Integrand size = 20, antiderivative size = 93 \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {(A b-a B) x^{1+m}}{3 a b \left (a+b x^3\right )}+\frac {(A b (2-m)+a B (1+m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{3 a^2 b (1+m)} \] Output:
1/3*(A*b-B*a)*x^(1+m)/a/b/(b*x^3+a)+1/3*(A*b*(2-m)+a*B*(1+m))*x^(1+m)*hype rgeom([1, 1/3+1/3*m],[4/3+1/3*m],-b*x^3/a)/a^2/b/(1+m)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {x^{1+m} \left (a B \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )+(A b-a B) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )\right )}{a^2 b (1+m)} \] Input:
Integrate[(x^m*(A + B*x^3))/(a + b*x^3)^2,x]
Output:
(x^(1 + m)*(a*B*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]))/(a ^2*b*(1 + m))
Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {957, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(a B (m+1)+A b (2-m)) \int \frac {x^m}{b x^3+a}dx}{3 a b}+\frac {x^{m+1} (A b-a B)}{3 a b \left (a+b x^3\right )}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{m+1} (a B (m+1)+A b (2-m)) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{3 a^2 b (m+1)}+\frac {x^{m+1} (A b-a B)}{3 a b \left (a+b x^3\right )}\) |
Input:
Int[(x^m*(A + B*x^3))/(a + b*x^3)^2,x]
Output:
((A*b - a*B)*x^(1 + m))/(3*a*b*(a + b*x^3)) + ((A*b*(2 - m) + a*B*(1 + m)) *x^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(3*a^ 2*b*(1 + m))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
\[\int \frac {x^{m} \left (B \,x^{3}+A \right )}{\left (b \,x^{3}+a \right )^{2}}d x\]
Input:
int(x^m*(B*x^3+A)/(b*x^3+a)^2,x)
Output:
int(x^m*(B*x^3+A)/(b*x^3+a)^2,x)
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(x^m*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="fricas")
Output:
integral((B*x^3 + A)*x^m/(b^2*x^6 + 2*a*b*x^3 + a^2), x)
Result contains complex when optimal does not.
Time = 152.79 (sec) , antiderivative size = 1049, normalized size of antiderivative = 11.28 \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x**m*(B*x**3+A)/(b*x**3+a)**2,x)
Output:
A*(-a*m**2*x**(m + 1)*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gam ma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + a*m*x**(m + 1)*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m /3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + 3 *a*m*x**(m + 1)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x** 3*gamma(m/3 + 4/3)) + 2*a*x**(m + 1)*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*ga mma(m/3 + 4/3)) + 3*a*x**(m + 1)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3 ) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) - b*m**2*x**3*x**(m + 1)*lerchphi(b*x **3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + b*m*x**3*x**(m + 1)*lerchphi(b* x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + 2*b*x**3*x**(m + 1)*lerchphi(b *x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3))) + B*(-a*m**2*x**(m + 4)*lerchp hi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 4/3)*gamma(m/3 + 4/3)/(27*a**3*gamma (m/3 + 7/3) + 27*a**2*b*x**3*gamma(m/3 + 7/3)) - 5*a*m*x**(m + 4)*lerchphi (b*x**3*exp_polar(I*pi)/a, 1, m/3 + 4/3)*gamma(m/3 + 4/3)/(27*a**3*gamma(m /3 + 7/3) + 27*a**2*b*x**3*gamma(m/3 + 7/3)) + 3*a*m*x**(m + 4)*gamma(m/3 + 4/3)/(27*a**3*gamma(m/3 + 7/3) + 27*a**2*b*x**3*gamma(m/3 + 7/3)) - 4...
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(x^m*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*x^m/(b*x^3 + a)^2, x)
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(x^m*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*x^m/(b*x^3 + a)^2, x)
Timed out. \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\int \frac {x^m\,\left (B\,x^3+A\right )}{{\left (b\,x^3+a\right )}^2} \,d x \] Input:
int((x^m*(A + B*x^3))/(a + b*x^3)^2,x)
Output:
int((x^m*(A + B*x^3))/(a + b*x^3)^2, x)
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\int \frac {x^{m}}{b \,x^{3}+a}d x \] Input:
int(x^m*(B*x^3+A)/(b*x^3+a)^2,x)
Output:
int(x**m/(a + b*x**3),x)