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

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

[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} \[ \frac {(4 x+3)^{p+1}}{4 (p+1)} \]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.88, size = 19, normalized size = 1.06 \[ \frac {{\left (4 \, x + 3\right )}^{p} {\left (4 \, x + 3\right )}}{4 \, {\left (p + 1\right )}} \]

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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giac [A]  time = 0.62, size = 16, normalized size = 0.89 \[ \frac {{\left (4 \, x + 3\right )}^{p + 1}}{4 \, {\left (p + 1\right )}} \]

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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maple [A]  time = 0.04, size = 17, normalized size = 0.94 \[ \frac {\left (4 x +3\right )^{p +1}}{4 p +4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.25, size = 16, normalized size = 0.89 \[ \frac {{\left (4 \, x + 3\right )}^{p + 1}}{4 \, {\left (p + 1\right )}} \]

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

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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sympy [A]  time = 0.06, size = 20, normalized size = 1.11 \[ \frac {\begin {cases} \frac {\left (4 x + 3\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (4 x + 3 \right )} & \text {otherwise} \end {cases}}{4} \]

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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