3.138 \(\int (3+4 x)^p \, dx\)

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

[Out]

(3 + 4*x)^(1 + p)/(4*(1 + p))

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Rubi [A]  time = 0.0018257, antiderivative size = 18, 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} \[ \frac{(4 x+3)^{p+1}}{4 (p+1)} \]

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*}

Mathematica [A]  time = 0.0072413, size = 17, normalized size = 0.94 \[ \frac{(4 x+3)^{p+1}}{4 p+4} \]

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 = 0.044, size = 17, normalized size = 0.9 \begin{align*}{\frac{ \left ( 4\,x+3 \right ) ^{1+p}}{4\,p+4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x+3)^p,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.95592, size = 47, normalized size = 2.61 \begin{align*} \frac{{\left (4 \, x + 3\right )}^{p}{\left (4 \, x + 3\right )}}{4 \,{\left (p + 1\right )}} \end{align*}

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 = 0.062529, size = 20, normalized size = 1.11 \begin{align*} \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{align*}

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 = 1.28638, size = 22, normalized size = 1.22 \begin{align*} \frac{{\left (4 \, x + 3\right )}^{p + 1}}{4 \,{\left (p + 1\right )}} \end{align*}

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)