\(\int \sqrt [3]{x} (a+b x) \, dx\) [653]

Optimal result

Integrand size = 11, antiderivative size = 21 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3}{4} a x^{4/3}+\frac {3}{7} b x^{7/3} \]

[Out]

3/4*a*x^(4/3)+3/7*b*x^(7/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3}{4} a x^{4/3}+\frac {3}{7} b x^{7/3} \]

[In]

Int[x^(1/3)*(a + b*x),x]

[Out]

(3*a*x^(4/3))/4 + (3*b*x^(7/3))/7

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a \sqrt [3]{x}+b x^{4/3}\right ) \, dx \\ & = \frac {3}{4} a x^{4/3}+\frac {3}{7} b x^{7/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3}{28} x^{4/3} (7 a+4 b x) \]

[In]

Integrate[x^(1/3)*(a + b*x),x]

[Out]

(3*x^(4/3)*(7*a + 4*b*x))/28

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
gosper	\(\frac {3 x^{\frac {4}{3}} \left (4 b x +7 a \right )}{28}\)	\(14\)
derivativedivides	\(\frac {3 a \,x^{\frac {4}{3}}}{4}+\frac {3 b \,x^{\frac {7}{3}}}{7}\)	\(14\)
default	\(\frac {3 a \,x^{\frac {4}{3}}}{4}+\frac {3 b \,x^{\frac {7}{3}}}{7}\)	\(14\)
trager	\(\frac {3 x^{\frac {4}{3}} \left (4 b x +7 a \right )}{28}\)	\(14\)
risch	\(\frac {3 x^{\frac {4}{3}} \left (4 b x +7 a \right )}{28}\)	\(14\)

[In]

int(x^(1/3)*(b*x+a),x,method=_RETURNVERBOSE)

[Out]

3/28*x^(4/3)*(4*b*x+7*a)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3}{28} \, {\left (4 \, b x^{2} + 7 \, a x\right )} x^{\frac {1}{3}} \]

[In]

integrate(x^(1/3)*(b*x+a),x, algorithm="fricas")

[Out]

3/28*(4*b*x^2 + 7*a*x)*x^(1/3)

Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3 a x^{\frac {4}{3}}}{4} + \frac {3 b x^{\frac {7}{3}}}{7} \]

[In]

integrate(x**(1/3)*(b*x+a),x)

[Out]

3*a*x**(4/3)/4 + 3*b*x**(7/3)/7

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3}{7} \, b x^{\frac {7}{3}} + \frac {3}{4} \, a x^{\frac {4}{3}} \]

[In]

integrate(x^(1/3)*(b*x+a),x, algorithm="maxima")

[Out]

3/7*b*x^(7/3) + 3/4*a*x^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3}{7} \, b x^{\frac {7}{3}} + \frac {3}{4} \, a x^{\frac {4}{3}} \]

[In]

integrate(x^(1/3)*(b*x+a),x, algorithm="giac")

[Out]

3/7*b*x^(7/3) + 3/4*a*x^(4/3)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{x} (a+b x) \, dx=\frac {3\,x^{4/3}\,\left (7\,a+4\,b\,x\right )}{28} \]

[In]

int(x^(1/3)*(a + b*x),x)

[Out]

(3*x^(4/3)*(7*a + 4*b*x))/28