\(\int x^2 (a+b x)^7 \, dx\) [104]

Optimal result

Integrand size = 11, antiderivative size = 47 \[ \int x^2 (a+b x)^7 \, dx=\frac {a^2 (a+b x)^8}{8 b^3}-\frac {2 a (a+b x)^9}{9 b^3}+\frac {(a+b x)^{10}}{10 b^3} \]

[Out]

1/8*a^2*(b*x+a)^8/b^3-2/9*a*(b*x+a)^9/b^3+1/10*(b*x+a)^10/b^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, 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} \[ \int x^2 (a+b x)^7 \, dx=\frac {a^2 (a+b x)^8}{8 b^3}+\frac {(a+b x)^{10}}{10 b^3}-\frac {2 a (a+b x)^9}{9 b^3} \]

[In]

Int[x^2*(a + b*x)^7,x]

[Out]

(a^2*(a + b*x)^8)/(8*b^3) - (2*a*(a + b*x)^9)/(9*b^3) + (a + b*x)^10/(10*b^3)

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.98 \[ \int x^2 (a+b x)^7 \, dx=\frac {a^7 x^3}{3}+\frac {7}{4} a^6 b x^4+\frac {21}{5} a^5 b^2 x^5+\frac {35}{6} a^4 b^3 x^6+5 a^3 b^4 x^7+\frac {21}{8} a^2 b^5 x^8+\frac {7}{9} a b^6 x^9+\frac {b^7 x^{10}}{10} \]

[In]

Integrate[x^2*(a + b*x)^7,x]

[Out]

(a^7*x^3)/3 + (7*a^6*b*x^4)/4 + (21*a^5*b^2*x^5)/5 + (35*a^4*b^3*x^6)/6 + 5*a^3*b^4*x^7 + (21*a^2*b^5*x^8)/8 +
(7*a*b^6*x^9)/9 + (b^7*x^10)/10

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.70

method	result	size
gosper	\(\frac {1}{10} b^{7} x^{10}+\frac {7}{9} a \,b^{6} x^{9}+\frac {21}{8} a^{2} b^{5} x^{8}+5 a^{3} b^{4} x^{7}+\frac {35}{6} a^{4} b^{3} x^{6}+\frac {21}{5} a^{5} b^{2} x^{5}+\frac {7}{4} a^{6} b \,x^{4}+\frac {1}{3} a^{7} x^{3}\)	\(80\)
default	\(\frac {1}{10} b^{7} x^{10}+\frac {7}{9} a \,b^{6} x^{9}+\frac {21}{8} a^{2} b^{5} x^{8}+5 a^{3} b^{4} x^{7}+\frac {35}{6} a^{4} b^{3} x^{6}+\frac {21}{5} a^{5} b^{2} x^{5}+\frac {7}{4} a^{6} b \,x^{4}+\frac {1}{3} a^{7} x^{3}\)	\(80\)
norman	\(\frac {1}{10} b^{7} x^{10}+\frac {7}{9} a \,b^{6} x^{9}+\frac {21}{8} a^{2} b^{5} x^{8}+5 a^{3} b^{4} x^{7}+\frac {35}{6} a^{4} b^{3} x^{6}+\frac {21}{5} a^{5} b^{2} x^{5}+\frac {7}{4} a^{6} b \,x^{4}+\frac {1}{3} a^{7} x^{3}\)	\(80\)
risch	\(\frac {1}{10} b^{7} x^{10}+\frac {7}{9} a \,b^{6} x^{9}+\frac {21}{8} a^{2} b^{5} x^{8}+5 a^{3} b^{4} x^{7}+\frac {35}{6} a^{4} b^{3} x^{6}+\frac {21}{5} a^{5} b^{2} x^{5}+\frac {7}{4} a^{6} b \,x^{4}+\frac {1}{3} a^{7} x^{3}\)	\(80\)
parallelrisch	\(\frac {1}{10} b^{7} x^{10}+\frac {7}{9} a \,b^{6} x^{9}+\frac {21}{8} a^{2} b^{5} x^{8}+5 a^{3} b^{4} x^{7}+\frac {35}{6} a^{4} b^{3} x^{6}+\frac {21}{5} a^{5} b^{2} x^{5}+\frac {7}{4} a^{6} b \,x^{4}+\frac {1}{3} a^{7} x^{3}\)	\(80\)

[In]

int(x^2*(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/10*b^7*x^10+7/9*a*b^6*x^9+21/8*a^2*b^5*x^8+5*a^3*b^4*x^7+35/6*a^4*b^3*x^6+21/5*a^5*b^2*x^5+7/4*a^6*b*x^4+1/3
*a^7*x^3

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.68 \[ \int x^2 (a+b x)^7 \, dx=\frac {1}{10} \, b^{7} x^{10} + \frac {7}{9} \, a b^{6} x^{9} + \frac {21}{8} \, a^{2} b^{5} x^{8} + 5 \, a^{3} b^{4} x^{7} + \frac {35}{6} \, a^{4} b^{3} x^{6} + \frac {21}{5} \, a^{5} b^{2} x^{5} + \frac {7}{4} \, a^{6} b x^{4} + \frac {1}{3} \, a^{7} x^{3} \]

[In]

integrate(x^2*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/10*b^7*x^10 + 7/9*a*b^6*x^9 + 21/8*a^2*b^5*x^8 + 5*a^3*b^4*x^7 + 35/6*a^4*b^3*x^6 + 21/5*a^5*b^2*x^5 + 7/4*a
^6*b*x^4 + 1/3*a^7*x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.96 \[ \int x^2 (a+b x)^7 \, dx=\frac {a^{7} x^{3}}{3} + \frac {7 a^{6} b x^{4}}{4} + \frac {21 a^{5} b^{2} x^{5}}{5} + \frac {35 a^{4} b^{3} x^{6}}{6} + 5 a^{3} b^{4} x^{7} + \frac {21 a^{2} b^{5} x^{8}}{8} + \frac {7 a b^{6} x^{9}}{9} + \frac {b^{7} x^{10}}{10} \]

[In]

integrate(x**2*(b*x+a)**7,x)

[Out]

a**7*x**3/3 + 7*a**6*b*x**4/4 + 21*a**5*b**2*x**5/5 + 35*a**4*b**3*x**6/6 + 5*a**3*b**4*x**7 + 21*a**2*b**5*x*
*8/8 + 7*a*b**6*x**9/9 + b**7*x**10/10

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.68 \[ \int x^2 (a+b x)^7 \, dx=\frac {1}{10} \, b^{7} x^{10} + \frac {7}{9} \, a b^{6} x^{9} + \frac {21}{8} \, a^{2} b^{5} x^{8} + 5 \, a^{3} b^{4} x^{7} + \frac {35}{6} \, a^{4} b^{3} x^{6} + \frac {21}{5} \, a^{5} b^{2} x^{5} + \frac {7}{4} \, a^{6} b x^{4} + \frac {1}{3} \, a^{7} x^{3} \]

[In]

integrate(x^2*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/10*b^7*x^10 + 7/9*a*b^6*x^9 + 21/8*a^2*b^5*x^8 + 5*a^3*b^4*x^7 + 35/6*a^4*b^3*x^6 + 21/5*a^5*b^2*x^5 + 7/4*a
^6*b*x^4 + 1/3*a^7*x^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.68 \[ \int x^2 (a+b x)^7 \, dx=\frac {1}{10} \, b^{7} x^{10} + \frac {7}{9} \, a b^{6} x^{9} + \frac {21}{8} \, a^{2} b^{5} x^{8} + 5 \, a^{3} b^{4} x^{7} + \frac {35}{6} \, a^{4} b^{3} x^{6} + \frac {21}{5} \, a^{5} b^{2} x^{5} + \frac {7}{4} \, a^{6} b x^{4} + \frac {1}{3} \, a^{7} x^{3} \]

[In]

integrate(x^2*(b*x+a)^7,x, algorithm="giac")

[Out]

1/10*b^7*x^10 + 7/9*a*b^6*x^9 + 21/8*a^2*b^5*x^8 + 5*a^3*b^4*x^7 + 35/6*a^4*b^3*x^6 + 21/5*a^5*b^2*x^5 + 7/4*a
^6*b*x^4 + 1/3*a^7*x^3

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^7 \, dx=\frac {{\left (a+b\,x\right )}^8\,\left (8\,a^2-64\,a\,b\,x+288\,b^2\,x^2\right )}{2880\,b^3} \]

[In]

int(x^2*(a + b*x)^7,x)

[Out]

((a + b*x)^8*(8*a^2 + 288*b^2*x^2 - 64*a*b*x))/(2880*b^3)