\(\int (a+b x)^7 \, dx\) [106]

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int (a+b x)^7 \, dx=\frac {(a+b x)^8}{8 b} \]

[Out]

1/8*(b*x+a)^8/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (a+b x)^7 \, dx=\frac {(a+b x)^8}{8 b} \]

[In]

Int[(a + b*x)^7,x]

[Out]

(a + b*x)^8/(8*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^8}{8 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b x)^7 \, dx=\frac {(a+b x)^8}{8 b} \]

[In]

Integrate[(a + b*x)^7,x]

[Out]

(a + b*x)^8/(8*b)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\left (b x +a \right )^{8}}{8 b}\)	\(13\)
gosper	\(\frac {1}{8} b^{7} x^{8}+a \,b^{6} x^{7}+\frac {7}{2} a^{2} b^{5} x^{6}+7 a^{3} b^{4} x^{5}+\frac {35}{4} a^{4} b^{3} x^{4}+7 a^{5} b^{2} x^{3}+\frac {7}{2} a^{6} b \,x^{2}+a^{7} x\)	\(76\)
norman	\(\frac {1}{8} b^{7} x^{8}+a \,b^{6} x^{7}+\frac {7}{2} a^{2} b^{5} x^{6}+7 a^{3} b^{4} x^{5}+\frac {35}{4} a^{4} b^{3} x^{4}+7 a^{5} b^{2} x^{3}+\frac {7}{2} a^{6} b \,x^{2}+a^{7} x\)	\(76\)
parallelrisch	\(\frac {1}{8} b^{7} x^{8}+a \,b^{6} x^{7}+\frac {7}{2} a^{2} b^{5} x^{6}+7 a^{3} b^{4} x^{5}+\frac {35}{4} a^{4} b^{3} x^{4}+7 a^{5} b^{2} x^{3}+\frac {7}{2} a^{6} b \,x^{2}+a^{7} x\)	\(76\)
risch	\(\frac {b^{7} x^{8}}{8}+a \,b^{6} x^{7}+\frac {7 a^{2} b^{5} x^{6}}{2}+7 a^{3} b^{4} x^{5}+\frac {35 a^{4} b^{3} x^{4}}{4}+7 a^{5} b^{2} x^{3}+\frac {7 a^{6} b \,x^{2}}{2}+a^{7} x +\frac {a^{8}}{8 b}\)	\(84\)

[In]

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

[Out]

1/8*(b*x+a)^8/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (12) = 24\).

Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 5.36 \[ \int (a+b x)^7 \, dx=\frac {1}{8} \, b^{7} x^{8} + a b^{6} x^{7} + \frac {7}{2} \, a^{2} b^{5} x^{6} + 7 \, a^{3} b^{4} x^{5} + \frac {35}{4} \, a^{4} b^{3} x^{4} + 7 \, a^{5} b^{2} x^{3} + \frac {7}{2} \, a^{6} b x^{2} + a^{7} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (8) = 16\).

Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 5.93 \[ \int (a+b x)^7 \, dx=a^{7} x + \frac {7 a^{6} b x^{2}}{2} + 7 a^{5} b^{2} x^{3} + \frac {35 a^{4} b^{3} x^{4}}{4} + 7 a^{3} b^{4} x^{5} + \frac {7 a^{2} b^{5} x^{6}}{2} + a b^{6} x^{7} + \frac {b^{7} x^{8}}{8} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (a+b x)^7 \, dx=\frac {{\left (b x + a\right )}^{8}}{8 \, b} \]

[In]

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

[Out]

1/8*(b*x + a)^8/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (a+b x)^7 \, dx=\frac {{\left (b x + a\right )}^{8}}{8 \, b} \]

[In]

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

[Out]

1/8*(b*x + a)^8/b

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 5.36 \[ \int (a+b x)^7 \, dx=a^7\,x+\frac {7\,a^6\,b\,x^2}{2}+7\,a^5\,b^2\,x^3+\frac {35\,a^4\,b^3\,x^4}{4}+7\,a^3\,b^4\,x^5+\frac {7\,a^2\,b^5\,x^6}{2}+a\,b^6\,x^7+\frac {b^7\,x^8}{8} \]

[In]

int((a + b*x)^7,x)

[Out]

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