3.13.82 \(\int (c+d x)^7 \, dx\) [1282]

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

[Out]

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

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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 {(c+d x)^8}{8 d} \end {gather*}

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*}

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

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 = 0.12, size = 13, normalized size = 0.93

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (d x + c\right )}^{8}}{8 \, d} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (12) = 24\).
time = 1.04, size = 75, normalized size = 5.36 \begin {gather*} \frac {1}{8} \, d^{7} x^{8} + c d^{6} x^{7} + \frac {7}{2} \, c^{2} d^{5} x^{6} + 7 \, c^{3} d^{4} x^{5} + \frac {35}{4} \, c^{4} d^{3} x^{4} + 7 \, c^{5} d^{2} x^{3} + \frac {7}{2} \, c^{6} d x^{2} + c^{7} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (8) = 16\).
time = 0.01, size = 83, normalized size = 5.93 \begin {gather*} 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 {gather*}

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 = 0.87, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (d x + c\right )}^{8}}{8 \, d} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7,x)

[Out]

c^7*x + (d^7*x^8)/8 + (7*c^6*d*x^2)/2 + c*d^6*x^7 + 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

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