Integrand size = 26, antiderivative size = 127 \[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\frac {2 B \sqrt {e x}}{b e (1-3 n) \left (a+b x^n\right )^{3/2}}-\frac {2 (a B-A b (1-3 n)) \sqrt {e x} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{a^2 b e (1-3 n) \sqrt {a+b x^n}} \] Output:
2*B*(e*x)^(1/2)/b/e/(1-3*n)/(a+b*x^n)^(3/2)-2*(B*a-A*b*(1-3*n))*(e*x)^(1/2 )*(1+b*x^n/a)^(1/2)*hypergeom([5/2, 1/2/n],[1+1/2/n],-b*x^n/a)/a^2/b/e/(1- 3*n)/(a+b*x^n)^(1/2)
Time = 0.58 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\frac {2 x \sqrt {1+\frac {b x^n}{a}} \left ((A+2 A n) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )+B x^n \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {\frac {1}{2}+n}{n},\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )\right )}{a^2 (1+2 n) \sqrt {e x} \sqrt {a+b x^n}} \] Input:
Integrate[(A + B*x^n)/(Sqrt[e*x]*(a + b*x^n)^(5/2)),x]
Output:
(2*x*Sqrt[1 + (b*x^n)/a]*((A + 2*A*n)*Hypergeometric2F1[5/2, 1/(2*n), (2 + n^(-1))/2, -((b*x^n)/a)] + B*x^n*Hypergeometric2F1[5/2, (1/2 + n)/n, (4 + n^(-1))/2, -((b*x^n)/a)]))/(a^2*(1 + 2*n)*Sqrt[e*x]*Sqrt[a + b*x^n])
Time = 0.43 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {959, 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.
\(\displaystyle \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \left (A-\frac {a B}{b-3 b n}\right ) \int \frac {1}{\sqrt {e x} \left (b x^n+a\right )^{5/2}}dx+\frac {2 B \sqrt {e x}}{b e (1-3 n) \left (a+b x^n\right )^{3/2}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{b-3 b n}\right ) \int \frac {1}{\sqrt {e x} \left (\frac {b x^n}{a}+1\right )^{5/2}}dx}{a^2 \sqrt {a+b x^n}}+\frac {2 B \sqrt {e x}}{b e (1-3 n) \left (a+b x^n\right )^{3/2}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 \sqrt {e x} \sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{b-3 b n}\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{a^2 e \sqrt {a+b x^n}}+\frac {2 B \sqrt {e x}}{b e (1-3 n) \left (a+b x^n\right )^{3/2}}\) |
Input:
Int[(A + B*x^n)/(Sqrt[e*x]*(a + b*x^n)^(5/2)),x]
Output:
(2*B*Sqrt[e*x])/(b*e*(1 - 3*n)*(a + b*x^n)^(3/2)) + (2*(A - (a*B)/(b - 3*b *n))*Sqrt[e*x]*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[5/2, 1/(2*n), (2 + n^ (-1))/2, -((b*x^n)/a)])/(a^2*e*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[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
\[\int \frac {A +B \,x^{n}}{\sqrt {e x}\, \left (a +b \,x^{n}\right )^{\frac {5}{2}}}d x\]
Input:
int((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(5/2),x)
Output:
int((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(5/2),x)
Exception generated. \[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(5/2),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 = 133.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\frac {A a^{\frac {1}{2 n}} a^{- \frac {5}{2} - \frac {1}{2 n}} \sqrt {x} \Gamma \left (\frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {1}{2 n} \\ 1 + \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {e} n \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {B a^{- \frac {7}{2} - \frac {1}{2 n}} a^{1 + \frac {1}{2 n}} x^{n + \frac {1}{2}} \Gamma \left (1 + \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, 1 + \frac {1}{2 n} \\ 2 + \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {e} n \Gamma \left (2 + \frac {1}{2 n}\right )} \] Input:
integrate((A+B*x**n)/(e*x)**(1/2)/(a+b*x**n)**(5/2),x)
Output:
A*a**(1/(2*n))*a**(-5/2 - 1/(2*n))*sqrt(x)*gamma(1/(2*n))*hyper((5/2, 1/(2 *n)), (1 + 1/(2*n),), b*x**n*exp_polar(I*pi)/a)/(sqrt(e)*n*gamma(1 + 1/(2* n))) + B*a**(-7/2 - 1/(2*n))*a**(1 + 1/(2*n))*x**(n + 1/2)*gamma(1 + 1/(2* n))*hyper((5/2, 1 + 1/(2*n)), (2 + 1/(2*n),), b*x**n*exp_polar(I*pi)/a)/(s qrt(e)*n*gamma(2 + 1/(2*n)))
\[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(5/2)*sqrt(e*x)), x)
\[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(5/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(5/2)*sqrt(e*x)), x)
Timed out. \[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\int \frac {A+B\,x^n}{\sqrt {e\,x}\,{\left (a+b\,x^n\right )}^{5/2}} \,d x \] Input:
int((A + B*x^n)/((e*x)^(1/2)*(a + b*x^n)^(5/2)),x)
Output:
int((A + B*x^n)/((e*x)^(1/2)*(a + b*x^n)^(5/2)), x)
\[ \int \frac {A+B x^n}{\sqrt {e x} \left (a+b x^n\right )^{5/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n +\frac {1}{2}} b^{2}+2 x^{n +\frac {1}{2}} a b +\sqrt {x}\, a^{2}}d x \right )}{e} \] Input:
int((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(5/2),x)
Output:
(sqrt(e)*int(sqrt(x**n*b + a)/(x**((4*n + 1)/2)*b**2 + 2*x**((2*n + 1)/2)* a*b + sqrt(x)*a**2),x))/e