3.6.90 \(\int \frac {x^m}{(a+b x^3)^3} \, dx\) [590]
Optimal. Leaf size=39 \[ \frac {x^{1+m} \, _2F_1\left (3,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a^3 (1+m)} \]
[Out]
x^(1+m)*hypergeom([3, 1/3+1/3*m],[4/3+1/3*m],-b*x^3/a)/a^3/(1+m)
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {371}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (3,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a^3 (m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^m/(a + b*x^3)^3,x]
[Out]
(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^3*(1 + m))
Rule 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^3\right )^3} \, dx &=\frac {x^{1+m} \, _2F_1\left (3,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a^3 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 41, normalized size = 1.05 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (3,\frac {1+m}{3};1+\frac {1+m}{3};-\frac {b x^3}{a}\right )}{a^3 (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^m/(a + b*x^3)^3,x]
[Out]
(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/3, 1 + (1 + m)/3, -((b*x^3)/a)])/(a^3*(1 + m))
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (b \,x^{3}+a \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m/(b*x^3+a)^3,x)
[Out]
int(x^m/(b*x^3+a)^3,x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(b*x^3+a)^3,x, algorithm="maxima")
[Out]
integrate(x^m/(b*x^3 + a)^3, x)
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Fricas [F]
time = 0.36, size = 37, normalized size = 0.95 \begin {gather*} {\rm integral}\left (\frac {x^{m}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(b*x^3+a)^3,x, algorithm="fricas")
[Out]
integral(x^m/(b^3*x^9 + 3*a*b^2*x^6 + 3*a^2*b*x^3 + a^3), x)
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Sympy [C] Result contains complex when optimal does not.
time = 111.17, size = 1559, normalized size = 39.97 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m/(b*x**3+a)**3,x)
[Out]
a**2*m**3*x*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3)
+ 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) - 6*a**2*m**2*x*x**m*lerchphi(b*x**3
*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/
3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) - 3*a**2*m**2*x*x**m*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 3
24*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 3*a**2*m*x*x**m*lerchphi(b*x**3*exp_p
olar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 1
62*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 21*a**2*m*x*x**m*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*
b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 10*a**2*x*x**m*lerchphi(b*x**3*exp_polar(I*pi
)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b
**2*x**6*gamma(m/3 + 4/3)) + 24*a**2*x*x**m*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamm
a(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 2*a*b*m**3*x**4*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a,
1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x
**6*gamma(m/3 + 4/3)) - 12*a*b*m**2*x**4*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3
)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) - 3*a*b
*m**2*x**4*x**m*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2
*x**6*gamma(m/3 + 4/3)) + 6*a*b*m*x**4*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/
(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 12*a*b*
m*x**4*x**m*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**
6*gamma(m/3 + 4/3)) + 20*a*b*x**4*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*
a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 15*a*b*x**4*
x**m*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma
(m/3 + 4/3)) + b**2*m**3*x**7*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5
*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) - 6*b**2*m**2*x**7
*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*
b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3)) + 3*b**2*m*x**7*x**m*lerchphi(b*x**3*exp_polar(
I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a*
*3*b**2*x**6*gamma(m/3 + 4/3)) + 10*b**2*x**7*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3
+ 1/3)/(162*a**5*gamma(m/3 + 4/3) + 324*a**4*b*x**3*gamma(m/3 + 4/3) + 162*a**3*b**2*x**6*gamma(m/3 + 4/3))
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(b*x^3+a)^3,x, algorithm="giac")
[Out]
integrate(x^m/(b*x^3 + a)^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^m}{{\left (b\,x^3+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m/(a + b*x^3)^3,x)
[Out]
int(x^m/(a + b*x^3)^3, x)
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