3.2.6 \(\int (a+b x)^7 \, dx\)

Optimal. Leaf size=14 \[ \frac {(a+b x)^8}{8 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} \frac {(a+b x)^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (a+b x)^7 \, dx &=\frac {(a+b x)^8}{8 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7,x]

[Out]

(a + b*x)^8/(8*b)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^7 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^7,x]

[Out]

IntegrateAlgebraic[(a + b*x)^7, x]

________________________________________________________________________________________

fricas [B]  time = 0.89, size = 75, normalized size = 5.36 \begin {gather*} \frac {1}{8} x^{8} b^{7} + x^{7} b^{6} a + \frac {7}{2} x^{6} b^{5} a^{2} + 7 x^{5} b^{4} a^{3} + \frac {35}{4} x^{4} b^{3} a^{4} + 7 x^{3} b^{2} a^{5} + \frac {7}{2} x^{2} b a^{6} + x a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.06, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )}^{8}}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x + a)^8/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} \frac {\left (b x +a \right )^{8}}{8 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.39, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )}^{8}}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x + a)^8/b

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 75, normalized size = 5.36 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [B]  time = 0.08, size = 83, normalized size = 5.93 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________