\(\int x^2 (a+b x)^{10} \, dx\) [132]

Optimal result

Integrand size = 11, antiderivative size = 47 \[ \int x^2 (a+b x)^{10} \, dx=\frac {a^2 (a+b x)^{11}}{11 b^3}-\frac {a (a+b x)^{12}}{6 b^3}+\frac {(a+b x)^{13}}{13 b^3} \]

[Out]

1/11*a^2*(b*x+a)^11/b^3-1/6*a*(b*x+a)^12/b^3+1/13*(b*x+a)^13/b^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^2 (a+b x)^{10} \, dx=\frac {a^2 (a+b x)^{11}}{11 b^3}+\frac {(a+b x)^{13}}{13 b^3}-\frac {a (a+b x)^{12}}{6 b^3} \]

[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 45

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*} \text {integral}& = \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(47)=94\).

Time = 0.00 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.68 \[ \int x^2 (a+b x)^{10} \, dx=\frac {a^{10} x^3}{3}+\frac {5}{2} a^9 b x^4+9 a^8 b^2 x^5+20 a^7 b^3 x^6+30 a^6 b^4 x^7+\frac {63}{2} a^5 b^5 x^8+\frac {70}{3} a^4 b^6 x^9+12 a^3 b^7 x^{10}+\frac {45}{11} a^2 b^8 x^{11}+\frac {5}{6} a b^9 x^{12}+\frac {b^{10} x^{13}}{13} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(41)=82\).

Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.40

method	result	size
gosper	\(\frac {1}{13} b^{10} x^{13}+\frac {5}{6} a \,b^{9} x^{12}+\frac {45}{11} a^{2} b^{8} x^{11}+12 a^{3} b^{7} x^{10}+\frac {70}{3} a^{4} b^{6} x^{9}+\frac {63}{2} a^{5} b^{5} x^{8}+30 a^{6} b^{4} x^{7}+20 a^{7} b^{3} x^{6}+9 a^{8} b^{2} x^{5}+\frac {5}{2} a^{9} b \,x^{4}+\frac {1}{3} a^{10} x^{3}\)	\(113\)
default	\(\frac {1}{13} b^{10} x^{13}+\frac {5}{6} a \,b^{9} x^{12}+\frac {45}{11} a^{2} b^{8} x^{11}+12 a^{3} b^{7} x^{10}+\frac {70}{3} a^{4} b^{6} x^{9}+\frac {63}{2} a^{5} b^{5} x^{8}+30 a^{6} b^{4} x^{7}+20 a^{7} b^{3} x^{6}+9 a^{8} b^{2} x^{5}+\frac {5}{2} a^{9} b \,x^{4}+\frac {1}{3} a^{10} x^{3}\)	\(113\)
norman	\(\frac {1}{13} b^{10} x^{13}+\frac {5}{6} a \,b^{9} x^{12}+\frac {45}{11} a^{2} b^{8} x^{11}+12 a^{3} b^{7} x^{10}+\frac {70}{3} a^{4} b^{6} x^{9}+\frac {63}{2} a^{5} b^{5} x^{8}+30 a^{6} b^{4} x^{7}+20 a^{7} b^{3} x^{6}+9 a^{8} b^{2} x^{5}+\frac {5}{2} a^{9} b \,x^{4}+\frac {1}{3} a^{10} x^{3}\)	\(113\)
risch	\(\frac {1}{13} b^{10} x^{13}+\frac {5}{6} a \,b^{9} x^{12}+\frac {45}{11} a^{2} b^{8} x^{11}+12 a^{3} b^{7} x^{10}+\frac {70}{3} a^{4} b^{6} x^{9}+\frac {63}{2} a^{5} b^{5} x^{8}+30 a^{6} b^{4} x^{7}+20 a^{7} b^{3} x^{6}+9 a^{8} b^{2} x^{5}+\frac {5}{2} a^{9} b \,x^{4}+\frac {1}{3} a^{10} x^{3}\)	\(113\)
parallelrisch	\(\frac {1}{13} b^{10} x^{13}+\frac {5}{6} a \,b^{9} x^{12}+\frac {45}{11} a^{2} b^{8} x^{11}+12 a^{3} b^{7} x^{10}+\frac {70}{3} a^{4} b^{6} x^{9}+\frac {63}{2} a^{5} b^{5} x^{8}+30 a^{6} b^{4} x^{7}+20 a^{7} b^{3} x^{6}+9 a^{8} b^{2} x^{5}+\frac {5}{2} a^{9} b \,x^{4}+\frac {1}{3} a^{10} x^{3}\)	\(113\)

[In]

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

[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] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.38 \[ \int x^2 (a+b x)^{10} \, dx=\frac {1}{13} \, b^{10} x^{13} + \frac {5}{6} \, a b^{9} x^{12} + \frac {45}{11} \, a^{2} b^{8} x^{11} + 12 \, a^{3} b^{7} x^{10} + \frac {70}{3} \, a^{4} b^{6} x^{9} + \frac {63}{2} \, a^{5} b^{5} x^{8} + 30 \, a^{6} b^{4} x^{7} + 20 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac {5}{2} \, a^{9} b x^{4} + \frac {1}{3} \, a^{10} x^{3} \]

[In]

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

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (39) = 78\).

Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.72 \[ \int x^2 (a+b x)^{10} \, dx=\frac {a^{10} x^{3}}{3} + \frac {5 a^{9} b x^{4}}{2} + 9 a^{8} b^{2} x^{5} + 20 a^{7} b^{3} x^{6} + 30 a^{6} b^{4} x^{7} + \frac {63 a^{5} b^{5} x^{8}}{2} + \frac {70 a^{4} b^{6} x^{9}}{3} + 12 a^{3} b^{7} x^{10} + \frac {45 a^{2} b^{8} x^{11}}{11} + \frac {5 a b^{9} x^{12}}{6} + \frac {b^{10} x^{13}}{13} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.38 \[ \int x^2 (a+b x)^{10} \, dx=\frac {1}{13} \, b^{10} x^{13} + \frac {5}{6} \, a b^{9} x^{12} + \frac {45}{11} \, a^{2} b^{8} x^{11} + 12 \, a^{3} b^{7} x^{10} + \frac {70}{3} \, a^{4} b^{6} x^{9} + \frac {63}{2} \, a^{5} b^{5} x^{8} + 30 \, a^{6} b^{4} x^{7} + 20 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac {5}{2} \, a^{9} b x^{4} + \frac {1}{3} \, a^{10} x^{3} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.38 \[ \int x^2 (a+b x)^{10} \, dx=\frac {1}{13} \, b^{10} x^{13} + \frac {5}{6} \, a b^{9} x^{12} + \frac {45}{11} \, a^{2} b^{8} x^{11} + 12 \, a^{3} b^{7} x^{10} + \frac {70}{3} \, a^{4} b^{6} x^{9} + \frac {63}{2} \, a^{5} b^{5} x^{8} + 30 \, a^{6} b^{4} x^{7} + 20 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac {5}{2} \, a^{9} b x^{4} + \frac {1}{3} \, a^{10} x^{3} \]

[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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^{10} \, dx=\frac {{\left (a+b\,x\right )}^{11}\,\left (8\,a^2-88\,a\,b\,x+528\,b^2\,x^2\right )}{6864\,b^3} \]

[In]

int(x^2*(a + b*x)^10,x)

[Out]

((a + b*x)^11*(8*a^2 + 528*b^2*x^2 - 88*a*b*x))/(6864*b^3)