3.5.6 \(\int \frac {x^3}{(a+b x)^{2/3}} \, dx\) [406]

Optimal. Leaf size=70 \[ -\frac {3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac {9 a^2 (a+b x)^{4/3}}{4 b^4}-\frac {9 a (a+b x)^{7/3}}{7 b^4}+\frac {3 (a+b x)^{10/3}}{10 b^4} \]

[Out]

-3*a^3*(b*x+a)^(1/3)/b^4+9/4*a^2*(b*x+a)^(4/3)/b^4-9/7*a*(b*x+a)^(7/3)/b^4+3/10*(b*x+a)^(10/3)/b^4

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Rubi [A]
time = 0.01, antiderivative size = 70, 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 \sqrt [3]{a+b x}}{b^4}+\frac {9 a^2 (a+b x)^{4/3}}{4 b^4}+\frac {3 (a+b x)^{10/3}}{10 b^4}-\frac {9 a (a+b x)^{7/3}}{7 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)^(1/3))/b^4 + (9*a^2*(a + b*x)^(4/3))/(4*b^4) - (9*a*(a + b*x)^(7/3))/(7*b^4) + (3*(a + b*x)^
(10/3))/(10*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 \frac {x^3}{(a+b x)^{2/3}} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^{2/3}}+\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{b^3}+\frac {(a+b x)^{7/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac {9 a^2 (a+b x)^{4/3}}{4 b^4}-\frac {9 a (a+b x)^{7/3}}{7 b^4}+\frac {3 (a+b x)^{10/3}}{10 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.66 \begin {gather*} \frac {3 \sqrt [3]{a+b x} \left (-81 a^3+27 a^2 b x-18 a b^2 x^2+14 b^3 x^3\right )}{140 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(1/3)*(-81*a^3 + 27*a^2*b*x - 18*a*b^2*x^2 + 14*b^3*x^3))/(140*b^4)

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Maple [A]
time = 0.11, size = 50, normalized size = 0.71

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +81 a^{3}\right )}{140 b^{4}}\)	\(43\)
trager	\(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +81 a^{3}\right )}{140 b^{4}}\)	\(43\)
risch	\(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +81 a^{3}\right )}{140 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {9 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {9 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a^{3} \left (b x +a \right )^{\frac {1}{3}}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {9 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {9 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a^{3} \left (b x +a \right )^{\frac {1}{3}}}{b^{4}}\)	\(50\)

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*(1/10*(b*x+a)^(10/3)-3/7*a*(b*x+a)^(7/3)+3/4*a^2*(b*x+a)^(4/3)-a^3*(b*x+a)^(1/3))

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Maxima [A]
time = 0.28, size = 56, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}}}{10 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2}}{4 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}}{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]

3/10*(b*x + a)^(10/3)/b^4 - 9/7*(b*x + a)^(7/3)*a/b^4 + 9/4*(b*x + a)^(4/3)*a^2/b^4 - 3*(b*x + a)^(1/3)*a^3/b^
4

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Fricas [A]
time = 0.68, size = 42, normalized size = 0.60 \begin {gather*} \frac {3 \, {\left (14 \, b^{3} x^{3} - 18 \, a b^{2} x^{2} + 27 \, a^{2} b x - 81 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{140 \, 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]

3/140*(14*b^3*x^3 - 18*a*b^2*x^2 + 27*a^2*b*x - 81*a^3)*(b*x + a)^(1/3)/b^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1640 vs. \(2 (66) = 132\).
time = 1.20, size = 1640, normalized size = 23.43 \begin {gather*} - \frac {243 a^{\frac {70}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {243 a^{\frac {70}{3}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} - \frac {1377 a^{\frac {67}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {1458 a^{\frac {67}{3}} b x}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} - \frac {3213 a^{\frac {64}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {3645 a^{\frac {64}{3}} b^{2} x^{2}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} - \frac {3927 a^{\frac {61}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {4860 a^{\frac {61}{3}} b^{3} x^{3}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} - \frac {2583 a^{\frac {58}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {3645 a^{\frac {58}{3}} b^{4} x^{4}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} - \frac {693 a^{\frac {55}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {1458 a^{\frac {55}{3}} b^{5} x^{5}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {273 a^{\frac {52}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {243 a^{\frac {52}{3}} b^{6} x^{6}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {387 a^{\frac {49}{3}} b^{7} x^{7} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {198 a^{\frac {46}{3}} b^{8} x^{8} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 a^{14} b^{10} x^{6}} + \frac {42 a^{\frac {43}{3}} b^{9} x^{9} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{20} b^{4} + 840 a^{19} b^{5} x + 2100 a^{18} b^{6} x^{2} + 2800 a^{17} b^{7} x^{3} + 2100 a^{16} b^{8} x^{4} + 840 a^{15} b^{9} x^{5} + 140 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**(70/3)*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*
x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 243*a**(70/3)/(140*a**20*b**4 + 84
0*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 14
0*a**14*b**10*x**6) - 1377*a**(67/3)*b*x*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b*
*6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 1458*a**
(67/3)*b*x/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*
x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) - 3213*a**(64/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(140*a**20*b*
*4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x*
*5 + 140*a**14*b**10*x**6) + 3645*a**(64/3)*b**2*x**2/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**
2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) - 3927*a**(61/3)
*b**3*x**3*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3
+ 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 4860*a**(61/3)*b**3*x**3/(140*a**20*b*
*4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x*
*5 + 140*a**14*b**10*x**6) - 2583*a**(58/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x +
2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**
6) + 3645*a**(58/3)*b**4*x**4/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3
+ 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) - 693*a**(55/3)*b**5*x**5*(1 + b*x/a)**(
1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 +
840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 1458*a**(55/3)*b**5*x**5/(140*a**20*b**4 + 840*a**19*b**5*x + 2
100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6
) + 273*a**(52/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 280
0*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 243*a**(52/3)*b**6*x*
*6/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 8
40*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 387*a**(49/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*
a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*
a**14*b**10*x**6) + 198*a**(46/3)*b**8*x**8*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18
*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 42*a*
*(43/3)*b**9*x**9*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b*
*7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6)

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Giac [A]
time = 1.34, size = 49, normalized size = 0.70 \begin {gather*} \frac {3 \, {\left (14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}\right )}}{140 \, 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="giac")

[Out]

3/140*(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)/b^4

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Mupad [B]
time = 0.05, size = 56, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{10/3}}{10\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{1/3}}{b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{4/3}}{4\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{7/3}}{7\,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)^(10/3))/(10*b^4) - (3*a^3*(a + b*x)^(1/3))/b^4 + (9*a^2*(a + b*x)^(4/3))/(4*b^4) - (9*a*(a + b*x)
^(7/3))/(7*b^4)

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