3.4.78 \(\int x^3 (a+b x)^{2/3} \, dx\) [378]

Optimal. Leaf size=72 \[ -\frac {3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac {9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac {9 a (a+b x)^{11/3}}{11 b^4}+\frac {3 (a+b x)^{14/3}}{14 b^4} \]

[Out]

-3/5*a^3*(b*x+a)^(5/3)/b^4+9/8*a^2*(b*x+a)^(8/3)/b^4-9/11*a*(b*x+a)^(11/3)/b^4+3/14*(b*x+a)^(14/3)/b^4

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Rubi [A]
time = 0.01, antiderivative size = 72, 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 = {45} \begin {gather*} -\frac {3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac {9 a^2 (a+b x)^{8/3}}{8 b^4}+\frac {3 (a+b x)^{14/3}}{14 b^4}-\frac {9 a (a+b x)^{11/3}}{11 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(2/3),x]

[Out]

(-3*a^3*(a + b*x)^(5/3))/(5*b^4) + (9*a^2*(a + b*x)^(8/3))/(8*b^4) - (9*a*(a + b*x)^(11/3))/(11*b^4) + (3*(a +
b*x)^(14/3))/(14*b^4)

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^3 (a+b x)^{2/3} \, dx &=\int \left (-\frac {a^3 (a+b x)^{2/3}}{b^3}+\frac {3 a^2 (a+b x)^{5/3}}{b^3}-\frac {3 a (a+b x)^{8/3}}{b^3}+\frac {(a+b x)^{11/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac {9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac {9 a (a+b x)^{11/3}}{11 b^4}+\frac {3 (a+b x)^{14/3}}{14 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.64 \begin {gather*} \frac {3 (a+b x)^{5/3} \left (-81 a^3+135 a^2 b x-180 a b^2 x^2+220 b^3 x^3\right )}{3080 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(5/3)*(-81*a^3 + 135*a^2*b*x - 180*a*b^2*x^2 + 220*b^3*x^3))/(3080*b^4)

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Maple [A]
time = 0.11, size = 50, normalized size = 0.69

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (-220 b^{3} x^{3}+180 a \,b^{2} x^{2}-135 a^{2} b x +81 a^{3}\right )}{3080 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {14}{3}}}{14}-\frac {9 a \left (b x +a \right )^{\frac {11}{3}}}{11}+\frac {9 a^{2} \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a^{3} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {14}{3}}}{14}-\frac {9 a \left (b x +a \right )^{\frac {11}{3}}}{11}+\frac {9 a^{2} \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a^{3} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{4}}\)	\(50\)
trager	\(-\frac {3 \left (-220 b^{4} x^{4}-40 a \,b^{3} x^{3}+45 a^{2} b^{2} x^{2}-54 a^{3} b x +81 a^{4}\right ) \left (b x +a \right )^{\frac {2}{3}}}{3080 b^{4}}\)	\(54\)
risch	\(-\frac {3 \left (-220 b^{4} x^{4}-40 a \,b^{3} x^{3}+45 a^{2} b^{2} x^{2}-54 a^{3} b x +81 a^{4}\right ) \left (b x +a \right )^{\frac {2}{3}}}{3080 b^{4}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3/b^4*(1/14*(b*x+a)^(14/3)-3/11*a*(b*x+a)^(11/3)+3/8*a^2*(b*x+a)^(8/3)-1/5*a^3*(b*x+a)^(5/3))

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Maxima [A]
time = 0.28, size = 56, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {14}{3}}}{14 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {11}{3}} a}{11 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {8}{3}} a^{2}}{8 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{3}}{5 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(2/3),x, algorithm="maxima")

[Out]

3/14*(b*x + a)^(14/3)/b^4 - 9/11*(b*x + a)^(11/3)*a/b^4 + 9/8*(b*x + a)^(8/3)*a^2/b^4 - 3/5*(b*x + a)^(5/3)*a^
3/b^4

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Fricas [A]
time = 0.43, size = 53, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (220 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} - 45 \, a^{2} b^{2} x^{2} + 54 \, a^{3} b x - 81 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3080 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(2/3),x, algorithm="fricas")

[Out]

3/3080*(220*b^4*x^4 + 40*a*b^3*x^3 - 45*a^2*b^2*x^2 + 54*a^3*b*x - 81*a^4)*(b*x + a)^(2/3)/b^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (68) = 136\).
time = 1.27, size = 1742, normalized size = 24.19 \begin {gather*} - \frac {243 a^{\frac {74}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {243 a^{\frac {74}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} - \frac {1296 a^{\frac {71}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {1458 a^{\frac {71}{3}} b x}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} - \frac {2808 a^{\frac {68}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {3645 a^{\frac {68}{3}} b^{2} x^{2}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} - \frac {3120 a^{\frac {65}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {4860 a^{\frac {65}{3}} b^{3} x^{3}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} - \frac {1050 a^{\frac {62}{3}} b^{4} x^{4} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {3645 a^{\frac {62}{3}} b^{4} x^{4}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {4032 a^{\frac {59}{3}} b^{5} x^{5} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {1458 a^{\frac {59}{3}} b^{5} x^{5}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {11004 a^{\frac {56}{3}} b^{6} x^{6} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {243 a^{\frac {56}{3}} b^{6} x^{6}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {14352 a^{\frac {53}{3}} b^{7} x^{7} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {10485 a^{\frac {50}{3}} b^{8} x^{8} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {4080 a^{\frac {47}{3}} b^{9} x^{9} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} + \frac {660 a^{\frac {44}{3}} b^{10} x^{10} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{3080 a^{20} b^{4} + 18480 a^{19} b^{5} x + 46200 a^{18} b^{6} x^{2} + 61600 a^{17} b^{7} x^{3} + 46200 a^{16} b^{8} x^{4} + 18480 a^{15} b^{9} x^{5} + 3080 a^{14} b^{10} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**(2/3),x)

[Out]

-243*a**(74/3)*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*
b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 243*a**(74/3)/(3080*a**20
*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**
15*b**9*x**5 + 3080*a**14*b**10*x**6) - 1296*a**(71/3)*b*x*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b
**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a
**14*b**10*x**6) + 1458*a**(71/3)*b*x/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a*
*17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) - 2808*a**(68/3)*b**2*x
**2*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 +
46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 3645*a**(68/3)*b**2*x**2/(3080*a**20
*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**
15*b**9*x**5 + 3080*a**14*b**10*x**6) - 3120*a**(65/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a
**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 +
3080*a**14*b**10*x**6) + 4860*a**(65/3)*b**3*x**3/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**
2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) - 1050*a**(
62/3)*b**4*x**4*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17
*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 3645*a**(62/3)*b**4*x**4
/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4
+ 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 4032*a**(59/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(3080*a**20*b*
*4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*
b**9*x**5 + 3080*a**14*b**10*x**6) + 1458*a**(59/3)*b**5*x**5/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a*
*18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6)
+ 11004*a**(56/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2
+ 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 243*a**(56/
3)*b**6*x**6/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**
16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 14352*a**(53/3)*b**7*x**7*(1 + b*x/a)**(2/3)/(
3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 +
18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 10485*a**(50/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(3080*a**20*b**
4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b
**9*x**5 + 3080*a**14*b**10*x**6) + 4080*a**(47/3)*b**9*x**9*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19
*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080
*a**14*b**10*x**6) + 660*a**(44/3)*b**10*x**10*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 4620
0*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x
**6)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (56) = 112\).
time = 1.12, size = 117, normalized size = 1.62 \begin {gather*} \frac {3 \, {\left (\frac {7 \, {\left (40 \, {\left (b x + a\right )}^{\frac {11}{3}} - 165 \, {\left (b x + a\right )}^{\frac {8}{3}} a + 264 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{2} - 220 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{3}\right )} a}{b^{3}} + \frac {2 \, {\left (110 \, {\left (b x + a\right )}^{\frac {14}{3}} - 560 \, {\left (b x + a\right )}^{\frac {11}{3}} a + 1155 \, {\left (b x + a\right )}^{\frac {8}{3}} a^{2} - 1232 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{3} + 770 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{4}\right )}}{b^{3}}\right )}}{3080 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(2/3),x, algorithm="giac")

[Out]

3/3080*(7*(40*(b*x + a)^(11/3) - 165*(b*x + a)^(8/3)*a + 264*(b*x + a)^(5/3)*a^2 - 220*(b*x + a)^(2/3)*a^3)*a/
b^3 + 2*(110*(b*x + a)^(14/3) - 560*(b*x + a)^(11/3)*a + 1155*(b*x + a)^(8/3)*a^2 - 1232*(b*x + a)^(5/3)*a^3 +
770*(b*x + a)^(2/3)*a^4)/b^3)/b

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Mupad [B]
time = 0.05, size = 56, normalized size = 0.78 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{14/3}}{14\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{5/3}}{5\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{8/3}}{8\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{11/3}}{11\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^(2/3),x)

[Out]

(3*(a + b*x)^(14/3))/(14*b^4) - (3*a^3*(a + b*x)^(5/3))/(5*b^4) + (9*a^2*(a + b*x)^(8/3))/(8*b^4) - (9*a*(a +
b*x)^(11/3))/(11*b^4)

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