3.7.69 \(\int \sqrt [3]{x} (a+b x)^3 \, dx\) [669]
Optimal. Leaf size=51 \[ \frac {3}{4} a^3 x^{4/3}+\frac {9}{7} a^2 b x^{7/3}+\frac {9}{10} a b^2 x^{10/3}+\frac {3}{13} b^3 x^{13/3} \]
[Out]
3/4*a^3*x^(4/3)+9/7*a^2*b*x^(7/3)+9/10*a*b^2*x^(10/3)+3/13*b^3*x^(13/3)
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Rubi [A]
time = 0.01, antiderivative size = 51, 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}{4} a^3 x^{4/3}+\frac {9}{7} a^2 b x^{7/3}+\frac {9}{10} a b^2 x^{10/3}+\frac {3}{13} b^3 x^{13/3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^(1/3)*(a + b*x)^3,x]
[Out]
(3*a^3*x^(4/3))/4 + (9*a^2*b*x^(7/3))/7 + (9*a*b^2*x^(10/3))/10 + (3*b^3*x^(13/3))/13
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 \sqrt [3]{x} (a+b x)^3 \, dx &=\int \left (a^3 \sqrt [3]{x}+3 a^2 b x^{4/3}+3 a b^2 x^{7/3}+b^3 x^{10/3}\right ) \, dx\\ &=\frac {3}{4} a^3 x^{4/3}+\frac {9}{7} a^2 b x^{7/3}+\frac {9}{10} a b^2 x^{10/3}+\frac {3}{13} b^3 x^{13/3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.76 \begin {gather*} \frac {3 x^{4/3} \left (455 a^3+780 a^2 b x+546 a b^2 x^2+140 b^3 x^3\right )}{1820} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^(1/3)*(a + b*x)^3,x]
[Out]
(3*x^(4/3)*(455*a^3 + 780*a^2*b*x + 546*a*b^2*x^2 + 140*b^3*x^3))/1820
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Maple [A]
time = 0.09, size = 36, normalized size = 0.71
| | |
| method |
result |
size |
| | |
| gosper |
\(\frac {3 x^{\frac {4}{3}} \left (140 b^{3} x^{3}+546 a \,b^{2} x^{2}+780 a^{2} b x +455 a^{3}\right )}{1820}\) |
\(36\) |
| derivativedivides |
\(\frac {3 a^{3} x^{\frac {4}{3}}}{4}+\frac {9 a^{2} b \,x^{\frac {7}{3}}}{7}+\frac {9 a \,b^{2} x^{\frac {10}{3}}}{10}+\frac {3 b^{3} x^{\frac {13}{3}}}{13}\) |
\(36\) |
| default |
\(\frac {3 a^{3} x^{\frac {4}{3}}}{4}+\frac {9 a^{2} b \,x^{\frac {7}{3}}}{7}+\frac {9 a \,b^{2} x^{\frac {10}{3}}}{10}+\frac {3 b^{3} x^{\frac {13}{3}}}{13}\) |
\(36\) |
| trager |
\(\frac {3 x^{\frac {4}{3}} \left (140 b^{3} x^{3}+546 a \,b^{2} x^{2}+780 a^{2} b x +455 a^{3}\right )}{1820}\) |
\(36\) |
| risch |
\(\frac {3 x^{\frac {4}{3}} \left (140 b^{3} x^{3}+546 a \,b^{2} x^{2}+780 a^{2} b x +455 a^{3}\right )}{1820}\) |
\(36\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(1/3)*(b*x+a)^3,x,method=_RETURNVERBOSE)
[Out]
3/4*a^3*x^(4/3)+9/7*a^2*b*x^(7/3)+9/10*a*b^2*x^(10/3)+3/13*b^3*x^(13/3)
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Maxima [A]
time = 0.26, size = 35, normalized size = 0.69 \begin {gather*} \frac {3}{13} \, b^{3} x^{\frac {13}{3}} + \frac {9}{10} \, a b^{2} x^{\frac {10}{3}} + \frac {9}{7} \, a^{2} b x^{\frac {7}{3}} + \frac {3}{4} \, a^{3} x^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(1/3)*(b*x+a)^3,x, algorithm="maxima")
[Out]
3/13*b^3*x^(13/3) + 9/10*a*b^2*x^(10/3) + 9/7*a^2*b*x^(7/3) + 3/4*a^3*x^(4/3)
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Fricas [A]
time = 1.38, size = 38, normalized size = 0.75 \begin {gather*} \frac {3}{1820} \, {\left (140 \, b^{3} x^{4} + 546 \, a b^{2} x^{3} + 780 \, a^{2} b x^{2} + 455 \, a^{3} x\right )} x^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(1/3)*(b*x+a)^3,x, algorithm="fricas")
[Out]
3/1820*(140*b^3*x^4 + 546*a*b^2*x^3 + 780*a^2*b*x^2 + 455*a^3*x)*x^(1/3)
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Sympy [C] Result contains complex when optimal does not.
time = 1.50, size = 5012, normalized size = 98.27 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**(1/3)*(b*x+a)**3,x)
[Out]
Piecewise((-243*a**(73/3)*(-1 + b*(a/b + x)/a)**(1/3)/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) +
27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 -
10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) + 243*a**(73/3)*exp(I*pi/3)/(1820*a**
20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b
+ x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b
+ x)**6) + 1377*a**(70/3)*b*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)
*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(
a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) - 1458*a**(70/3)*b*(a/b
+ x)*exp(I*pi/3)/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 -
36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 +
1820*a**14*b**(22/3)*(a/b + x)**6) - 3213*a**(67/3)*b**2*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**2/(1820*a**20
*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b +
x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b +
x)**6) + 3645*a**(67/3)*b**2*(a/b + x)**2*exp(I*pi/3)/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) +
27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4
- 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) + 3927*a**(64/3)*b**3*(-1 + b*(a/b +
x)/a)**(1/3)*(a/b + x)**3/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b
+ x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b
+ x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) - 4860*a**(64/3)*b**3*(a/b + x)**3*exp(I*pi/3)/(1820*a**20*b**(4
/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3
+ 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6
) - 2163*a**(61/3)*b**4*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**4/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(
a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/
b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) + 3645*a**(61/3)*b**4*(a/b
+ x)**4*exp(I*pi/3)/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**
2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)*
*5 + 1820*a**14*b**(22/3)*(a/b + x)**6) - 1827*a**(58/3)*b**5*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**5/(1820*a
**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a
/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a
/b + x)**6) - 1458*a**(58/3)*b**5*(a/b + x)**5*exp(I*pi/3)/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b +
x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)
**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) + 6573*a**(55/3)*b**6*(-1 + b*(a
/b + x)/a)**(1/3)*(a/b + x)**6/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(
a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*
(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) + 243*a**(55/3)*b**6*(a/b + x)**6*exp(I*pi/3)/(1820*a**20*b*
*(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)
**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)
**6) - 8787*a**(52/3)*b**7*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**7/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3
)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*
(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) + 6498*a**(49/3)*b**8*(
-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**8/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b*
*(10/3)*(a/b + x)**2 - 36400*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b
**(19/3)*(a/b + x)**5 + 1820*a**14*b**(22/3)*(a/b + x)**6) - 2562*a**(46/3)*b**9*(-1 + b*(a/b + x)/a)**(1/3)*(
a/b + x)**9/(1820*a**20*b**(4/3) - 10920*a**19*b**(7/3)*(a/b + x) + 27300*a**18*b**(10/3)*(a/b + x)**2 - 36400
*a**17*b**(13/3)*(a/b + x)**3 + 27300*a**16*b**(16/3)*(a/b + x)**4 - 10920*a**15*b**(19/3)*(a/b + x)**5 + 1820
*a**14*b**(22/3)*(a/b + x)**6) + 420*a**(43/3)*...
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Giac [A]
time = 2.35, size = 35, normalized size = 0.69 \begin {gather*} \frac {3}{13} \, b^{3} x^{\frac {13}{3}} + \frac {9}{10} \, a b^{2} x^{\frac {10}{3}} + \frac {9}{7} \, a^{2} b x^{\frac {7}{3}} + \frac {3}{4} \, a^{3} x^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(1/3)*(b*x+a)^3,x, algorithm="giac")
[Out]
3/13*b^3*x^(13/3) + 9/10*a*b^2*x^(10/3) + 9/7*a^2*b*x^(7/3) + 3/4*a^3*x^(4/3)
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Mupad [B]
time = 0.05, size = 35, normalized size = 0.69 \begin {gather*} \frac {3\,a^3\,x^{4/3}}{4}+\frac {3\,b^3\,x^{13/3}}{13}+\frac {9\,a^2\,b\,x^{7/3}}{7}+\frac {9\,a\,b^2\,x^{10/3}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(1/3)*(a + b*x)^3,x)
[Out]
(3*a^3*x^(4/3))/4 + (3*b^3*x^(13/3))/13 + (9*a^2*b*x^(7/3))/7 + (9*a*b^2*x^(10/3))/10
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