\(\int x (a+b x)^7 \, dx\) [105]

Optimal result

Integrand size = 9, antiderivative size = 30 \[ \int x (a+b x)^7 \, dx=-\frac {a (a+b x)^8}{8 b^2}+\frac {(a+b x)^9}{9 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int x (a+b x)^7 \, dx=\frac {(a+b x)^9}{9 b^2}-\frac {a (a+b x)^8}{8 b^2} \]

[In]

Int[x*(a + b*x)^7,x]

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(30)=60\).

Time = 0.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.03 \[ \int x (a+b x)^7 \, dx=\frac {a^7 x^2}{2}+\frac {7}{3} a^6 b x^3+\frac {21}{4} a^5 b^2 x^4+7 a^4 b^3 x^5+\frac {35}{6} a^3 b^4 x^6+3 a^2 b^5 x^7+\frac {7}{8} a b^6 x^8+\frac {b^7 x^9}{9} \]

[In]

Integrate[x*(a + b*x)^7,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(26)=52\).

Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (26) = 52\).

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int x (a+b x)^7 \, dx=\frac {1}{9} \, b^{7} x^{9} + \frac {7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac {35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac {21}{4} \, a^{5} b^{2} x^{4} + \frac {7}{3} \, a^{6} b x^{3} + \frac {1}{2} \, a^{7} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).

Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.00 \[ \int x (a+b x)^7 \, dx=\frac {a^{7} x^{2}}{2} + \frac {7 a^{6} b x^{3}}{3} + \frac {21 a^{5} b^{2} x^{4}}{4} + 7 a^{4} b^{3} x^{5} + \frac {35 a^{3} b^{4} x^{6}}{6} + 3 a^{2} b^{5} x^{7} + \frac {7 a b^{6} x^{8}}{8} + \frac {b^{7} x^{9}}{9} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int x (a+b x)^7 \, dx=\frac {1}{9} \, b^{7} x^{9} + \frac {7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac {35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac {21}{4} \, a^{5} b^{2} x^{4} + \frac {7}{3} \, a^{6} b x^{3} + \frac {1}{2} \, a^{7} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (26) = 52\).

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int x (a+b x)^7 \, dx=\frac {1}{9} \, b^{7} x^{9} + \frac {7}{8} \, a b^{6} x^{8} + 3 \, a^{2} b^{5} x^{7} + \frac {35}{6} \, a^{3} b^{4} x^{6} + 7 \, a^{4} b^{3} x^{5} + \frac {21}{4} \, a^{5} b^{2} x^{4} + \frac {7}{3} \, a^{6} b x^{3} + \frac {1}{2} \, a^{7} x^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int x (a+b x)^7 \, dx=-\frac {2\,\left (\frac {a\,{\left (a+b\,x\right )}^8}{16}-\frac {{\left (a+b\,x\right )}^9}{18}\right )}{b^2} \]

[In]

int(x*(a + b*x)^7,x)

[Out]

-(2*((a*(a + b*x)^8)/16 - (a + b*x)^9/18))/b^2