\(\int x^4 (a+b x)^5 \, dx\) [79]

Optimal result

Integrand size = 11, antiderivative size = 69 \[ \int x^4 (a+b x)^5 \, dx=\frac {a^5 x^5}{5}+\frac {5}{6} a^4 b x^6+\frac {10}{7} a^3 b^2 x^7+\frac {5}{4} a^2 b^3 x^8+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{10}}{10} \]

[Out]

1/5*a^5*x^5+5/6*a^4*b*x^6+10/7*a^3*b^2*x^7+5/4*a^2*b^3*x^8+5/9*a*b^4*x^9+1/10*b^5*x^10

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int x^4 (a+b x)^5 \, dx=\frac {a^5 x^5}{5}+\frac {5}{6} a^4 b x^6+\frac {10}{7} a^3 b^2 x^7+\frac {5}{4} a^2 b^3 x^8+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{10}}{10} \]

[In]

Int[x^4*(a + b*x)^5,x]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^6)/6 + (10*a^3*b^2*x^7)/7 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^9)/9 + (b^5*x^10)/10

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 (a^5 x^4+5 a^4 b x^5+10 a^3 b^2 x^6+10 a^2 b^3 x^7+5 a b^4 x^8+b^5 x^9\right ) \, dx \\ & = \frac {a^5 x^5}{5}+\frac {5}{6} a^4 b x^6+\frac {10}{7} a^3 b^2 x^7+\frac {5}{4} a^2 b^3 x^8+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{10}}{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x)^5 \, dx=\frac {a^5 x^5}{5}+\frac {5}{6} a^4 b x^6+\frac {10}{7} a^3 b^2 x^7+\frac {5}{4} a^2 b^3 x^8+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{10}}{10} \]

[In]

Integrate[x^4*(a + b*x)^5,x]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^6)/6 + (10*a^3*b^2*x^7)/7 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^9)/9 + (b^5*x^10)/10

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {1}{5} a^{5} x^{5}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{7} a^{3} b^{2} x^{7}+\frac {5}{4} a^{2} b^{3} x^{8}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{10} b^{5} x^{10}\)	\(58\)
default	\(\frac {1}{5} a^{5} x^{5}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{7} a^{3} b^{2} x^{7}+\frac {5}{4} a^{2} b^{3} x^{8}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{10} b^{5} x^{10}\)	\(58\)
norman	\(\frac {1}{5} a^{5} x^{5}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{7} a^{3} b^{2} x^{7}+\frac {5}{4} a^{2} b^{3} x^{8}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{10} b^{5} x^{10}\)	\(58\)
risch	\(\frac {1}{5} a^{5} x^{5}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{7} a^{3} b^{2} x^{7}+\frac {5}{4} a^{2} b^{3} x^{8}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{10} b^{5} x^{10}\)	\(58\)
parallelrisch	\(\frac {1}{5} a^{5} x^{5}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{7} a^{3} b^{2} x^{7}+\frac {5}{4} a^{2} b^{3} x^{8}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{10} b^{5} x^{10}\)	\(58\)

[In]

int(x^4*(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/5*a^5*x^5+5/6*a^4*b*x^6+10/7*a^3*b^2*x^7+5/4*a^2*b^3*x^8+5/9*a*b^4*x^9+1/10*b^5*x^10

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int x^4 (a+b x)^5 \, dx=\frac {1}{10} \, b^{5} x^{10} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} \]

[In]

integrate(x^4*(b*x+a)^5,x, algorithm="fricas")

[Out]

1/10*b^5*x^10 + 5/9*a*b^4*x^9 + 5/4*a^2*b^3*x^8 + 10/7*a^3*b^2*x^7 + 5/6*a^4*b*x^6 + 1/5*a^5*x^5

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int x^4 (a+b x)^5 \, dx=\frac {a^{5} x^{5}}{5} + \frac {5 a^{4} b x^{6}}{6} + \frac {10 a^{3} b^{2} x^{7}}{7} + \frac {5 a^{2} b^{3} x^{8}}{4} + \frac {5 a b^{4} x^{9}}{9} + \frac {b^{5} x^{10}}{10} \]

[In]

integrate(x**4*(b*x+a)**5,x)

[Out]

a**5*x**5/5 + 5*a**4*b*x**6/6 + 10*a**3*b**2*x**7/7 + 5*a**2*b**3*x**8/4 + 5*a*b**4*x**9/9 + b**5*x**10/10

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int x^4 (a+b x)^5 \, dx=\frac {1}{10} \, b^{5} x^{10} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} \]

[In]

integrate(x^4*(b*x+a)^5,x, algorithm="maxima")

[Out]

1/10*b^5*x^10 + 5/9*a*b^4*x^9 + 5/4*a^2*b^3*x^8 + 10/7*a^3*b^2*x^7 + 5/6*a^4*b*x^6 + 1/5*a^5*x^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int x^4 (a+b x)^5 \, dx=\frac {1}{10} \, b^{5} x^{10} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} \]

[In]

integrate(x^4*(b*x+a)^5,x, algorithm="giac")

[Out]

1/10*b^5*x^10 + 5/9*a*b^4*x^9 + 5/4*a^2*b^3*x^8 + 10/7*a^3*b^2*x^7 + 5/6*a^4*b*x^6 + 1/5*a^5*x^5

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int x^4 (a+b x)^5 \, dx=\frac {a^5\,x^5}{5}+\frac {5\,a^4\,b\,x^6}{6}+\frac {10\,a^3\,b^2\,x^7}{7}+\frac {5\,a^2\,b^3\,x^8}{4}+\frac {5\,a\,b^4\,x^9}{9}+\frac {b^5\,x^{10}}{10} \]

[In]

int(x^4*(a + b*x)^5,x)

[Out]

(a^5*x^5)/5 + (b^5*x^10)/10 + (5*a^4*b*x^6)/6 + (5*a*b^4*x^9)/9 + (10*a^3*b^2*x^7)/7 + (5*a^2*b^3*x^8)/4