3.2.26 \(\int x^8 (a+b x)^{10} \, dx\)

Optimal. Leaf size=147 \[ \frac {a^8 (a+b x)^{11}}{11 b^9}-\frac {2 a^7 (a+b x)^{12}}{3 b^9}+\frac {28 a^6 (a+b x)^{13}}{13 b^9}-\frac {4 a^5 (a+b x)^{14}}{b^9}+\frac {14 a^4 (a+b x)^{15}}{3 b^9}-\frac {7 a^3 (a+b x)^{16}}{2 b^9}+\frac {28 a^2 (a+b x)^{17}}{17 b^9}+\frac {(a+b x)^{19}}{19 b^9}-\frac {4 a (a+b x)^{18}}{9 b^9} \]

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Rubi [A]  time = 0.06, antiderivative size = 147, 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 = {43} \begin {gather*} \frac {28 a^2 (a+b x)^{17}}{17 b^9}-\frac {7 a^3 (a+b x)^{16}}{2 b^9}+\frac {14 a^4 (a+b x)^{15}}{3 b^9}-\frac {4 a^5 (a+b x)^{14}}{b^9}+\frac {28 a^6 (a+b x)^{13}}{13 b^9}-\frac {2 a^7 (a+b x)^{12}}{3 b^9}+\frac {a^8 (a+b x)^{11}}{11 b^9}+\frac {(a+b x)^{19}}{19 b^9}-\frac {4 a (a+b x)^{18}}{9 b^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x)^10,x]

[Out]

(a^8*(a + b*x)^11)/(11*b^9) - (2*a^7*(a + b*x)^12)/(3*b^9) + (28*a^6*(a + b*x)^13)/(13*b^9) - (4*a^5*(a + b*x)
^14)/b^9 + (14*a^4*(a + b*x)^15)/(3*b^9) - (7*a^3*(a + b*x)^16)/(2*b^9) + (28*a^2*(a + b*x)^17)/(17*b^9) - (4*
a*(a + b*x)^18)/(9*b^9) + (a + b*x)^19/(19*b^9)

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^8 (a+b x)^{10} \, dx &=\int \left (\frac {a^8 (a+b x)^{10}}{b^8}-\frac {8 a^7 (a+b x)^{11}}{b^8}+\frac {28 a^6 (a+b x)^{12}}{b^8}-\frac {56 a^5 (a+b x)^{13}}{b^8}+\frac {70 a^4 (a+b x)^{14}}{b^8}-\frac {56 a^3 (a+b x)^{15}}{b^8}+\frac {28 a^2 (a+b x)^{16}}{b^8}-\frac {8 a (a+b x)^{17}}{b^8}+\frac {(a+b x)^{18}}{b^8}\right ) \, dx\\ &=\frac {a^8 (a+b x)^{11}}{11 b^9}-\frac {2 a^7 (a+b x)^{12}}{3 b^9}+\frac {28 a^6 (a+b x)^{13}}{13 b^9}-\frac {4 a^5 (a+b x)^{14}}{b^9}+\frac {14 a^4 (a+b x)^{15}}{3 b^9}-\frac {7 a^3 (a+b x)^{16}}{2 b^9}+\frac {28 a^2 (a+b x)^{17}}{17 b^9}-\frac {4 a (a+b x)^{18}}{9 b^9}+\frac {(a+b x)^{19}}{19 b^9}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 125, normalized size = 0.85 \begin {gather*} \frac {a^{10} x^9}{9}+a^9 b x^{10}+\frac {45}{11} a^8 b^2 x^{11}+10 a^7 b^3 x^{12}+\frac {210}{13} a^6 b^4 x^{13}+18 a^5 b^5 x^{14}+14 a^4 b^6 x^{15}+\frac {15}{2} a^3 b^7 x^{16}+\frac {45}{17} a^2 b^8 x^{17}+\frac {5}{9} a b^9 x^{18}+\frac {b^{10} x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x)^10,x]

[Out]

(a^10*x^9)/9 + a^9*b*x^10 + (45*a^8*b^2*x^11)/11 + 10*a^7*b^3*x^12 + (210*a^6*b^4*x^13)/13 + 18*a^5*b^5*x^14 +
14*a^4*b^6*x^15 + (15*a^3*b^7*x^16)/2 + (45*a^2*b^8*x^17)/17 + (5*a*b^9*x^18)/9 + (b^10*x^19)/19

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^8 (a+b x)^{10} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^8*(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[x^8*(a + b*x)^10, x]

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fricas [A]  time = 1.09, size = 111, normalized size = 0.76 \begin {gather*} \frac {1}{19} x^{19} b^{10} + \frac {5}{9} x^{18} b^{9} a + \frac {45}{17} x^{17} b^{8} a^{2} + \frac {15}{2} x^{16} b^{7} a^{3} + 14 x^{15} b^{6} a^{4} + 18 x^{14} b^{5} a^{5} + \frac {210}{13} x^{13} b^{4} a^{6} + 10 x^{12} b^{3} a^{7} + \frac {45}{11} x^{11} b^{2} a^{8} + x^{10} b a^{9} + \frac {1}{9} x^{9} a^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/19*x^19*b^10 + 5/9*x^18*b^9*a + 45/17*x^17*b^8*a^2 + 15/2*x^16*b^7*a^3 + 14*x^15*b^6*a^4 + 18*x^14*b^5*a^5 +
210/13*x^13*b^4*a^6 + 10*x^12*b^3*a^7 + 45/11*x^11*b^2*a^8 + x^10*b*a^9 + 1/9*x^9*a^10

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giac [A]  time = 1.01, size = 111, normalized size = 0.76 \begin {gather*} \frac {1}{19} \, b^{10} x^{19} + \frac {5}{9} \, a b^{9} x^{18} + \frac {45}{17} \, a^{2} b^{8} x^{17} + \frac {15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac {210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac {45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac {1}{9} \, a^{10} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10,x, algorithm="giac")

[Out]

1/19*b^10*x^19 + 5/9*a*b^9*x^18 + 45/17*a^2*b^8*x^17 + 15/2*a^3*b^7*x^16 + 14*a^4*b^6*x^15 + 18*a^5*b^5*x^14 +
210/13*a^6*b^4*x^13 + 10*a^7*b^3*x^12 + 45/11*a^8*b^2*x^11 + a^9*b*x^10 + 1/9*a^10*x^9

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maple [A]  time = 0.00, size = 112, normalized size = 0.76 \begin {gather*} \frac {1}{19} b^{10} x^{19}+\frac {5}{9} a \,b^{9} x^{18}+\frac {45}{17} a^{2} b^{8} x^{17}+\frac {15}{2} a^{3} b^{7} x^{16}+14 a^{4} b^{6} x^{15}+18 a^{5} b^{5} x^{14}+\frac {210}{13} a^{6} b^{4} x^{13}+10 a^{7} b^{3} x^{12}+\frac {45}{11} a^{8} b^{2} x^{11}+a^{9} b \,x^{10}+\frac {1}{9} a^{10} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x+a)^10,x)

[Out]

1/19*b^10*x^19+5/9*a*b^9*x^18+45/17*a^2*b^8*x^17+15/2*a^3*b^7*x^16+14*a^4*b^6*x^15+18*a^5*b^5*x^14+210/13*a^6*
b^4*x^13+10*a^7*b^3*x^12+45/11*a^8*b^2*x^11+a^9*b*x^10+1/9*a^10*x^9

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maxima [A]  time = 1.36, size = 111, normalized size = 0.76 \begin {gather*} \frac {1}{19} \, b^{10} x^{19} + \frac {5}{9} \, a b^{9} x^{18} + \frac {45}{17} \, a^{2} b^{8} x^{17} + \frac {15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac {210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac {45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac {1}{9} \, a^{10} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/19*b^10*x^19 + 5/9*a*b^9*x^18 + 45/17*a^2*b^8*x^17 + 15/2*a^3*b^7*x^16 + 14*a^4*b^6*x^15 + 18*a^5*b^5*x^14 +
210/13*a^6*b^4*x^13 + 10*a^7*b^3*x^12 + 45/11*a^8*b^2*x^11 + a^9*b*x^10 + 1/9*a^10*x^9

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mupad [B]  time = 0.09, size = 111, normalized size = 0.76 \begin {gather*} \frac {a^{10}\,x^9}{9}+a^9\,b\,x^{10}+\frac {45\,a^8\,b^2\,x^{11}}{11}+10\,a^7\,b^3\,x^{12}+\frac {210\,a^6\,b^4\,x^{13}}{13}+18\,a^5\,b^5\,x^{14}+14\,a^4\,b^6\,x^{15}+\frac {15\,a^3\,b^7\,x^{16}}{2}+\frac {45\,a^2\,b^8\,x^{17}}{17}+\frac {5\,a\,b^9\,x^{18}}{9}+\frac {b^{10}\,x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x)^10,x)

[Out]

(a^10*x^9)/9 + (b^10*x^19)/19 + a^9*b*x^10 + (5*a*b^9*x^18)/9 + (45*a^8*b^2*x^11)/11 + 10*a^7*b^3*x^12 + (210*
a^6*b^4*x^13)/13 + 18*a^5*b^5*x^14 + 14*a^4*b^6*x^15 + (15*a^3*b^7*x^16)/2 + (45*a^2*b^8*x^17)/17

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sympy [A]  time = 0.11, size = 126, normalized size = 0.86 \begin {gather*} \frac {a^{10} x^{9}}{9} + a^{9} b x^{10} + \frac {45 a^{8} b^{2} x^{11}}{11} + 10 a^{7} b^{3} x^{12} + \frac {210 a^{6} b^{4} x^{13}}{13} + 18 a^{5} b^{5} x^{14} + 14 a^{4} b^{6} x^{15} + \frac {15 a^{3} b^{7} x^{16}}{2} + \frac {45 a^{2} b^{8} x^{17}}{17} + \frac {5 a b^{9} x^{18}}{9} + \frac {b^{10} x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x+a)**10,x)

[Out]

a**10*x**9/9 + a**9*b*x**10 + 45*a**8*b**2*x**11/11 + 10*a**7*b**3*x**12 + 210*a**6*b**4*x**13/13 + 18*a**5*b*
*5*x**14 + 14*a**4*b**6*x**15 + 15*a**3*b**7*x**16/2 + 45*a**2*b**8*x**17/17 + 5*a*b**9*x**18/9 + b**10*x**19/
19

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