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

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int (a+b x)^{4/3} \, dx=\frac {3 (a+b x)^{7/3}}{7 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int (a+b x)^{4/3} \, dx=\frac {3 (a+b x)^{7/3}}{7 b} \]

[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*} \text {integral}& = \frac {3 (a+b x)^{7/3}}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int (a+b x)^{4/3} \, dx=\frac {3 (a+b x)^{7/3}}{7 b} \]

[In]

Integrate[(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3))/(7*b)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7 b}\)	\(13\)
pseudoelliptic	\(\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7 b}\)	\(13\)
trager	\(\frac {3 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7 b}\)	\(29\)
risch	\(\frac {3 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7 b}\)	\(29\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int (a+b x)^{4/3} \, dx=\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{7 \, b} \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (a+b x)^{4/3} \, dx=\frac {3 \left (a + b x\right )^{\frac {7}{3}}}{7 b} \]

[In]

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

[Out]

3*(a + b*x)**(7/3)/(7*b)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (a+b x)^{4/3} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}}}{7 \, b} \]

[In]

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

[Out]

3/7*(b*x + a)^(7/3)/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).

Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.62 \[ \int (a+b x)^{4/3} \, dx=\frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 28 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2} + 7 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a\right )}}{14 \, b} \]

[In]

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

[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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (a+b x)^{4/3} \, dx=\frac {3\,{\left (a+b\,x\right )}^{7/3}}{7\,b} \]

[In]

int((a + b*x)^(4/3),x)

[Out]

(3*(a + b*x)^(7/3))/(7*b)