3.2.29 \(\int x^5 (a+b x)^{10} \, dx\) [129]

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

[Out]

-1/11*a^5*(b*x+a)^11/b^6+5/12*a^4*(b*x+a)^12/b^6-10/13*a^3*(b*x+a)^13/b^6+5/7*a^2*(b*x+a)^14/b^6-1/3*a*(b*x+a)
^15/b^6+1/16*(b*x+a)^16/b^6

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Rubi [A]
time = 0.03, antiderivative size = 98, 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^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+b x)^{16}}{16 b^6}-\frac {a (a+b x)^{15}}{3 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^10,x]

[Out]

-1/11*(a^5*(a + b*x)^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 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^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*}

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Mathematica [A]
time = 0.00, size = 132, normalized size = 1.35 \begin {gather*} \frac {a^{10} x^6}{6}+\frac {10}{7} a^9 b x^7+\frac {45}{8} a^8 b^2 x^8+\frac {40}{3} a^7 b^3 x^9+21 a^6 b^4 x^{10}+\frac {252}{11} a^5 b^5 x^{11}+\frac {35}{2} a^4 b^6 x^{12}+\frac {120}{13} a^3 b^7 x^{13}+\frac {45}{14} a^2 b^8 x^{14}+\frac {2}{3} a b^9 x^{15}+\frac {b^{10} x^{16}}{16} \end {gather*}

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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Mathics [A]
time = 2.14, size = 112, normalized size = 1.14 \begin {gather*} \frac {x^6 \left (8008 a^{10}+68640 a^9 b x+270270 a^8 b^2 x^2+640640 a^7 b^3 x^3+1009008 a^6 b^4 x^4+1100736 a^5 b^5 x^5+840840 a^4 b^6 x^6+443520 a^3 b^7 x^7+154440 a^2 b^8 x^8+32032 a b^9 x^9+3003 b^{10} x^{10}\right )}{48048} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^5*(a + b*x)^10,x]')

[Out]

x ^ 6 (8008 a ^ 10 + 68640 a ^ 9 b x + 270270 a ^ 8 b ^ 2 x ^ 2 + 640640 a ^ 7 b ^ 3 x ^ 3 + 1009008 a ^ 6 b ^
4 x ^ 4 + 1100736 a ^ 5 b ^ 5 x ^ 5 + 840840 a ^ 4 b ^ 6 x ^ 6 + 443520 a ^ 3 b ^ 7 x ^ 7 + 154440 a ^ 2 b ^
8 x ^ 8 + 32032 a b ^ 9 x ^ 9 + 3003 b ^ 10 x ^ 10) / 48048

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Maple [A]
time = 0.08, size = 113, normalized size = 1.15

method	result	size
gosper	\(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\)	\(113\)
default	\(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\)	\(113\)
norman	\(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\)	\(113\)
risch	\(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.24, size = 112, normalized size = 1.14 \begin {gather*} \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 {gather*}

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 = 0.30, size = 112, normalized size = 1.14 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Sympy [A]
time = 0.05, size = 133, normalized size = 1.36 \begin {gather*} \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 {gather*}

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 = 0.00, size = 132, normalized size = 1.35 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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Mupad [B]
time = 0.12, size = 112, normalized size = 1.14 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x)^10,x)

[Out]

(a^10*x^6)/6 + (b^10*x^16)/16 + (10*a^9*b*x^7)/7 + (2*a*b^9*x^15)/3 + (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

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