{11 b^3}+\frac{(a+b x)^{13}}{13 b^3}-\frac{a (a+b x)^{12}}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^10,x]

[Out]

(a^2*(a + b*x)^11)/(11*b^3) - (a*(a + b*x)^12)/(6*b^3) + (a + b*x)^13/(13*b^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 (a+b x)^{10} \, dx &=\int \left (\frac{a^2 (a+b x)^{10}}{b^2}-\frac{2 a (a+b x)^{11}}{b^2}+\frac{(a+b x)^{12}}{b^2}\right ) \, dx\\ &=\frac{a^2 (a+b x)^{11}}{11 b^3}-\frac{a (a+b x)^{12}}{6 b^3}+\frac{(a+b x)^{13}}{13 b^3}\\ \end{align*}

Mathematica [B]  time = 0.0030878, size = 126, normalized size = 2.68 \[ \frac{45}{11} a^2 b^8 x^{11}+12 a^3 b^7 x^{10}+\frac{70}{3} a^4 b^6 x^9+\frac{63}{2} a^5 b^5 x^8+30 a^6 b^4 x^7+20 a^7 b^3 x^6+9 a^8 b^2 x^5+\frac{5}{2} a^9 b x^4+\frac{a^{10} x^3}{3}+\frac{5}{6} a b^9 x^{12}+\frac{b^{10} x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^10,x]

[Out]

(a^10*x^3)/3 + (5*a^9*b*x^4)/2 + 9*a^8*b^2*x^5 + 20*a^7*b^3*x^6 + 30*a^6*b^4*x^7 + (63*a^5*b^5*x^8)/2 + (70*a^
4*b^6*x^9)/3 + 12*a^3*b^7*x^10 + (45*a^2*b^8*x^11)/11 + (5*a*b^9*x^12)/6 + (b^10*x^13)/13

________________________________________________________________________________________

Maple [B]  time = 0., size = 113, normalized size = 2.4 \begin{align*}{\frac{{b}^{10}{x}^{13}}{13}}+{\frac{5\,a{b}^{9}{x}^{12}}{6}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{11}}{11}}+12\,{a}^{3}{b}^{7}{x}^{10}+{\frac{70\,{a}^{4}{b}^{6}{x}^{9}}{3}}+{\frac{63\,{a}^{5}{b}^{5}{x}^{8}}{2}}+30\,{a}^{6}{b}^{4}{x}^{7}+20\,{a}^{7}{b}^{3}{x}^{6}+9\,{a}^{8}{b}^{2}{x}^{5}+{\frac{5\,{a}^{9}b{x}^{4}}{2}}+{\frac{{a}^{10}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^10,x)

[Out]

1/13*b^10*x^13+5/6*a*b^9*x^12+45/11*a^2*b^8*x^11+12*a^3*b^7*x^10+70/3*a^4*b^6*x^9+63/2*a^5*b^5*x^8+30*a^6*b^4*
x^7+20*a^7*b^3*x^6+9*a^8*b^2*x^5+5/2*a^9*b*x^4+1/3*a^10*x^3

________________________________________________________________________________________

Maxima [B]  time = 1.02456, size = 151, normalized size = 3.21 \begin{align*} \frac{1}{13} \, b^{10} x^{13} + \frac{5}{6} \, a b^{9} x^{12} + \frac{45}{11} \, a^{2} b^{8} x^{11} + 12 \, a^{3} b^{7} x^{10} + \frac{70}{3} \, a^{4} b^{6} x^{9} + \frac{63}{2} \, a^{5} b^{5} x^{8} + 30 \, a^{6} b^{4} x^{7} + 20 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac{5}{2} \, a^{9} b x^{4} + \frac{1}{3} \, a^{10} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/13*b^10*x^13 + 5/6*a*b^9*x^12 + 45/11*a^2*b^8*x^11 + 12*a^3*b^7*x^10 + 70/3*a^4*b^6*x^9 + 63/2*a^5*b^5*x^8 +
30*a^6*b^4*x^7 + 20*a^7*b^3*x^6 + 9*a^8*b^2*x^5 + 5/2*a^9*b*x^4 + 1/3*a^10*x^3

________________________________________________________________________________________

Fricas [B]  time = 1.70969, size = 258, normalized size = 5.49 \begin{align*} \frac{1}{13} x^{13} b^{10} + \frac{5}{6} x^{12} b^{9} a + \frac{45}{11} x^{11} b^{8} a^{2} + 12 x^{10} b^{7} a^{3} + \frac{70}{3} x^{9} b^{6} a^{4} + \frac{63}{2} x^{8} b^{5} a^{5} + 30 x^{7} b^{4} a^{6} + 20 x^{6} b^{3} a^{7} + 9 x^{5} b^{2} a^{8} + \frac{5}{2} x^{4} b a^{9} + \frac{1}{3} x^{3} a^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/13*x^13*b^10 + 5/6*x^12*b^9*a + 45/11*x^11*b^8*a^2 + 12*x^10*b^7*a^3 + 70/3*x^9*b^6*a^4 + 63/2*x^8*b^5*a^5 +
30*x^7*b^4*a^6 + 20*x^6*b^3*a^7 + 9*x^5*b^2*a^8 + 5/2*x^4*b*a^9 + 1/3*x^3*a^10

________________________________________________________________________________________

Sympy [B]  time = 0.105326, size = 128, normalized size = 2.72 \begin{align*} \frac{a^{10} x^{3}}{3} + \frac{5 a^{9} b x^{4}}{2} + 9 a^{8} b^{2} x^{5} + 20 a^{7} b^{3} x^{6} + 30 a^{6} b^{4} x^{7} + \frac{63 a^{5} b^{5} x^{8}}{2} + \frac{70 a^{4} b^{6} x^{9}}{3} + 12 a^{3} b^{7} x^{10} + \frac{45 a^{2} b^{8} x^{11}}{11} + \frac{5 a b^{9} x^{12}}{6} + \frac{b^{10} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)**10,x)

[Out]

a**10*x**3/3 + 5*a**9*b*x**4/2 + 9*a**8*b**2*x**5 + 20*a**7*b**3*x**6 + 30*a**6*b**4*x**7 + 63*a**5*b**5*x**8/
2 + 70*a**4*b**6*x**9/3 + 12*a**3*b**7*x**10 + 45*a**2*b**8*x**11/11 + 5*a*b**9*x**12/6 + b**10*x**13/13

________________________________________________________________________________________

Giac [B]  time = 1.19456, size = 151, normalized size = 3.21 \begin{align*} \frac{1}{13} \, b^{10} x^{13} + \frac{5}{6} \, a b^{9} x^{12} + \frac{45}{11} \, a^{2} b^{8} x^{11} + 12 \, a^{3} b^{7} x^{10} + \frac{70}{3} \, a^{4} b^{6} x^{9} + \frac{63}{2} \, a^{5} b^{5} x^{8} + 30 \, a^{6} b^{4} x^{7} + 20 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac{5}{2} \, a^{9} b x^{4} + \frac{1}{3} \, a^{10} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^10,x, algorithm="giac")

[Out]

1/13*b^10*x^13 + 5/6*a*b^9*x^12 + 45/11*a^2*b^8*x^11 + 12*a^3*b^7*x^10 + 70/3*a^4*b^6*x^9 + 63/2*a^5*b^5*x^8 +
30*a^6*b^4*x^7 + 20*a^7*b^3*x^6 + 9*a^8*b^2*x^5 + 5/2*a^9*b*x^4 + 1/3*a^10*x^3