3.1282 \(\int (c+d x)^7 \, dx\)

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

[Out]

(c + d*x)^8/(8*d)

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Rubi [A]  time = 0.0015101, antiderivative size = 14, normalized size of antiderivative = 1., 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} \[ \frac{(c+d x)^8}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7,x]

[Out]

(c + d*x)^8/(8*d)

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 (c+d x)^7 \, dx &=\frac{(c+d x)^8}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.0015981, size = 14, normalized size = 1. \[ \frac{(c+d x)^8}{8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7,x]

[Out]

(c + d*x)^8/(8*d)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( dx+c \right ) ^{8}}{8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7,x)

[Out]

1/8*(d*x+c)^8/d

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Maxima [A]  time = 0.943771, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (d x + c\right )}^{8}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7,x, algorithm="maxima")

[Out]

1/8*(d*x + c)^8/d

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Fricas [B]  time = 1.97808, size = 159, normalized size = 11.36 \begin{align*} \frac{1}{8} x^{8} d^{7} + x^{7} d^{6} c + \frac{7}{2} x^{6} d^{5} c^{2} + 7 x^{5} d^{4} c^{3} + \frac{35}{4} x^{4} d^{3} c^{4} + 7 x^{3} d^{2} c^{5} + \frac{7}{2} x^{2} d c^{6} + x c^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7,x, algorithm="fricas")

[Out]

1/8*x^8*d^7 + x^7*d^6*c + 7/2*x^6*d^5*c^2 + 7*x^5*d^4*c^3 + 35/4*x^4*d^3*c^4 + 7*x^3*d^2*c^5 + 7/2*x^2*d*c^6 +
x*c^7

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Sympy [B]  time = 0.070815, size = 83, normalized size = 5.93 \begin{align*} c^{7} x + \frac{7 c^{6} d x^{2}}{2} + 7 c^{5} d^{2} x^{3} + \frac{35 c^{4} d^{3} x^{4}}{4} + 7 c^{3} d^{4} x^{5} + \frac{7 c^{2} d^{5} x^{6}}{2} + c d^{6} x^{7} + \frac{d^{7} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**7,x)

[Out]

c**7*x + 7*c**6*d*x**2/2 + 7*c**5*d**2*x**3 + 35*c**4*d**3*x**4/4 + 7*c**3*d**4*x**5 + 7*c**2*d**5*x**6/2 + c*
d**6*x**7 + d**7*x**8/8

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Giac [A]  time = 1.04431, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (d x + c\right )}^{8}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7,x, algorithm="giac")

[Out]

1/8*(d*x + c)^8/d