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

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

[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

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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} \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} \end {gather*}

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 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*} \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(47)=94\).
time = 0.00, size = 126, normalized size = 2.68 \begin {gather*} \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} \end {gather*}

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

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(47)=94\).
time = 2.21, size = 112, normalized size = 2.38 \begin {gather*} \frac {x^3 \left (286 a^{10}+2145 a^9 b x+7722 a^8 b^2 x^2+17160 a^7 b^3 x^3+25740 a^6 b^4 x^4+27027 a^5 b^5 x^5+20020 a^4 b^6 x^6+10296 a^3 b^7 x^7+3510 a^2 b^8 x^8+715 a b^9 x^9+66 b^{10} x^{10}\right )}{858} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^2*(a + b*x)^10,x]')

[Out]

x ^ 3 (286 a ^ 10 + 2145 a ^ 9 b x + 7722 a ^ 8 b ^ 2 x ^ 2 + 17160 a ^ 7 b ^ 3 x ^ 3 + 25740 a ^ 6 b ^ 4 x ^
4 + 27027 a ^ 5 b ^ 5 x ^ 5 + 20020 a ^ 4 b ^ 6 x ^ 6 + 10296 a ^ 3 b ^ 7 x ^ 7 + 3510 a ^ 2 b ^ 8 x ^ 8 + 715
a b ^ 9 x ^ 9 + 66 b ^ 10 x ^ 10) / 858

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(41)=82\).
time = 0.08, size = 113, normalized size = 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	\(\frac {1}{13} b^{10} x^{13}+\frac {5}{6} a \,b^{9} x^{12}+\frac {45}{11} a^{2} b^{8} x^{11}+12 a^{3} b^{7} x^{10}+\frac {70}{3} a^{4} b^{6} x^{9}+\frac {63}{2} a^{5} b^{5} x^{8}+30 a^{6} b^{4} x^{7}+20 a^{7} b^{3} x^{6}+9 a^{8} b^{2} x^{5}+\frac {5}{2} a^{9} b \,x^{4}+\frac {1}{3} a^{10} x^{3}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).
time = 0.24, size = 112, normalized size = 2.38 \begin {gather*} \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 {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).
time = 0.30, size = 112, normalized size = 2.38 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (39) = 78\).
time = 0.04, size = 128, normalized size = 2.72 \begin {gather*} \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 {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).
time = 0.00, size = 126, normalized size = 2.68 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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

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Mupad [B]
time = 0.07, size = 31, normalized size = 0.66 \begin {gather*} \frac {{\left (a+b\,x\right )}^{11}\,\left (8\,a^2-88\,a\,b\,x+528\,b^2\,x^2\right )}{6864\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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