3.23.86 \(\int (a+b \sqrt [3]{x}) x^3 \, dx\) [2286]

Optimal. Leaf size=19 \[ \frac {a x^4}{4}+\frac {3}{13} b x^{13/3} \]

[Out]

1/4*a*x^4+3/13*b*x^(13/3)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \begin {gather*} \frac {a x^4}{4}+\frac {3}{13} b x^{13/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))*x^3,x]

[Out]

(a*x^4)/4 + (3*b*x^(13/3))/13

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \left (a+b \sqrt [3]{x}\right ) x^3 \, dx &=\int \left (a x^3+b x^{10/3}\right ) \, dx\\ &=\frac {a x^4}{4}+\frac {3}{13} b x^{13/3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{52} \left (13 a+12 b \sqrt [3]{x}\right ) x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))*x^3,x]

[Out]

((13*a + 12*b*x^(1/3))*x^4)/52

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Maple [A]
time = 0.03, size = 14, normalized size = 0.74

method	result	size
derivativedivides	\(\frac {a \,x^{4}}{4}+\frac {3 b \,x^{\frac {13}{3}}}{13}\)	\(14\)
default	\(\frac {a \,x^{4}}{4}+\frac {3 b \,x^{\frac {13}{3}}}{13}\)	\(14\)
trager	\(\frac {a \left (x^{3}+x^{2}+x +1\right ) \left (x -1\right )}{4}+\frac {3 b \,x^{\frac {13}{3}}}{13}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4+3/13*b*x^(13/3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (13) = 26\).
time = 0.30, size = 200, normalized size = 10.53 \begin {gather*} \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13}}{13 \, b^{12}} - \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a}{4 \, b^{12}} + \frac {15 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{2}}{b^{12}} - \frac {99 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} a^{3}}{2 \, b^{12}} + \frac {110 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a^{4}}{b^{12}} - \frac {693 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{5}}{4 \, b^{12}} + \frac {198 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{6}}{b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{7}}{b^{12}} + \frac {99 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{8}}{b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{9}}{4 \, b^{12}} + \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{10}}{b^{12}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{11}}{2 \, b^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*(b*x^(1/3) + a)^13/b^12 - 11/4*(b*x^(1/3) + a)^12*a/b^12 + 15*(b*x^(1/3) + a)^11*a^2/b^12 - 99/2*(b*x^(1/
3) + a)^10*a^3/b^12 + 110*(b*x^(1/3) + a)^9*a^4/b^12 - 693/4*(b*x^(1/3) + a)^8*a^5/b^12 + 198*(b*x^(1/3) + a)^
7*a^6/b^12 - 165*(b*x^(1/3) + a)^6*a^7/b^12 + 99*(b*x^(1/3) + a)^5*a^8/b^12 - 165/4*(b*x^(1/3) + a)^4*a^9/b^12
+ 11*(b*x^(1/3) + a)^3*a^10/b^12 - 3/2*(b*x^(1/3) + a)^2*a^11/b^12

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Fricas [A]
time = 0.35, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{13} \, b x^{\frac {13}{3}} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*b*x^(13/3) + 1/4*a*x^4

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Sympy [A]
time = 0.69, size = 15, normalized size = 0.79 \begin {gather*} \frac {a x^{4}}{4} + \frac {3 b x^{\frac {13}{3}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 + 3*b*x**(13/3)/13

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Giac [A]
time = 1.35, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{13} \, b x^{\frac {13}{3}} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*b*x^(13/3) + 1/4*a*x^4

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.68 \begin {gather*} \frac {a\,x^4}{4}+\frac {3\,b\,x^{13/3}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^(1/3)),x)

[Out]

(a*x^4)/4 + (3*b*x^(13/3))/13

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