3.7.73 \(\int \frac {(a+b x)^3}{x^{5/3}} \, dx\) [673]
Optimal. Leaf size=49 \[ -\frac {3 a^3}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac {9}{4} a b^2 x^{4/3}+\frac {3}{7} b^3 x^{7/3} \]
[Out]
-3/2*a^3/x^(2/3)+9*a^2*b*x^(1/3)+9/4*a*b^2*x^(4/3)+3/7*b^3*x^(7/3)
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Rubi [A]
time = 0.01, antiderivative size = 49, 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}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac {9}{4} a b^2 x^{4/3}+\frac {3}{7} b^3 x^{7/3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3/x^(5/3),x]
[Out]
(-3*a^3)/(2*x^(2/3)) + 9*a^2*b*x^(1/3) + (9*a*b^2*x^(4/3))/4 + (3*b^3*x^(7/3))/7
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 {(a+b x)^3}{x^{5/3}} \, dx &=\int \left (\frac {a^3}{x^{5/3}}+\frac {3 a^2 b}{x^{2/3}}+3 a b^2 \sqrt [3]{x}+b^3 x^{4/3}\right ) \, dx\\ &=-\frac {3 a^3}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac {9}{4} a b^2 x^{4/3}+\frac {3}{7} b^3 x^{7/3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.80 \begin {gather*} -\frac {3 \left (14 a^3-84 a^2 b x-21 a b^2 x^2-4 b^3 x^3\right )}{28 x^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3/x^(5/3),x]
[Out]
(-3*(14*a^3 - 84*a^2*b*x - 21*a*b^2*x^2 - 4*b^3*x^3))/(28*x^(2/3))
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 23.09, size = 2278, normalized size = 46.49
result too large to display
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[(a + b*x)^3/x^(5/3),x]')
[Out]
Piecewise[{{3 a ^ (1 / 3) (-14 a ^ 3 (b x / a) ^ (1 / 3) - 81 -1 ^ (1 / 3) a ^ 2 b x + 84 a ^ 2 b x (b x / a)
^ (1 / 3) + 21 a b ^ 2 x ^ 2 (b x / a) ^ (1 / 3) + 4 b ^ 3 x ^ 3 (b x / a) ^ (1 / 3)) / (28 b ^ (1 / 3) x), Ab
s[(a + b x) / a] > 1}}, -243 E ^ (I Pi / 3) a ^ (67 / 3) b ^ (2 / 3) / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) +
420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168
a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) + 243 E ^ (I Pi / 3) a ^ (67 / 3) b ^ (2 / 3)
(1 - b (a / b + x) / a) ^ (1 / 3) / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 -
560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a
^ 14 b ^ 6 (a / b + x) ^ 6) - 1377 E ^ (I Pi / 3) a ^ (64 / 3) b ^ (5 / 3) (a / b + x) (1 - b (a / b + x) / a
) ^ (1 / 3) / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a /
b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x
) ^ 6) + 1458 E ^ (I Pi / 3) a ^ (64 / 3) b ^ (5 / 3) (a / b + x) / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 42
0 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^
15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) - 3645 E ^ (I Pi / 3) a ^ (61 / 3) b ^ (8 / 3) (a
/ b + x) ^ 2 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a
/ b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b +
x) ^ 6) + 3213 E ^ (I Pi / 3) a ^ (61 / 3) b ^ (8 / 3) (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 2 / (2
8 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 42
0 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) - 3927 E
^ (I Pi / 3) a ^ (58 / 3) b ^ (11 / 3) (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 3 / (28 a ^ 20 - 168 a
^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a
/ b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) + 4860 E ^ (I Pi / 3) a ^
(58 / 3) b ^ (11 / 3) (a / b + x) ^ 3 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^
2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 +
28 a ^ 14 b ^ 6 (a / b + x) ^ 6) - 3645 E ^ (I Pi / 3) a ^ (55 / 3) b ^ (14 / 3) (a / b + x) ^ 4 / (28 a ^ 20
- 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16
b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) + 2625 E ^ (I Pi /
3) a ^ (55 / 3) b ^ (14 / 3) (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 4 / (28 a ^ 20 - 168 a ^ 19 b (a
/ b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x)
^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) - 903 E ^ (I Pi / 3) a ^ (52 / 3) b
^ (17 / 3) (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 5 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a
^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15
b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) + 1458 E ^ (I Pi / 3) a ^ (52 / 3) b ^ (17 / 3) (a /
b + x) ^ 5 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a /
b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x)
^ 6) - 243 E ^ (I Pi / 3) a ^ (49 / 3) b ^ (20 / 3) (a / b + x) ^ 6 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) +
420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168
a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) + 147 E ^ (I Pi / 3) a ^ (49 / 3) b ^ (20 / 3)
(1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 6 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2
(a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a /
b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b + x) ^ 6) - 33 E ^ (I Pi / 3) a ^ (46 / 3) b ^ (23 / 3) (1 - b (a / b + x)
/ a) ^ (1 / 3) (a / b + x) ^ 7 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 5
60 a ^ 17 b ^ 3 (a / b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^
14 b ^ 6 (a / b + x) ^ 6) + 12 E ^ (I Pi / 3) a ^ (43 / 3) b ^ (26 / 3) (1 - b (a / b + x) / a) ^ (1 / 3) (a
/ b + x) ^ 8 / (28 a ^ 20 - 168 a ^ 19 b (a / b + x) + 420 a ^ 18 b ^ 2 (a / b + x) ^ 2 - 560 a ^ 17 b ^ 3 (a
/ b + x) ^ 3 + 420 a ^ 16 b ^ 4 (a / b + x) ^ 4 - 168 a ^ 15 b ^ 5 (a / b + x) ^ 5 + 28 a ^ 14 b ^ 6 (a / b +
x) ^ 6)]
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Maple [A]
time = 0.11, size = 36, normalized size = 0.73
| | |
| method |
result |
size |
| | |
| gosper |
\(-\frac {3 \left (-4 b^{3} x^{3}-21 a \,b^{2} x^{2}-84 a^{2} b x +14 a^{3}\right )}{28 x^{\frac {2}{3}}}\) |
\(36\) |
| derivativedivides |
\(-\frac {3 a^{3}}{2 x^{\frac {2}{3}}}+9 a^{2} b \,x^{\frac {1}{3}}+\frac {9 a \,b^{2} x^{\frac {4}{3}}}{4}+\frac {3 b^{3} x^{\frac {7}{3}}}{7}\) |
\(36\) |
| default |
\(-\frac {3 a^{3}}{2 x^{\frac {2}{3}}}+9 a^{2} b \,x^{\frac {1}{3}}+\frac {9 a \,b^{2} x^{\frac {4}{3}}}{4}+\frac {3 b^{3} x^{\frac {7}{3}}}{7}\) |
\(36\) |
| trager |
\(-\frac {3 \left (-4 b^{3} x^{3}-21 a \,b^{2} x^{2}-84 a^{2} b x +14 a^{3}\right )}{28 x^{\frac {2}{3}}}\) |
\(36\) |
| risch |
\(-\frac {3 \left (-4 b^{3} x^{3}-21 a \,b^{2} x^{2}-84 a^{2} b x +14 a^{3}\right )}{28 x^{\frac {2}{3}}}\) |
\(36\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3/x^(5/3),x,method=_RETURNVERBOSE)
[Out]
-3/2*a^3/x^(2/3)+9*a^2*b*x^(1/3)+9/4*a*b^2*x^(4/3)+3/7*b^3*x^(7/3)
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Maxima [A]
time = 0.27, size = 35, normalized size = 0.71 \begin {gather*} \frac {3}{7} \, b^{3} x^{\frac {7}{3}} + \frac {9}{4} \, a b^{2} x^{\frac {4}{3}} + 9 \, a^{2} b x^{\frac {1}{3}} - \frac {3 \, a^{3}}{2 \, x^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/x^(5/3),x, algorithm="maxima")
[Out]
3/7*b^3*x^(7/3) + 9/4*a*b^2*x^(4/3) + 9*a^2*b*x^(1/3) - 3/2*a^3/x^(2/3)
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Fricas [A]
time = 0.31, size = 35, normalized size = 0.71 \begin {gather*} \frac {3 \, {\left (4 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 84 \, a^{2} b x - 14 \, a^{3}\right )}}{28 \, x^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/x^(5/3),x, algorithm="fricas")
[Out]
3/28*(4*b^3*x^3 + 21*a*b^2*x^2 + 84*a^2*b*x - 14*a^3)/x^(2/3)
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Sympy [C] Result contains complex when optimal does not.
time = 1.63, size = 3964, normalized size = 80.90
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**3/x**(5/3),x)
[Out]
Piecewise((243*a**(67/3)*b**(2/3)*(-1 + b*(a/b + x)/a)**(1/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b*
*2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28
*a**14*b**6*(a/b + x)**6) - 243*a**(67/3)*b**(2/3)*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b
**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 2
8*a**14*b**6*(a/b + x)**6) - 1377*a**(64/3)*b**(5/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)/(28*a**20 - 168*a**
19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168
*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 1458*a**(64/3)*b**(5/3)*(a/b + x)*exp(I*pi/3)/(28*a**
20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b +
x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 3213*a**(61/3)*b**(8/3)*(-1 + b*(a/b + x)
/a)**(1/3)*(a/b + x)**2/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b
+ x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 3645*a**(6
1/3)*b**(8/3)*(a/b + x)**2*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a
**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**
6) - 3927*a**(58/3)*b**(11/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**3/(28*a**20 - 168*a**19*b*(a/b + x) + 420
*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x
)**5 + 28*a**14*b**6*(a/b + x)**6) + 4860*a**(58/3)*b**(11/3)*(a/b + x)**3*exp(I*pi/3)/(28*a**20 - 168*a**19*b
*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**
15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 2625*a**(55/3)*b**(14/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b
+ x)**4/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a
**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 3645*a**(55/3)*b**(14/3)*
(a/b + x)**4*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b
+ x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 903*a**(5
2/3)*b**(17/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**5/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/
b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14
*b**6*(a/b + x)**6) + 1458*a**(52/3)*b**(17/3)*(a/b + x)**5*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 42
0*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b +
x)**5 + 28*a**14*b**6*(a/b + x)**6) + 147*a**(49/3)*b**(20/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**6/(28*a**
20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b +
x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 243*a**(49/3)*b**(20/3)*(a/b + x)**6*exp(
I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a*
*16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 33*a**(46/3)*b**(23/3)*(-1
+ b*(a/b + x)/a)**(1/3)*(a/b + x)**7/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a*
*17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6
) + 12*a**(43/3)*b**(26/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**8/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a*
*18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**
5 + 28*a**14*b**6*(a/b + x)**6), Abs(b*(a/b + x)/a) > 1), (243*a**(67/3)*b**(2/3)*(1 - b*(a/b + x)/a)**(1/3)*e
xp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420
*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 243*a**(67/3)*b**(2/3)*
exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 42
0*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 1377*a**(64/3)*b**(5/3
)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x
)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6
*(a/b + x)**6) + 1458*a**(64/3)*b**(5/3)*(a/b + x)*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b
**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 2
8*a**14*b**6*(a/b + x)**6) + 3213*a**(61/3)*b**(8/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**2*exp(I*pi/3)/(28*a
**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b
+ x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 3645*a**(61/3)*b**(8/3)*(a/b + x)**2*ex
p(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*
a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 3927*a**(58/3)*b**(11/3)
*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**3*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b +
x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b*
*6*(a/b + x)**6) + 4860*a**(58/3)*b**(11/3)*(a/b + x)**3*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a
**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)*
*5 + 28*a**14*b**6*(a/b + x)**6) + 2625*a**(55/3)*b**(14/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**4*exp(I*pi/3
)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b*
*4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 3645*a**(55/3)*b**(14/3)*(a/b +
x)**4*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**
3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 903*a**(52/3)*b*
*(17/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**5*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2
*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a
**14*b**6*(a/b + x)**6) + 1458*a**(52/3)*b**(17/3)*(a/b + x)**5*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x)
+ 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/
b + x)**5 + 28*a**14*b**6*(a/b + x)**6) + 147*a**(49/3)*b**(20/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**6*exp(
I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a*
*16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 243*a**(49/3)*b**(20/3)*(a
/b + x)**6*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b +
x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6) - 33*a**(46/3
)*b**(23/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**7*exp(I*pi/3)/(28*a**20 - 168*a**19*b*(a/b + x) + 420*a**18*
b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b + x)**4 - 168*a**15*b**5*(a/b + x)**5 +
28*a**14*b**6*(a/b + x)**6) + 12*a**(43/3)*b**(26/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**8*exp(I*pi/3)/(28*a
**20 - 168*a**19*b*(a/b + x) + 420*a**18*b**2*(a/b + x)**2 - 560*a**17*b**3*(a/b + x)**3 + 420*a**16*b**4*(a/b
+ x)**4 - 168*a**15*b**5*(a/b + x)**5 + 28*a**14*b**6*(a/b + x)**6), True))
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Giac [A]
time = 0.00, size = 58, normalized size = 1.18 \begin {gather*} \frac {3}{7} x^{\frac {1}{3}} x^{2} b^{3}+\frac {9}{4} x^{\frac {1}{3}} x b^{2} a+9 x^{\frac {1}{3}} b a^{2}-\frac {\frac {1}{2}\cdot 3 a^{3}}{\left (x^{\frac {1}{3}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/x^(5/3),x)
[Out]
3/7*b^3*x^(7/3) + 9/4*a*b^2*x^(4/3) + 9*a^2*b*x^(1/3) - 3/2*a^3/x^(2/3)
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Mupad [B]
time = 0.04, size = 35, normalized size = 0.71 \begin {gather*} \frac {3\,b^3\,x^{7/3}}{7}-\frac {3\,a^3}{2\,x^{2/3}}+9\,a^2\,b\,x^{1/3}+\frac {9\,a\,b^2\,x^{4/3}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^3/x^(5/3),x)
[Out]
(3*b^3*x^(7/3))/7 - (3*a^3)/(2*x^(2/3)) + 9*a^2*b*x^(1/3) + (9*a*b^2*x^(4/3))/4
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