3.2.38 \(\int (3+4 x)^p \, dx\) [138]

Optimal. Leaf size=18 \[ \frac {(3+4 x)^{1+p}}{4 (1+p)} \]

[Out]

1/4*(3+4*x)^(1+p)/(1+p)

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Rubi [A]
time = 0.00, antiderivative size = 18, 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 {(4 x+3)^{p+1}}{4 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x)^p,x]

[Out]

(3 + 4*x)^(1 + p)/(4*(1 + p))

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 (3+4 x)^p \, dx &=\frac {(3+4 x)^{1+p}}{4 (1+p)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {(3+4 x)^{1+p}}{4+4 p} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 4*x)^p,x]

[Out]

(3 + 4*x)^(1 + p)/(4 + 4*p)

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Maple [A]
time = 0.40, size = 17, normalized size = 0.94

method	result	size
gosper	\(\frac {\left (3+4 x \right )^{1+p}}{4 p +4}\)	\(17\)
default	\(\frac {\left (3+4 x \right )^{1+p}}{4 p +4}\)	\(17\)
meijerg	\(3^{p} x \hypergeom \left (\left [1, -p \right ], \left [2\right ], -\frac {4 x}{3}\right )\)	\(17\)
risch	\(\frac {\left (3+4 x \right ) \left (3+4 x \right )^{p}}{4 p +4}\)	\(20\)
norman	\(\frac {x \,{\mathrm e}^{p \ln \left (3+4 x \right )}}{1+p}+\frac {3 \,{\mathrm e}^{p \ln \left (3+4 x \right )}}{4 \left (1+p \right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+4*x)^p,x,method=_RETURNVERBOSE)

[Out]

1/4*(3+4*x)^(1+p)/(1+p)

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Maxima [A]
time = 0.29, size = 16, normalized size = 0.89 \begin {gather*} \frac {{\left (4 \, x + 3\right )}^{p + 1}}{4 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+4*x)^p,x, algorithm="maxima")

[Out]

1/4*(4*x + 3)^(p + 1)/(p + 1)

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Fricas [A]
time = 1.28, size = 19, normalized size = 1.06 \begin {gather*} \frac {{\left (4 \, x + 3\right )}^{p} {\left (4 \, x + 3\right )}}{4 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+4*x)^p,x, algorithm="fricas")

[Out]

1/4*(4*x + 3)^p*(4*x + 3)/(p + 1)

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Sympy [A]
time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \frac {\begin {cases} \frac {\left (4 x + 3\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (4 x + 3 \right )} & \text {otherwise} \end {cases}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+4*x)**p,x)

[Out]

Piecewise(((4*x + 3)**(p + 1)/(p + 1), Ne(p, -1)), (log(4*x + 3), True))/4

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Giac [A]
time = 3.32, size = 16, normalized size = 0.89 \begin {gather*} \frac {{\left (4 \, x + 3\right )}^{p + 1}}{4 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+4*x)^p,x, algorithm="giac")

[Out]

1/4*(4*x + 3)^(p + 1)/(p + 1)

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Mupad [B]
time = 0.39, size = 32, normalized size = 1.78 \begin {gather*} \left \{\begin {array}{cl} \frac {\ln \left (4\,x+3\right )}{4} & \text {\ if\ \ }p=-1\\ \frac {{\left (4\,x+3\right )}^{p+1}}{4\,\left (p+1\right )} & \text {\ if\ \ }p\neq -1 \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + 3)^p,x)

[Out]

piecewise(p == -1, log(4*x + 3)/4, p ~= -1, (4*x + 3)^(p + 1)/(4*(p + 1)))

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