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

Optimal. Leaf size=30 \[ \frac {(a+b x)^{12}}{12 b^2}-\frac {a (a+b x)^{11}}{11 b^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} \frac {(a+b x)^{12}}{12 b^2}-\frac {a (a+b x)^{11}}{11 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^10,x]

[Out]

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

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

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Mathematica [B]  time = 0.00, size = 128, normalized size = 4.27 \begin {gather*} \frac {a^{10} x^2}{2}+\frac {10}{3} a^9 b x^3+\frac {45}{4} a^8 b^2 x^4+24 a^7 b^3 x^5+35 a^6 b^4 x^6+36 a^5 b^5 x^7+\frac {105}{4} a^4 b^6 x^8+\frac {40}{3} a^3 b^7 x^9+\frac {9}{2} a^2 b^8 x^{10}+\frac {10}{11} a b^9 x^{11}+\frac {b^{10} x^{12}}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^10,x]

[Out]

(a^10*x^2)/2 + (10*a^9*b*x^3)/3 + (45*a^8*b^2*x^4)/4 + 24*a^7*b^3*x^5 + 35*a^6*b^4*x^6 + 36*a^5*b^5*x^7 + (105
*a^4*b^6*x^8)/4 + (40*a^3*b^7*x^9)/3 + (9*a^2*b^8*x^10)/2 + (10*a*b^9*x^11)/11 + (b^10*x^12)/12

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (a+b x)^{10} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[x*(a + b*x)^10, x]

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fricas [B]  time = 1.01, size = 112, normalized size = 3.73 \begin {gather*} \frac {1}{12} x^{12} b^{10} + \frac {10}{11} x^{11} b^{9} a + \frac {9}{2} x^{10} b^{8} a^{2} + \frac {40}{3} x^{9} b^{7} a^{3} + \frac {105}{4} x^{8} b^{6} a^{4} + 36 x^{7} b^{5} a^{5} + 35 x^{6} b^{4} a^{6} + 24 x^{5} b^{3} a^{7} + \frac {45}{4} x^{4} b^{2} a^{8} + \frac {10}{3} x^{3} b a^{9} + \frac {1}{2} x^{2} a^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*b^10 + 10/11*x^11*b^9*a + 9/2*x^10*b^8*a^2 + 40/3*x^9*b^7*a^3 + 105/4*x^8*b^6*a^4 + 36*x^7*b^5*a^5 +
35*x^6*b^4*a^6 + 24*x^5*b^3*a^7 + 45/4*x^4*b^2*a^8 + 10/3*x^3*b*a^9 + 1/2*x^2*a^10

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giac [B]  time = 1.44, size = 112, normalized size = 3.73 \begin {gather*} \frac {1}{12} \, b^{10} x^{12} + \frac {10}{11} \, a b^{9} x^{11} + \frac {9}{2} \, a^{2} b^{8} x^{10} + \frac {40}{3} \, a^{3} b^{7} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {10}{3} \, a^{9} b x^{3} + \frac {1}{2} \, a^{10} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^10*x^12 + 10/11*a*b^9*x^11 + 9/2*a^2*b^8*x^10 + 40/3*a^3*b^7*x^9 + 105/4*a^4*b^6*x^8 + 36*a^5*b^5*x^7 +
35*a^6*b^4*x^6 + 24*a^7*b^3*x^5 + 45/4*a^8*b^2*x^4 + 10/3*a^9*b*x^3 + 1/2*a^10*x^2

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maple [B]  time = 0.00, size = 113, normalized size = 3.77 \begin {gather*} \frac {1}{12} b^{10} x^{12}+\frac {10}{11} a \,b^{9} x^{11}+\frac {9}{2} a^{2} b^{8} x^{10}+\frac {40}{3} a^{3} b^{7} x^{9}+\frac {105}{4} a^{4} b^{6} x^{8}+36 a^{5} b^{5} x^{7}+35 a^{6} b^{4} x^{6}+24 a^{7} b^{3} x^{5}+\frac {45}{4} a^{8} b^{2} x^{4}+\frac {10}{3} a^{9} b \,x^{3}+\frac {1}{2} a^{10} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^10*x^12+10/11*a*b^9*x^11+9/2*a^2*b^8*x^10+40/3*a^3*b^7*x^9+105/4*a^4*b^6*x^8+36*a^5*b^5*x^7+35*a^6*b^4*
x^6+24*a^7*b^3*x^5+45/4*a^8*b^2*x^4+10/3*a^9*b*x^3+1/2*a^10*x^2

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maxima [B]  time = 1.35, size = 112, normalized size = 3.73 \begin {gather*} \frac {1}{12} \, b^{10} x^{12} + \frac {10}{11} \, a b^{9} x^{11} + \frac {9}{2} \, a^{2} b^{8} x^{10} + \frac {40}{3} \, a^{3} b^{7} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {10}{3} \, a^{9} b x^{3} + \frac {1}{2} \, a^{10} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^10*x^12 + 10/11*a*b^9*x^11 + 9/2*a^2*b^8*x^10 + 40/3*a^3*b^7*x^9 + 105/4*a^4*b^6*x^8 + 36*a^5*b^5*x^7 +
35*a^6*b^4*x^6 + 24*a^7*b^3*x^5 + 45/4*a^8*b^2*x^4 + 10/3*a^9*b*x^3 + 1/2*a^10*x^2

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mupad [B]  time = 0.09, size = 25, normalized size = 0.83 \begin {gather*} -\frac {2\,\left (\frac {a\,{\left (a+b\,x\right )}^{11}}{22}-\frac {{\left (a+b\,x\right )}^{12}}{24}\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^10,x)

[Out]

-(2*((a*(a + b*x)^11)/22 - (a + b*x)^12/24))/b^2

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sympy [B]  time = 0.11, size = 129, normalized size = 4.30 \begin {gather*} \frac {a^{10} x^{2}}{2} + \frac {10 a^{9} b x^{3}}{3} + \frac {45 a^{8} b^{2} x^{4}}{4} + 24 a^{7} b^{3} x^{5} + 35 a^{6} b^{4} x^{6} + 36 a^{5} b^{5} x^{7} + \frac {105 a^{4} b^{6} x^{8}}{4} + \frac {40 a^{3} b^{7} x^{9}}{3} + \frac {9 a^{2} b^{8} x^{10}}{2} + \frac {10 a b^{9} x^{11}}{11} + \frac {b^{10} x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*x**2/2 + 10*a**9*b*x**3/3 + 45*a**8*b**2*x**4/4 + 24*a**7*b**3*x**5 + 35*a**6*b**4*x**6 + 36*a**5*b**5*x
**7 + 105*a**4*b**6*x**8/4 + 40*a**3*b**7*x**9/3 + 9*a**2*b**8*x**10/2 + 10*a*b**9*x**11/11 + b**10*x**12/12

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