\(\int \frac {(a+b x)^3}{x^{5/3}} \, dx\) [673]

Optimal result

Integrand size = 13, antiderivative size = 49 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=-\frac {3 a^3}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac {9}{4} a b^2 x^{4/3}+\frac {3}{7} b^3 x^{7/3} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=-\frac {3 a^3}{2 x^{2/3}}+9 a^2 b \sqrt [3]{x}+\frac {9}{4} a b^2 x^{4/3}+\frac {3}{7} b^3 x^{7/3} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=-\frac {3 \left (14 a^3-84 a^2 b x-21 a b^2 x^2-4 b^3 x^3\right )}{28 x^{2/3}} \]

[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))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 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	\(-\frac {3 \left (-4 b^{3} x^{3}-21 a \,b^{2} x^{2}-84 a^{2} b x +14 a^{3}\right )}{28 x^{\frac {2}{3}}}\)	\(36\)

[In]

int((b*x+a)^3/x^(5/3),x,method=_RETURNVERBOSE)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=\frac {3 \, {\left (4 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 84 \, a^{2} b x - 14 \, a^{3}\right )}}{28 \, x^{\frac {2}{3}}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.57 (sec) , antiderivative size = 3964, normalized size of antiderivative = 80.90 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=\text {Too large to display} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=\frac {3}{7} \, b^{3} x^{\frac {7}{3}} + \frac {9}{4} \, a b^{2} x^{\frac {4}{3}} + 9 \, a^{2} b x^{\frac {1}{3}} - \frac {3 \, a^{3}}{2 \, x^{\frac {2}{3}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=\frac {3}{7} \, b^{3} x^{\frac {7}{3}} + \frac {9}{4} \, a b^{2} x^{\frac {4}{3}} + 9 \, a^{2} b x^{\frac {1}{3}} - \frac {3 \, a^{3}}{2 \, x^{\frac {2}{3}}} \]

[In]

integrate((b*x+a)^3/x^(5/3),x, algorithm="giac")

[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)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^3}{x^{5/3}} \, dx=\frac {3\,b^3\,x^{7/3}}{7}-\frac {3\,a^3}{2\,x^{2/3}}+9\,a^2\,b\,x^{1/3}+\frac {9\,a\,b^2\,x^{4/3}}{4} \]

[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