Integrand size = 20, antiderivative size = 93 \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {(A b-a B) x^{1+m}}{6 a b \left (a+b x^3\right )^2}+\frac {(A b (5-m)+a B (1+m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{6 a^3 b (1+m)} \] Output:
1/6*(A*b-B*a)*x^(1+m)/a/b/(b*x^3+a)^2+1/6*(A*b*(5-m)+a*B*(1+m))*x^(1+m)*hy pergeom([2, 1/3+1/3*m],[4/3+1/3*m],-b*x^3/a)/a^3/b/(1+m)
Time = 0.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {x^{1+m} \left (a B \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )+(A b-a B) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )\right )}{a^3 b (1+m)} \] Input:
Integrate[(x^m*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
(x^(1 + m)*(a*B*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]))/(a ^3*b*(1 + m))
Time = 0.37 (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.
\(\displaystyle \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(a B (m+1)+A b (5-m)) \int \frac {x^m}{\left (b x^3+a\right )^2}dx}{6 a b}+\frac {x^{m+1} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{m+1} (a B (m+1)+A b (5-m)) \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{6 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
Input:
Int[(x^m*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
((A*b - a*B)*x^(1 + m))/(6*a*b*(a + b*x^3)^2) + ((A*b*(5 - m) + a*B*(1 + m ))*x^(1 + m)*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(6* a^3*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)]))
\[\int \frac {x^{m} \left (B \,x^{3}+A \right )}{\left (b \,x^{3}+a \right )^{3}}d x\]
Input:
int(x^m*(B*x^3+A)/(b*x^3+a)^3,x)
Output:
int(x^m*(B*x^3+A)/(b*x^3+a)^3,x)
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:
integrate(x^m*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
integral((B*x^3 + A)*x^m/(b^3*x^9 + 3*a*b^2*x^6 + 3*a^2*b*x^3 + a^3), x)
Timed out. \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**m*(B*x**3+A)/(b*x**3+a)**3,x)
Output:
Timed out
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:
integrate(x^m*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*x^m/(b*x^3 + a)^3, x)
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:
integrate(x^m*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*x^m/(b*x^3 + a)^3, x)
Timed out. \[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\int \frac {x^m\,\left (B\,x^3+A\right )}{{\left (b\,x^3+a\right )}^3} \,d x \] Input:
int((x^m*(A + B*x^3))/(a + b*x^3)^3,x)
Output:
int((x^m*(A + B*x^3))/(a + b*x^3)^3, x)
\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\int \frac {x^{m}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \] Input:
int(x^m*(B*x^3+A)/(b*x^3+a)^3,x)
Output:
int(x**m/(a**2 + 2*a*b*x**3 + b**2*x**6),x)