3.4.86 \(\int x^2 (a+b x)^{4/3} \, dx\) [386]

Optimal. Leaf size=53 \[ \frac {3 a^2 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{10/3}}{5 b^3}+\frac {3 (a+b x)^{13/3}}{13 b^3} \]

[Out]

3/7*a^2*(b*x+a)^(7/3)/b^3-3/5*a*(b*x+a)^(10/3)/b^3+3/13*(b*x+a)^(13/3)/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)^{7/3}}{7 b^3}+\frac {3 (a+b x)^{13/3}}{13 b^3}-\frac {3 a (a+b x)^{10/3}}{5 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(4/3),x]

[Out]

(3*a^2*(a + b*x)^(7/3))/(7*b^3) - (3*a*(a + b*x)^(10/3))/(5*b^3) + (3*(a + b*x)^(13/3))/(13*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)^{4/3} \, dx &=\int \left (\frac {a^2 (a+b x)^{4/3}}{b^2}-\frac {2 a (a+b x)^{7/3}}{b^2}+\frac {(a+b x)^{10/3}}{b^2}\right ) \, dx\\ &=\frac {3 a^2 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{10/3}}{5 b^3}+\frac {3 (a+b x)^{13/3}}{13 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)^{7/3} \left (9 a^2-21 a b x+35 b^2 x^2\right )}{455 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3)*(9*a^2 - 21*a*b*x + 35*b^2*x^2))/(455*b^3)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(225\) vs. \(2(53)=106\).
time = 7.87, size = 207, normalized size = 3.91 \begin {gather*} \frac {3 a^{\frac {1}{3}} \left (9 a^7 \left (-1+\left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+3 a^6 b x \left (-9+8 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^5 b^2 x^2 \left (-27+20 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^2 b^3 x^3 \left (-9 a^2+254 b^2 x^2 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+5 a^3 b^3 x^3 \left (11 a+37 b x\right ) \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+154 a b^6 x^6 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+35 b^7 x^7 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )}{455 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)^(4/3),x]')

[Out]

3 a ^ (1 / 3) (9 a ^ 7 (-1 + ((a + b x) / a) ^ (1 / 3)) + 3 a ^ 6 b x (-9 + 8 ((a + b x) / a) ^ (1 / 3)) + a ^
5 b ^ 2 x ^ 2 (-27 + 20 ((a + b x) / a) ^ (1 / 3)) + a ^ 2 b ^ 3 x ^ 3 (-9 a ^ 2 + 254 b ^ 2 x ^ 2 ((a + b x)
/ a) ^ (1 / 3)) + 5 a ^ 3 b ^ 3 x ^ 3 (11 a + 37 b x) ((a + b x) / a) ^ (1 / 3) + 154 a b ^ 6 x ^ 6 ((a + b x
) / a) ^ (1 / 3) + 35 b ^ 7 x ^ 7 ((a + b x) / a) ^ (1 / 3)) / (455 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 = 0.10, size = 38, normalized size = 0.72

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (35 x^{2} b^{2}-21 a b x +9 a^{2}\right )}{455 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {13}{3}}}{13}-\frac {3 a \left (b x +a \right )^{\frac {10}{3}}}{5}+\frac {3 a^{2} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{3}}\)	\(38\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {13}{3}}}{13}-\frac {3 a \left (b x +a \right )^{\frac {10}{3}}}{5}+\frac {3 a^{2} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{3}}\)	\(38\)
trager	\(\frac {3 \left (35 b^{4} x^{4}+49 a \,b^{3} x^{3}+2 a^{2} b^{2} x^{2}-3 a^{3} b x +9 a^{4}\right ) \left (b x +a \right )^{\frac {1}{3}}}{455 b^{3}}\)	\(54\)
risch	\(\frac {3 \left (35 b^{4} x^{4}+49 a \,b^{3} x^{3}+2 a^{2} b^{2} x^{2}-3 a^{3} b x +9 a^{4}\right ) \left (b x +a \right )^{\frac {1}{3}}}{455 b^{3}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^(4/3),x,method=_RETURNVERBOSE)

[Out]

3/b^3*(1/13*(b*x+a)^(13/3)-1/5*a*(b*x+a)^(10/3)+1/7*a^2*(b*x+a)^(7/3))

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Maxima [A]
time = 0.26, size = 41, normalized size = 0.77 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {13}{3}}}{13 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}} a}{5 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2}}{7 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(4/3),x, algorithm="maxima")

[Out]

3/13*(b*x + a)^(13/3)/b^3 - 3/5*(b*x + a)^(10/3)*a/b^3 + 3/7*(b*x + a)^(7/3)*a^2/b^3

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Fricas [A]
time = 0.30, size = 53, normalized size = 1.00 \begin {gather*} \frac {3 \, {\left (35 \, b^{4} x^{4} + 49 \, a b^{3} x^{3} + 2 \, a^{2} b^{2} x^{2} - 3 \, a^{3} b x + 9 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{455 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(4/3),x, algorithm="fricas")

[Out]

3/455*(35*b^4*x^4 + 49*a*b^3*x^3 + 2*a^2*b^2*x^2 - 3*a^3*b*x + 9*a^4)*(b*x + a)^(1/3)/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (49) = 98\).
time = 1.03, size = 733, normalized size = 13.83 \begin {gather*} \frac {27 a^{\frac {37}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {37}{3}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {34}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {34}{3}} b x}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {31}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {31}{3}} b^{2} x^{2}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {165 a^{\frac {28}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {28}{3}} b^{3} x^{3}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {555 a^{\frac {25}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {762 a^{\frac {22}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {462 a^{\frac {19}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {105 a^{\frac {16}{3}} b^{7} x^{7} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 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)**(4/3),x)

[Out]

27*a**(37/3)*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3)
- 27*a**(37/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 72*a**(34/3)*b*
x*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) - 81*a**(34
/3)*b*x/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 60*a**(31/3)*b**2*x**2
*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) - 81*a**(31/
3)*b**2*x**2/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 165*a**(28/3)*b**
3*x**3*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) - 27*a
**(28/3)*b**3*x**3/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 555*a**(25/
3)*b**4*x**4*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3)
+ 762*a**(22/3)*b**5*x**5*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**
5*b**6*x**3) + 462*a**(19/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x
**2 + 455*a**5*b**6*x**3) + 105*a**(16/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 136
5*a**6*b**5*x**2 + 455*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. 157 vs. \(2 (41) = 82\).
time = 0.00, size = 256, normalized size = 4.83 \begin {gather*} \frac {\frac {3 b^{2} \left (\frac {1}{13} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{4}-\frac {2}{5} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3} a+\frac {6}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a^{2}-\left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{3}+\left (a+b x\right )^{\frac {1}{3}} a^{4}\right )}{b^{4}}+\frac {6 a b \left (\frac {1}{10} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3}-\frac {3}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a+\frac {3}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{2}-\left (a+b x\right )^{\frac {1}{3}} a^{3}\right )}{b^{3}}+\frac {3 a^{2} \left (\frac {1}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2}-\frac {1}{2} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a+\left (a+b x\right )^{\frac {1}{3}} 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)^(4/3),x)

[Out]

3/910*(65*(2*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a + 14*(b*x + a)^(1/3)*a^2)*a^2/b^2 + 13*(14*(b*x + a)^(10/3)
- 60*(b*x + a)^(7/3)*a + 105*(b*x + a)^(4/3)*a^2 - 140*(b*x + a)^(1/3)*a^3)*a/b^2 + 2*(35*(b*x + a)^(13/3) -
182*(b*x + a)^(10/3)*a + 390*(b*x + a)^(7/3)*a^2 - 455*(b*x + a)^(4/3)*a^3 + 455*(b*x + a)^(1/3)*a^4)/b^2)/b

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {105\,{\left (a+b\,x\right )}^{13/3}-273\,a\,{\left (a+b\,x\right )}^{10/3}+195\,a^2\,{\left (a+b\,x\right )}^{7/3}}{455\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(4/3),x)

[Out]

(105*(a + b*x)^(13/3) - 273*a*(a + b*x)^(10/3) + 195*a^2*(a + b*x)^(7/3))/(455*b^3)

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