3.7.71 \(\int \frac {(a+b x)^3}{x^{2/3}} \, dx\) [671]
Optimal. Leaf size=49 \[ 3 a^3 \sqrt [3]{x}+\frac {9}{4} a^2 b x^{4/3}+\frac {9}{7} a b^2 x^{7/3}+\frac {3}{10} b^3 x^{10/3} \]
[Out]
3*a^3*x^(1/3)+9/4*a^2*b*x^(4/3)+9/7*a*b^2*x^(7/3)+3/10*b^3*x^(10/3)
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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*} 3 a^3 \sqrt [3]{x}+\frac {9}{4} a^2 b x^{4/3}+\frac {9}{7} a b^2 x^{7/3}+\frac {3}{10} b^3 x^{10/3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3/x^(2/3),x]
[Out]
3*a^3*x^(1/3) + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(7/3))/7 + (3*b^3*x^(10/3))/10
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^{2/3}} \, dx &=\int \left (\frac {a^3}{x^{2/3}}+3 a^2 b \sqrt [3]{x}+3 a b^2 x^{4/3}+b^3 x^{7/3}\right ) \, dx\\ &=3 a^3 \sqrt [3]{x}+\frac {9}{4} a^2 b x^{4/3}+\frac {9}{7} a b^2 x^{7/3}+\frac {3}{10} b^3 x^{10/3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.80 \begin {gather*} \frac {3}{140} \sqrt [3]{x} \left (140 a^3+105 a^2 b x+60 a b^2 x^2+14 b^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3/x^(2/3),x]
[Out]
(3*x^(1/3)*(140*a^3 + 105*a^2*b*x + 60*a*b^2*x^2 + 14*b^3*x^3))/140
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Maple [A]
time = 0.09, size = 36, normalized size = 0.73
| | |
| method |
result |
size |
| | |
| trager |
\(\left (\frac {3}{10} b^{3} x^{3}+\frac {9}{7} a \,b^{2} x^{2}+\frac {9}{4} a^{2} b x +3 a^{3}\right ) x^{\frac {1}{3}}\) |
\(35\) |
| gosper |
\(\frac {3 x^{\frac {1}{3}} \left (14 b^{3} x^{3}+60 a \,b^{2} x^{2}+105 a^{2} b x +140 a^{3}\right )}{140}\) |
\(36\) |
| derivativedivides |
\(3 a^{3} x^{\frac {1}{3}}+\frac {9 a^{2} b \,x^{\frac {4}{3}}}{4}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {3 b^{3} x^{\frac {10}{3}}}{10}\) |
\(36\) |
| default |
\(3 a^{3} x^{\frac {1}{3}}+\frac {9 a^{2} b \,x^{\frac {4}{3}}}{4}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {3 b^{3} x^{\frac {10}{3}}}{10}\) |
\(36\) |
| risch |
\(\frac {3 x^{\frac {1}{3}} \left (14 b^{3} x^{3}+60 a \,b^{2} x^{2}+105 a^{2} b x +140 a^{3}\right )}{140}\) |
\(36\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3/x^(2/3),x,method=_RETURNVERBOSE)
[Out]
3*a^3*x^(1/3)+9/4*a^2*b*x^(4/3)+9/7*a*b^2*x^(7/3)+3/10*b^3*x^(10/3)
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Maxima [A]
time = 0.37, size = 35, normalized size = 0.71 \begin {gather*} \frac {3}{10} \, b^{3} x^{\frac {10}{3}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {9}{4} \, a^{2} b x^{\frac {4}{3}} + 3 \, a^{3} x^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/x^(2/3),x, algorithm="maxima")
[Out]
3/10*b^3*x^(10/3) + 9/7*a*b^2*x^(7/3) + 9/4*a^2*b*x^(4/3) + 3*a^3*x^(1/3)
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Fricas [A]
time = 0.57, size = 35, normalized size = 0.71 \begin {gather*} \frac {3}{140} \, {\left (14 \, b^{3} x^{3} + 60 \, a b^{2} x^{2} + 105 \, a^{2} b x + 140 \, a^{3}\right )} x^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/x^(2/3),x, algorithm="fricas")
[Out]
3/140*(14*b^3*x^3 + 60*a*b^2*x^2 + 105*a^2*b*x + 140*a^3)*x^(1/3)
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Sympy [C] Result contains complex when optimal does not.
time = 1.53, size = 6667, normalized size = 136.06 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**3/x**(2/3),x)
[Out]
Piecewise((243*a**(70/3)*(-1 + b*(a/b + x)/a)**(1/3)*exp(2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**
19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a
/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*
exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) + 243*a**(70/3)/(140*a**20*b**(1/3)*exp(2*I*pi
/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17
*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*
(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) - 1377*a**(67/3)*b*(-1 + b*(a/b +
x)/a)**(1/3)*(a/b + x)*exp(2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I
*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 21
00*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**
(19/3)*(a/b + x)**6*exp(2*I*pi/3)) - 1458*a**(67/3)*b*(a/b + x)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*
b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b
+ x)**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp
(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) + 3213*a**(64/3)*b**2*(-1 + b*(a/b + x)/a)**(1/3)
*(a/b + x)**2*exp(2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2
100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b
**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/
b + x)**6*exp(2*I*pi/3)) + 3645*a**(64/3)*b**2*(a/b + x)**2/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(
4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)
**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I
*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) - 3927*a**(61/3)*b**3*(-1 + b*(a/b + x)/a)**(1/3)*(a/
b + x)**3*exp(2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*
a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b**(1
3/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b +
x)**6*exp(2*I*pi/3)) - 4860*a**(61/3)*b**3*(a/b + x)**3/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)
*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*
exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/
3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) + 2583*a**(58/3)*b**4*(-1 + b*(a/b + x)/a)**(1/3)*(a/b +
x)**4*exp(2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**1
8*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)
*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**
6*exp(2*I*pi/3)) + 3645*a**(58/3)*b**4*(a/b + x)**4/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/
b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(
2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) +
140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) - 693*a**(55/3)*b**5*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**5
*exp(2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**
(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b
+ x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp
(2*I*pi/3)) - 1458*a**(55/3)*b**5*(a/b + x)**5/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x
)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*p
i/3) + 2100*a**16*b**(13/3)*(a/b + x)**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*
a**14*b**(19/3)*(a/b + x)**6*exp(2*I*pi/3)) - 273*a**(52/3)*b**6*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**6*exp(
2*I*pi/3)/(140*a**20*b**(1/3)*exp(2*I*pi/3) - 840*a**19*b**(4/3)*(a/b + x)*exp(2*I*pi/3) + 2100*a**18*b**(7/3)
*(a/b + x)**2*exp(2*I*pi/3) - 2800*a**17*b**(10/3)*(a/b + x)**3*exp(2*I*pi/3) + 2100*a**16*b**(13/3)*(a/b + x)
**4*exp(2*I*pi/3) - 840*a**15*b**(16/3)*(a/b + x)**5*exp(2*I*pi/3) + 140*a**14*b**(19/3)*(a/b + x)**6*exp(2*I*
pi/3)) + 243*a**(52/3)*b**6*(a/b + x)**6/(140*a...
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Giac [A]
time = 1.57, size = 35, normalized size = 0.71 \begin {gather*} \frac {3}{10} \, b^{3} x^{\frac {10}{3}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {9}{4} \, a^{2} b x^{\frac {4}{3}} + 3 \, a^{3} x^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/x^(2/3),x, algorithm="giac")
[Out]
3/10*b^3*x^(10/3) + 9/7*a*b^2*x^(7/3) + 9/4*a^2*b*x^(4/3) + 3*a^3*x^(1/3)
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Mupad [B]
time = 0.04, size = 35, normalized size = 0.71 \begin {gather*} 3\,a^3\,x^{1/3}+\frac {3\,b^3\,x^{10/3}}{10}+\frac {9\,a^2\,b\,x^{4/3}}{4}+\frac {9\,a\,b^2\,x^{7/3}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^3/x^(2/3),x)
[Out]
3*a^3*x^(1/3) + (3*b^3*x^(10/3))/10 + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(7/3))/7
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