3.7.50 \(\int x^{5/3} (a+b x) \, dx\)

Optimal. Leaf size=21 \[ \frac {3}{8} a x^{8/3}+\frac {3}{11} b x^{11/3} \]

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {3}{8} a x^{8/3}+\frac {3}{11} b x^{11/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/3)*(a + b*x),x]

[Out]

(3*a*x^(8/3))/8 + (3*b*x^(11/3))/11

Rule 43

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*} \int x^{5/3} (a+b x) \, dx &=\int \left (a x^{5/3}+b x^{8/3}\right ) \, dx\\ &=\frac {3}{8} a x^{8/3}+\frac {3}{11} b x^{11/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} \frac {3}{88} x^{8/3} (11 a+8 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/3)*(a + b*x),x]

[Out]

(3*x^(8/3)*(11*a + 8*b*x))/88

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IntegrateAlgebraic [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {3}{88} \left (11 a x^{8/3}+8 b x^{11/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/3)*(a + b*x),x]

[Out]

(3*(11*a*x^(8/3) + 8*b*x^(11/3)))/88

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fricas [A]  time = 1.25, size = 18, normalized size = 0.86 \begin {gather*} \frac {3}{88} \, {\left (8 \, b x^{3} + 11 \, a x^{2}\right )} x^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a),x, algorithm="fricas")

[Out]

3/88*(8*b*x^3 + 11*a*x^2)*x^(2/3)

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giac [A]  time = 0.98, size = 13, normalized size = 0.62 \begin {gather*} \frac {3}{11} \, b x^{\frac {11}{3}} + \frac {3}{8} \, a x^{\frac {8}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a),x, algorithm="giac")

[Out]

3/11*b*x^(11/3) + 3/8*a*x^(8/3)

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maple [A]  time = 0.00, size = 14, normalized size = 0.67 \begin {gather*} \frac {3 \left (8 b x +11 a \right ) x^{\frac {8}{3}}}{88} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/3)*(b*x+a),x)

[Out]

3/88*x^(8/3)*(8*b*x+11*a)

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maxima [A]  time = 1.30, size = 13, normalized size = 0.62 \begin {gather*} \frac {3}{11} \, b x^{\frac {11}{3}} + \frac {3}{8} \, a x^{\frac {8}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a),x, algorithm="maxima")

[Out]

3/11*b*x^(11/3) + 3/8*a*x^(8/3)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.62 \begin {gather*} \frac {3\,x^{8/3}\,\left (11\,a+8\,b\,x\right )}{88} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/3)*(a + b*x),x)

[Out]

(3*x^(8/3)*(11*a + 8*b*x))/88

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sympy [A]  time = 2.01, size = 19, normalized size = 0.90 \begin {gather*} \frac {3 a x^{\frac {8}{3}}}{8} + \frac {3 b x^{\frac {11}{3}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/3)*(b*x+a),x)

[Out]

3*a*x**(8/3)/8 + 3*b*x**(11/3)/11

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