3.129 \(\int x^5 (a+b x)^{10} \, dx\)

Optimal. Leaf size=98 \[ \frac{5 a^2 (a+b x)^{14}}{7 b^6}-\frac{10 a^3 (a+b x)^{13}}{13 b^6}+\frac{5 a^4 (a+b x)^{12}}{12 b^6}-\frac{a^5 (a+b x)^{11}}{11 b^6}+\frac{(a+b x)^{16}}{16 b^6}-\frac{a (a+b x)^{15}}{3 b^6} \]

[Out]

-(a^5*(a + b*x)^11)/(11*b^6) + (5*a^4*(a + b*x)^12)/(12*b^6) - (10*a^3*(a + b*x)^13)/(13*b^6) + (5*a^2*(a + b*
x)^14)/(7*b^6) - (a*(a + b*x)^15)/(3*b^6) + (a + b*x)^16/(16*b^6)

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Rubi [A]  time = 0.0452189, antiderivative size = 98, 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} \[ \frac{5 a^2 (a+b x)^{14}}{7 b^6}-\frac{10 a^3 (a+b x)^{13}}{13 b^6}+\frac{5 a^4 (a+b x)^{12}}{12 b^6}-\frac{a^5 (a+b x)^{11}}{11 b^6}+\frac{(a+b x)^{16}}{16 b^6}-\frac{a (a+b x)^{15}}{3 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^10,x]

[Out]

-(a^5*(a + b*x)^11)/(11*b^6) + (5*a^4*(a + b*x)^12)/(12*b^6) - (10*a^3*(a + b*x)^13)/(13*b^6) + (5*a^2*(a + b*
x)^14)/(7*b^6) - (a*(a + b*x)^15)/(3*b^6) + (a + b*x)^16/(16*b^6)

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

Mathematica [A]  time = 0.0080195, size = 132, normalized size = 1.35 \[ \frac{45}{14} a^2 b^8 x^{14}+\frac{120}{13} a^3 b^7 x^{13}+\frac{35}{2} a^4 b^6 x^{12}+\frac{252}{11} a^5 b^5 x^{11}+21 a^6 b^4 x^{10}+\frac{40}{3} a^7 b^3 x^9+\frac{45}{8} a^8 b^2 x^8+\frac{10}{7} a^9 b x^7+\frac{a^{10} x^6}{6}+\frac{2}{3} a b^9 x^{15}+\frac{b^{10} x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^10,x]

[Out]

(a^10*x^6)/6 + (10*a^9*b*x^7)/7 + (45*a^8*b^2*x^8)/8 + (40*a^7*b^3*x^9)/3 + 21*a^6*b^4*x^10 + (252*a^5*b^5*x^1
1)/11 + (35*a^4*b^6*x^12)/2 + (120*a^3*b^7*x^13)/13 + (45*a^2*b^8*x^14)/14 + (2*a*b^9*x^15)/3 + (b^10*x^16)/16

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Maple [A]  time = 0.002, size = 113, normalized size = 1.2 \begin{align*}{\frac{{b}^{10}{x}^{16}}{16}}+{\frac{2\,a{b}^{9}{x}^{15}}{3}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{14}}{14}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{13}}{13}}+{\frac{35\,{a}^{4}{b}^{6}{x}^{12}}{2}}+{\frac{252\,{a}^{5}{b}^{5}{x}^{11}}{11}}+21\,{a}^{6}{b}^{4}{x}^{10}+{\frac{40\,{a}^{7}{b}^{3}{x}^{9}}{3}}+{\frac{45\,{a}^{8}{b}^{2}{x}^{8}}{8}}+{\frac{10\,{a}^{9}b{x}^{7}}{7}}+{\frac{{a}^{10}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^10,x)

[Out]

1/16*b^10*x^16+2/3*a*b^9*x^15+45/14*a^2*b^8*x^14+120/13*a^3*b^7*x^13+35/2*a^4*b^6*x^12+252/11*a^5*b^5*x^11+21*
a^6*b^4*x^10+40/3*a^7*b^3*x^9+45/8*a^8*b^2*x^8+10/7*a^9*b*x^7+1/6*a^10*x^6

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Maxima [A]  time = 1.04719, size = 151, normalized size = 1.54 \begin{align*} \frac{1}{16} \, b^{10} x^{16} + \frac{2}{3} \, a b^{9} x^{15} + \frac{45}{14} \, a^{2} b^{8} x^{14} + \frac{120}{13} \, a^{3} b^{7} x^{13} + \frac{35}{2} \, a^{4} b^{6} x^{12} + \frac{252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac{40}{3} \, a^{7} b^{3} x^{9} + \frac{45}{8} \, a^{8} b^{2} x^{8} + \frac{10}{7} \, a^{9} b x^{7} + \frac{1}{6} \, a^{10} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/2*a^4*b^6*x^12 + 252/11*a^5*b^
5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^3*x^9 + 45/8*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6

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Fricas [A]  time = 1.5311, size = 278, normalized size = 2.84 \begin{align*} \frac{1}{16} x^{16} b^{10} + \frac{2}{3} x^{15} b^{9} a + \frac{45}{14} x^{14} b^{8} a^{2} + \frac{120}{13} x^{13} b^{7} a^{3} + \frac{35}{2} x^{12} b^{6} a^{4} + \frac{252}{11} x^{11} b^{5} a^{5} + 21 x^{10} b^{4} a^{6} + \frac{40}{3} x^{9} b^{3} a^{7} + \frac{45}{8} x^{8} b^{2} a^{8} + \frac{10}{7} x^{7} b a^{9} + \frac{1}{6} x^{6} a^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/16*x^16*b^10 + 2/3*x^15*b^9*a + 45/14*x^14*b^8*a^2 + 120/13*x^13*b^7*a^3 + 35/2*x^12*b^6*a^4 + 252/11*x^11*b
^5*a^5 + 21*x^10*b^4*a^6 + 40/3*x^9*b^3*a^7 + 45/8*x^8*b^2*a^8 + 10/7*x^7*b*a^9 + 1/6*x^6*a^10

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Sympy [A]  time = 0.127557, size = 133, normalized size = 1.36 \begin{align*} \frac{a^{10} x^{6}}{6} + \frac{10 a^{9} b x^{7}}{7} + \frac{45 a^{8} b^{2} x^{8}}{8} + \frac{40 a^{7} b^{3} x^{9}}{3} + 21 a^{6} b^{4} x^{10} + \frac{252 a^{5} b^{5} x^{11}}{11} + \frac{35 a^{4} b^{6} x^{12}}{2} + \frac{120 a^{3} b^{7} x^{13}}{13} + \frac{45 a^{2} b^{8} x^{14}}{14} + \frac{2 a b^{9} x^{15}}{3} + \frac{b^{10} x^{16}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x+a)**10,x)

[Out]

a**10*x**6/6 + 10*a**9*b*x**7/7 + 45*a**8*b**2*x**8/8 + 40*a**7*b**3*x**9/3 + 21*a**6*b**4*x**10 + 252*a**5*b*
*5*x**11/11 + 35*a**4*b**6*x**12/2 + 120*a**3*b**7*x**13/13 + 45*a**2*b**8*x**14/14 + 2*a*b**9*x**15/3 + b**10
*x**16/16

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Giac [A]  time = 1.18422, size = 151, normalized size = 1.54 \begin{align*} \frac{1}{16} \, b^{10} x^{16} + \frac{2}{3} \, a b^{9} x^{15} + \frac{45}{14} \, a^{2} b^{8} x^{14} + \frac{120}{13} \, a^{3} b^{7} x^{13} + \frac{35}{2} \, a^{4} b^{6} x^{12} + \frac{252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac{40}{3} \, a^{7} b^{3} x^{9} + \frac{45}{8} \, a^{8} b^{2} x^{8} + \frac{10}{7} \, a^{9} b x^{7} + \frac{1}{6} \, a^{10} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^10,x, algorithm="giac")

[Out]

1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/2*a^4*b^6*x^12 + 252/11*a^5*b^
5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^3*x^9 + 45/8*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6