Integrand size = 24, antiderivative size = 144 \[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {2 B (c x)^{1+m}}{b c (2+2 m-n) \sqrt {a+b x^n}}-\frac {(2 a B (1+m)-A b (2+2 m-n)) (c x)^{1+m} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a b c (1+m) (2+2 m-n) \sqrt {a+b x^n}} \] Output:
2*B*(c*x)^(1+m)/b/c/(2+2*m-n)/(a+b*x^n)^(1/2)-(2*a*B*(1+m)-A*b*(2+2*m-n))* (c*x)^(1+m)*(1+b*x^n/a)^(1/2)*hypergeom([3/2, (1+m)/n],[(1+m+n)/n],-b*x^n/ a)/a/b/c/(1+m)/(2+2*m-n)/(a+b*x^n)^(1/2)
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {x (c x)^m \sqrt {1+\frac {b x^n}{a}} \left (A (1+m+n) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+B (1+m) x^n \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m+n}{n},\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )\right )}{a (1+m) (1+m+n) \sqrt {a+b x^n}} \] Input:
Integrate[((c*x)^m*(A + B*x^n))/(a + b*x^n)^(3/2),x]
Output:
(x*(c*x)^m*Sqrt[1 + (b*x^n)/a]*(A*(1 + m + n)*Hypergeometric2F1[3/2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + B*(1 + m)*x^n*Hypergeometric2F1[3/2, (1 + m + n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)]))/(a*(1 + m)*(1 + m + n)*Sqr t[a + b*x^n])
Time = 0.48 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {957, 889, 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.
\(\displaystyle \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(2 a B (m+1)-A b (2 m-n+2)) \int \frac {(c x)^m}{\sqrt {b x^n+a}}dx}{a b n}+\frac {2 (c x)^{m+1} (A b-a B)}{a b c n \sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} (2 a B (m+1)-A b (2 m-n+2)) \int \frac {(c x)^m}{\sqrt {\frac {b x^n}{a}+1}}dx}{a b n \sqrt {a+b x^n}}+\frac {2 (c x)^{m+1} (A b-a B)}{a b c n \sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {(c x)^{m+1} \sqrt {\frac {b x^n}{a}+1} (2 a B (m+1)-A b (2 m-n+2)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a b c (m+1) n \sqrt {a+b x^n}}+\frac {2 (c x)^{m+1} (A b-a B)}{a b c n \sqrt {a+b x^n}}\) |
Input:
Int[((c*x)^m*(A + B*x^n))/(a + b*x^n)^(3/2),x]
Output:
(2*(A*b - a*B)*(c*x)^(1 + m))/(a*b*c*n*Sqrt[a + b*x^n]) + ((2*a*B*(1 + m) - A*b*(2 + 2*m - n))*(c*x)^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1 /2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*b*c*(1 + m)*n*Sqrt[a + b*x ^n])
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[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)]))
\[\int \frac {\left (c x \right )^{m} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{\frac {3}{2}}}d x\]
Input:
int((c*x)^m*(A+B*x^n)/(a+b*x^n)^(3/2),x)
Output:
int((c*x)^m*(A+B*x^n)/(a+b*x^n)^(3/2),x)
Exception generated. \[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x)^m*(A+B*x^n)/(a+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Result contains complex when optimal does not.
Time = 5.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.15 \[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {A a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} - \frac {3}{2} - \frac {1}{n}} c^{m} x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {B a^{- \frac {m}{n} - \frac {5}{2} - \frac {1}{n}} a^{\frac {m}{n} + 1 + \frac {1}{n}} c^{m} x^{m + n + 1} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} \] Input:
integrate((c*x)**m*(A+B*x**n)/(a+b*x**n)**(3/2),x)
Output:
A*a**(m/n + 1/n)*a**(-m/n - 3/2 - 1/n)*c**m*x**(m + 1)*gamma(m/n + 1/n)*hy per((3/2, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma (m/n + 1 + 1/n)) + B*a**(-m/n - 5/2 - 1/n)*a**(m/n + 1 + 1/n)*c**m*x**(m + n + 1)*gamma(m/n + 1 + 1/n)*hyper((3/2, m/n + 1 + 1/n), (m/n + 2 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 2 + 1/n))
\[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (c x\right )^{m}}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^m*(A+B*x^n)/(a+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^n + A)*(c*x)^m/(b*x^n + a)^(3/2), x)
\[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (c x\right )^{m}}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^m*(A+B*x^n)/(a+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(c*x)^m/(b*x^n + a)^(3/2), x)
Timed out. \[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^{3/2}} \,d x \] Input:
int(((c*x)^m*(A + B*x^n))/(a + b*x^n)^(3/2),x)
Output:
int(((c*x)^m*(A + B*x^n))/(a + b*x^n)^(3/2), x)
\[ \int \frac {(c x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=c^{m} \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{x^{n} b +a}d x \right ) \] Input:
int((c*x)^m*(A+B*x^n)/(a+b*x^n)^(3/2),x)
Output:
c**m*int((x**m*sqrt(x**n*b + a))/(x**n*b + a),x)