3.4.88 \(\int (a+b x)^{4/3} \, dx\) [388]

Optimal. Leaf size=16 \[ \frac {3 (a+b x)^{7/3}}{7 b} \]

[Out]

3/7*(b*x+a)^(7/3)/b

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \begin {gather*} \frac {3 (a+b x)^{7/3}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3))/(7*b)

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 (a+b x)^{4/3} \, dx &=\frac {3 (a+b x)^{7/3}}{7 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 (a+b x)^{7/3}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3))/(7*b)

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Mathics [A]
time = 1.59, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \left (a+b x\right )^{\frac {7}{3}}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^(4/3),x]')

[Out]

3 (a + b x) ^ (7 / 3) / (7 b)

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Maple [A]
time = 0.11, size = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7 b}\)	\(13\)
derivativedivides	\(\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7 b}\)	\(13\)
default	\(\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7 b}\)	\(13\)
trager	\(\frac {3 \left (x^{2} b^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7 b}\)	\(29\)
risch	\(\frac {3 \left (x^{2} b^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7 b}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(4/3),x,method=_RETURNVERBOSE)

[Out]

3/7*(b*x+a)^(7/3)/b

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Maxima [A]
time = 0.26, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}}}{7 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3),x, algorithm="maxima")

[Out]

3/7*(b*x + a)^(7/3)/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).
time = 0.31, size = 28, normalized size = 1.75 \begin {gather*} \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{7 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3),x, algorithm="fricas")

[Out]

3/7*(b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)^(1/3)/b

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Sympy [A]
time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \left (a + b x\right )^{\frac {7}{3}}}{7 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(4/3),x)

[Out]

3*(a + b*x)**(7/3)/(7*b)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).
time = 0.00, size = 112, normalized size = 7.00 \begin {gather*} \frac {6 a \left (\frac {1}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )-a \left (a+b x\right )^{\frac {1}{3}}\right )+\frac {3 b^{2} \left (\frac {1}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2}-\frac {1}{2} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a+\left (a+b x\right )^{\frac {1}{3}} a^{2}\right )}{b^{2}}+3 a^{2} \left (a+b x\right )^{\frac {1}{3}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3),x)

[Out]

3/14*(2*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a + 28*(b*x + a)^(1/3)*a^2 + 7*((b*x + a)^(4/3) - 4*(b*x + a)^(1/3
)*a)*a)/b

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{7/3}}{7\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(4/3),x)

[Out]

(3*(a + b*x)^(7/3))/(7*b)

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