3.4.79 \(\int x^2 (a+b x)^{2/3} \, dx\) [379]

Optimal. Leaf size=53 \[ \frac {3 a^2 (a+b x)^{5/3}}{5 b^3}-\frac {3 a (a+b x)^{8/3}}{4 b^3}+\frac {3 (a+b x)^{11/3}}{11 b^3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 53, 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^2 (a+b x)^{5/3}}{5 b^3}+\frac {3 (a+b x)^{11/3}}{11 b^3}-\frac {3 a (a+b x)^{8/3}}{4 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(2/3),x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {3 (a+b x)^{5/3} \left (9 a^2-15 a b x+20 b^2 x^2\right )}{220 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(5/3)*(9*a^2 - 15*a*b*x + 20*b^2*x^2))/(220*b^3)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(53)=106\).
time = 7.41, size = 194, normalized size = 3.66 \begin {gather*} \frac {3 a^{\frac {2}{3}} \left (9 a^6 \left (-1+\left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+3 a^5 b x \left (-9+7 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+a^4 b^2 x^2 \left (-27+14 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+3 a^2 b^3 x^3 \left (-3 a+23 b x \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+26 a^3 b^3 x^3 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}+65 a b^5 x^5 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}+20 b^6 x^6 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )}{220 b^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^2*(a + b*x)^(2/3),x]')

[Out]

3 a ^ (2 / 3) (9 a ^ 6 (-1 + ((a + b x) / a) ^ (2 / 3)) + 3 a ^ 5 b x (-9 + 7 ((a + b x) / a) ^ (2 / 3)) + a ^
4 b ^ 2 x ^ 2 (-27 + 14 ((a + b x) / a) ^ (2 / 3)) + 3 a ^ 2 b ^ 3 x ^ 3 (-3 a + 23 b x ((a + b x) / a) ^ (2
/ 3)) + 26 a ^ 3 b ^ 3 x ^ 3 ((a + b x) / a) ^ (2 / 3) + 65 a b ^ 5 x ^ 5 ((a + b x) / a) ^ (2 / 3) + 20 b ^ 6
x ^ 6 ((a + b x) / a) ^ (2 / 3)) / (220 b ^ 3 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

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Maple [A]
time = 0.11, size = 38, normalized size = 0.72

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (20 x^{2} b^{2}-15 a b x +9 a^{2}\right )}{220 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {11}{3}}}{11}-\frac {3 a \left (b x +a \right )^{\frac {8}{3}}}{4}+\frac {3 a^{2} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{3}}\)	\(38\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {11}{3}}}{11}-\frac {3 a \left (b x +a \right )^{\frac {8}{3}}}{4}+\frac {3 a^{2} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{3}}\)	\(38\)
trager	\(\frac {3 \left (20 b^{3} x^{3}+5 a \,b^{2} x^{2}-6 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {2}{3}}}{220 b^{3}}\)	\(43\)
risch	\(\frac {3 \left (20 b^{3} x^{3}+5 a \,b^{2} x^{2}-6 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {2}{3}}}{220 b^{3}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 41, normalized size = 0.77 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {11}{3}}}{11 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}} a}{4 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{2}}{5 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 42, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (20 \, b^{3} x^{3} + 5 \, a b^{2} x^{2} - 6 \, a^{2} b x + 9 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{220 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/220*(20*b^3*x^3 + 5*a*b^2*x^2 - 6*a^2*b*x + 9*a^3)*(b*x + a)^(2/3)/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (49) = 98\).
time = 0.92, size = 666, normalized size = 12.57 \begin {gather*} \frac {27 a^{\frac {35}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {35}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {63 a^{\frac {32}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {32}{3}} b x}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {42 a^{\frac {29}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {29}{3}} b^{2} x^{2}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {78 a^{\frac {26}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {26}{3}} b^{3} x^{3}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {207 a^{\frac {23}{3}} b^{4} x^{4} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {195 a^{\frac {20}{3}} b^{5} x^{5} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {17}{3}} b^{6} x^{6} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*a**(35/3)*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) -
27*a**(35/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 63*a**(32/3)*b*x*(1
+ b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 81*a**(32/3)*b*
x/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 42*a**(29/3)*b**2*x**2*(1 + b*
x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 81*a**(29/3)*b**2*x*
*2/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 78*a**(26/3)*b**3*x**3*(1 + b
*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 27*a**(26/3)*b**3*x
**3/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 207*a**(23/3)*b**4*x**4*(1 +
b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 195*a**(20/3)*b**
5*x**5*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 60*a**
(17/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).
time = 0.00, size = 166, normalized size = 3.13 \begin {gather*} \frac {\frac {3 b \left (\frac {1}{11} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{3}-\frac {3}{8} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{2} a+\frac {3}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right ) a^{2}-\frac {1}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a^{3}\right )}{b^{3}}+\frac {3 a \left (\frac {1}{8} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{2}-\frac {2}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right ) a+\frac {1}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a^{2}\right )}{b^{2}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/440*(11*(5*(b*x + a)^(8/3) - 16*(b*x + a)^(5/3)*a + 20*(b*x + a)^(2/3)*a^2)*a/b^2 + (40*(b*x + a)^(11/3) - 1
65*(b*x + a)^(8/3)*a + 264*(b*x + a)^(5/3)*a^2 - 220*(b*x + a)^(2/3)*a^3)/b^2)/b

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {60\,{\left (a+b\,x\right )}^{11/3}-165\,a\,{\left (a+b\,x\right )}^{8/3}+132\,a^2\,{\left (a+b\,x\right )}^{5/3}}{220\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(2/3),x)

[Out]

(60*(a + b*x)^(11/3) - 165*a*(a + b*x)^(8/3) + 132*a^2*(a + b*x)^(5/3))/(220*b^3)

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