3.7.51 \(\int x^{4/3} (a+b x) \, dx\) [651]

Optimal. Leaf size=21 \[ \frac {3}{7} a x^{7/3}+\frac {3}{10} b x^{10/3} \]

[Out]

3/7*a*x^(7/3)+3/10*b*x^(10/3)

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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 = {45} \begin {gather*} \frac {3}{7} a x^{7/3}+\frac {3}{10} b x^{10/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(4/3)*(a + b*x),x]

[Out]

(3*a*x^(7/3))/7 + (3*b*x^(10/3))/10

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*} \int x^{4/3} (a+b x) \, dx &=\int \left (a x^{4/3}+b x^{7/3}\right ) \, dx\\ &=\frac {3}{7} a x^{7/3}+\frac {3}{10} b x^{10/3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} \frac {3}{70} x^{7/3} (10 a+7 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(4/3)*(a + b*x),x]

[Out]

(3*x^(7/3)*(10*a + 7*b*x))/70

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Mathics [A]
time = 1.74, size = 13, normalized size = 0.62 \begin {gather*} \frac {3 x^{\frac {7}{3}} \left (10 a+7 b x\right )}{70} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3 x ^ (7 / 3) (10 a + 7 b x) / 70

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Maple [A]
time = 0.05, size = 14, normalized size = 0.67

method	result	size
gosper	\(\frac {3 x^{\frac {7}{3}} \left (7 b x +10 a \right )}{70}\)	\(14\)
derivativedivides	\(\frac {3 a \,x^{\frac {7}{3}}}{7}+\frac {3 b \,x^{\frac {10}{3}}}{10}\)	\(14\)
default	\(\frac {3 a \,x^{\frac {7}{3}}}{7}+\frac {3 b \,x^{\frac {10}{3}}}{10}\)	\(14\)
trager	\(\frac {3 x^{\frac {7}{3}} \left (7 b x +10 a \right )}{70}\)	\(14\)
risch	\(\frac {3 x^{\frac {7}{3}} \left (7 b x +10 a \right )}{70}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*a*x^(7/3)+3/10*b*x^(10/3)

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.62 \begin {gather*} \frac {3}{10} \, b x^{\frac {10}{3}} + \frac {3}{7} \, a x^{\frac {7}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*b*x^(10/3) + 3/7*a*x^(7/3)

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Fricas [A]
time = 0.30, size = 18, normalized size = 0.86 \begin {gather*} \frac {3}{70} \, {\left (7 \, b x^{3} + 10 \, a x^{2}\right )} x^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/70*(7*b*x^3 + 10*a*x^2)*x^(1/3)

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Sympy [A]
time = 0.24, size = 19, normalized size = 0.90 \begin {gather*} \frac {3 a x^{\frac {7}{3}}}{7} + \frac {3 b x^{\frac {10}{3}}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a*x**(7/3)/7 + 3*b*x**(10/3)/10

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Giac [A]
time = 0.00, size = 27, normalized size = 1.29 \begin {gather*} \frac {3}{10} b x^{\frac {1}{3}} x^{3}+\frac {3}{7} a x^{\frac {1}{3}} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*b*x^(10/3) + 3/7*a*x^(7/3)

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.62 \begin {gather*} \frac {3\,x^{7/3}\,\left (10\,a+7\,b\,x\right )}{70} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(4/3)*(a + b*x),x)

[Out]

(3*x^(7/3)*(10*a + 7*b*x))/70

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