\(\int x^4 (a+b x)^{10} \, dx\) [130]

Optimal result

Integrand size = 11, antiderivative size = 81 \[ \int x^4 (a+b x)^{10} \, dx=\frac {a^4 (a+b x)^{11}}{11 b^5}-\frac {a^3 (a+b x)^{12}}{3 b^5}+\frac {6 a^2 (a+b x)^{13}}{13 b^5}-\frac {2 a (a+b x)^{14}}{7 b^5}+\frac {(a+b x)^{15}}{15 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^4 (a+b x)^{10} \, dx=\frac {a^4 (a+b x)^{11}}{11 b^5}-\frac {a^3 (a+b x)^{12}}{3 b^5}+\frac {6 a^2 (a+b x)^{13}}{13 b^5}+\frac {(a+b x)^{15}}{15 b^5}-\frac {2 a (a+b x)^{14}}{7 b^5} \]

[In]

Int[x^4*(a + b*x)^10,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.60 \[ \int x^4 (a+b x)^{10} \, dx=\frac {a^{10} x^5}{5}+\frac {5}{3} a^9 b x^6+\frac {45}{7} a^8 b^2 x^7+15 a^7 b^3 x^8+\frac {70}{3} a^6 b^4 x^9+\frac {126}{5} a^5 b^5 x^{10}+\frac {210}{11} a^4 b^6 x^{11}+10 a^3 b^7 x^{12}+\frac {45}{13} a^2 b^8 x^{13}+\frac {5}{7} a b^9 x^{14}+\frac {b^{10} x^{15}}{15} \]

[In]

Integrate[x^4*(a + b*x)^10,x]

[Out]

(a^10*x^5)/5 + (5*a^9*b*x^6)/3 + (45*a^8*b^2*x^7)/7 + 15*a^7*b^3*x^8 + (70*a^6*b^4*x^9)/3 + (126*a^5*b^5*x^10)
/5 + (210*a^4*b^6*x^11)/11 + 10*a^3*b^7*x^12 + (45*a^2*b^8*x^13)/13 + (5*a*b^9*x^14)/7 + (b^10*x^15)/15

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.40

method	result	size
gosper	\(\frac {1}{5} a^{10} x^{5}+\frac {5}{3} a^{9} b \,x^{6}+\frac {45}{7} a^{8} b^{2} x^{7}+15 a^{7} b^{3} x^{8}+\frac {70}{3} a^{6} b^{4} x^{9}+\frac {126}{5} a^{5} b^{5} x^{10}+\frac {210}{11} a^{4} b^{6} x^{11}+10 a^{3} b^{7} x^{12}+\frac {45}{13} a^{2} b^{8} x^{13}+\frac {5}{7} a \,b^{9} x^{14}+\frac {1}{15} b^{10} x^{15}\)	\(113\)
default	\(\frac {1}{5} a^{10} x^{5}+\frac {5}{3} a^{9} b \,x^{6}+\frac {45}{7} a^{8} b^{2} x^{7}+15 a^{7} b^{3} x^{8}+\frac {70}{3} a^{6} b^{4} x^{9}+\frac {126}{5} a^{5} b^{5} x^{10}+\frac {210}{11} a^{4} b^{6} x^{11}+10 a^{3} b^{7} x^{12}+\frac {45}{13} a^{2} b^{8} x^{13}+\frac {5}{7} a \,b^{9} x^{14}+\frac {1}{15} b^{10} x^{15}\)	\(113\)
norman	\(\frac {1}{5} a^{10} x^{5}+\frac {5}{3} a^{9} b \,x^{6}+\frac {45}{7} a^{8} b^{2} x^{7}+15 a^{7} b^{3} x^{8}+\frac {70}{3} a^{6} b^{4} x^{9}+\frac {126}{5} a^{5} b^{5} x^{10}+\frac {210}{11} a^{4} b^{6} x^{11}+10 a^{3} b^{7} x^{12}+\frac {45}{13} a^{2} b^{8} x^{13}+\frac {5}{7} a \,b^{9} x^{14}+\frac {1}{15} b^{10} x^{15}\)	\(113\)
risch	\(\frac {1}{5} a^{10} x^{5}+\frac {5}{3} a^{9} b \,x^{6}+\frac {45}{7} a^{8} b^{2} x^{7}+15 a^{7} b^{3} x^{8}+\frac {70}{3} a^{6} b^{4} x^{9}+\frac {126}{5} a^{5} b^{5} x^{10}+\frac {210}{11} a^{4} b^{6} x^{11}+10 a^{3} b^{7} x^{12}+\frac {45}{13} a^{2} b^{8} x^{13}+\frac {5}{7} a \,b^{9} x^{14}+\frac {1}{15} b^{10} x^{15}\)	\(113\)
parallelrisch	\(\frac {1}{5} a^{10} x^{5}+\frac {5}{3} a^{9} b \,x^{6}+\frac {45}{7} a^{8} b^{2} x^{7}+15 a^{7} b^{3} x^{8}+\frac {70}{3} a^{6} b^{4} x^{9}+\frac {126}{5} a^{5} b^{5} x^{10}+\frac {210}{11} a^{4} b^{6} x^{11}+10 a^{3} b^{7} x^{12}+\frac {45}{13} a^{2} b^{8} x^{13}+\frac {5}{7} a \,b^{9} x^{14}+\frac {1}{15} b^{10} x^{15}\)	\(113\)

[In]

int(x^4*(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

1/5*a^10*x^5+5/3*a^9*b*x^6+45/7*a^8*b^2*x^7+15*a^7*b^3*x^8+70/3*a^6*b^4*x^9+126/5*a^5*b^5*x^10+210/11*a^4*b^6*
x^11+10*a^3*b^7*x^12+45/13*a^2*b^8*x^13+5/7*a*b^9*x^14+1/15*b^10*x^15

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int x^4 (a+b x)^{10} \, dx=\frac {1}{15} \, b^{10} x^{15} + \frac {5}{7} \, a b^{9} x^{14} + \frac {45}{13} \, a^{2} b^{8} x^{13} + 10 \, a^{3} b^{7} x^{12} + \frac {210}{11} \, a^{4} b^{6} x^{11} + \frac {126}{5} \, a^{5} b^{5} x^{10} + \frac {70}{3} \, a^{6} b^{4} x^{9} + 15 \, a^{7} b^{3} x^{8} + \frac {45}{7} \, a^{8} b^{2} x^{7} + \frac {5}{3} \, a^{9} b x^{6} + \frac {1}{5} \, a^{10} x^{5} \]

[In]

integrate(x^4*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/15*b^10*x^15 + 5/7*a*b^9*x^14 + 45/13*a^2*b^8*x^13 + 10*a^3*b^7*x^12 + 210/11*a^4*b^6*x^11 + 126/5*a^5*b^5*x
^10 + 70/3*a^6*b^4*x^9 + 15*a^7*b^3*x^8 + 45/7*a^8*b^2*x^7 + 5/3*a^9*b*x^6 + 1/5*a^10*x^5

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.62 \[ \int x^4 (a+b x)^{10} \, dx=\frac {a^{10} x^{5}}{5} + \frac {5 a^{9} b x^{6}}{3} + \frac {45 a^{8} b^{2} x^{7}}{7} + 15 a^{7} b^{3} x^{8} + \frac {70 a^{6} b^{4} x^{9}}{3} + \frac {126 a^{5} b^{5} x^{10}}{5} + \frac {210 a^{4} b^{6} x^{11}}{11} + 10 a^{3} b^{7} x^{12} + \frac {45 a^{2} b^{8} x^{13}}{13} + \frac {5 a b^{9} x^{14}}{7} + \frac {b^{10} x^{15}}{15} \]

[In]

integrate(x**4*(b*x+a)**10,x)

[Out]

a**10*x**5/5 + 5*a**9*b*x**6/3 + 45*a**8*b**2*x**7/7 + 15*a**7*b**3*x**8 + 70*a**6*b**4*x**9/3 + 126*a**5*b**5
*x**10/5 + 210*a**4*b**6*x**11/11 + 10*a**3*b**7*x**12 + 45*a**2*b**8*x**13/13 + 5*a*b**9*x**14/7 + b**10*x**1
5/15

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int x^4 (a+b x)^{10} \, dx=\frac {1}{15} \, b^{10} x^{15} + \frac {5}{7} \, a b^{9} x^{14} + \frac {45}{13} \, a^{2} b^{8} x^{13} + 10 \, a^{3} b^{7} x^{12} + \frac {210}{11} \, a^{4} b^{6} x^{11} + \frac {126}{5} \, a^{5} b^{5} x^{10} + \frac {70}{3} \, a^{6} b^{4} x^{9} + 15 \, a^{7} b^{3} x^{8} + \frac {45}{7} \, a^{8} b^{2} x^{7} + \frac {5}{3} \, a^{9} b x^{6} + \frac {1}{5} \, a^{10} x^{5} \]

[In]

integrate(x^4*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/15*b^10*x^15 + 5/7*a*b^9*x^14 + 45/13*a^2*b^8*x^13 + 10*a^3*b^7*x^12 + 210/11*a^4*b^6*x^11 + 126/5*a^5*b^5*x
^10 + 70/3*a^6*b^4*x^9 + 15*a^7*b^3*x^8 + 45/7*a^8*b^2*x^7 + 5/3*a^9*b*x^6 + 1/5*a^10*x^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int x^4 (a+b x)^{10} \, dx=\frac {1}{15} \, b^{10} x^{15} + \frac {5}{7} \, a b^{9} x^{14} + \frac {45}{13} \, a^{2} b^{8} x^{13} + 10 \, a^{3} b^{7} x^{12} + \frac {210}{11} \, a^{4} b^{6} x^{11} + \frac {126}{5} \, a^{5} b^{5} x^{10} + \frac {70}{3} \, a^{6} b^{4} x^{9} + 15 \, a^{7} b^{3} x^{8} + \frac {45}{7} \, a^{8} b^{2} x^{7} + \frac {5}{3} \, a^{9} b x^{6} + \frac {1}{5} \, a^{10} x^{5} \]

[In]

integrate(x^4*(b*x+a)^10,x, algorithm="giac")

[Out]

1/15*b^10*x^15 + 5/7*a*b^9*x^14 + 45/13*a^2*b^8*x^13 + 10*a^3*b^7*x^12 + 210/11*a^4*b^6*x^11 + 126/5*a^5*b^5*x
^10 + 70/3*a^6*b^4*x^9 + 15*a^7*b^3*x^8 + 45/7*a^8*b^2*x^7 + 5/3*a^9*b*x^6 + 1/5*a^10*x^5

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int x^4 (a+b x)^{10} \, dx=\frac {a^{10}\,x^5}{5}+\frac {5\,a^9\,b\,x^6}{3}+\frac {45\,a^8\,b^2\,x^7}{7}+15\,a^7\,b^3\,x^8+\frac {70\,a^6\,b^4\,x^9}{3}+\frac {126\,a^5\,b^5\,x^{10}}{5}+\frac {210\,a^4\,b^6\,x^{11}}{11}+10\,a^3\,b^7\,x^{12}+\frac {45\,a^2\,b^8\,x^{13}}{13}+\frac {5\,a\,b^9\,x^{14}}{7}+\frac {b^{10}\,x^{15}}{15} \]

[In]

int(x^4*(a + b*x)^10,x)

[Out]

(a^10*x^5)/5 + (b^10*x^15)/15 + (5*a^9*b*x^6)/3 + (5*a*b^9*x^14)/7 + (45*a^8*b^2*x^7)/7 + 15*a^7*b^3*x^8 + (70
*a^6*b^4*x^9)/3 + (126*a^5*b^5*x^10)/5 + (210*a^4*b^6*x^11)/11 + 10*a^3*b^7*x^12 + (45*a^2*b^8*x^13)/13