Integrand size = 26, antiderivative size = 122 \[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {2 B (e x)^{3/2}}{b e (3-n) \sqrt {a+b x^n}}+\frac {2 \left (\frac {A}{a}-\frac {3 B}{b (3-n)}\right ) (e x)^{3/2} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 e \sqrt {a+b x^n}} \] Output:
2*B*(e*x)^(3/2)/b/e/(3-n)/(a+b*x^n)^(1/2)+2/3*(A/a-3*B/b/(3-n))*(e*x)^(3/2 )*(1+b*x^n/a)^(1/2)*hypergeom([3/2, 3/2/n],[1+3/2/n],-b*x^n/a)/e/(a+b*x^n) ^(1/2)
Time = 0.57 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {2 x \sqrt {e x} \sqrt {1+\frac {b x^n}{a}} \left (A (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )+3 B x^n \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {\frac {3}{2}+n}{n},2+\frac {3}{2 n},-\frac {b x^n}{a}\right )\right )}{3 a (3+2 n) \sqrt {a+b x^n}} \] Input:
Integrate[(Sqrt[e*x]*(A + B*x^n))/(a + b*x^n)^(3/2),x]
Output:
(2*x*Sqrt[e*x]*Sqrt[1 + (b*x^n)/a]*(A*(3 + 2*n)*Hypergeometric2F1[3/2, 3/( 2*n), 1 + 3/(2*n), -((b*x^n)/a)] + 3*B*x^n*Hypergeometric2F1[3/2, (3/2 + n )/n, 2 + 3/(2*n), -((b*x^n)/a)]))/(3*a*(3 + 2*n)*Sqrt[a + b*x^n])
Time = 0.42 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \left (A-\frac {3 a B}{b (3-n)}\right ) \int \frac {\sqrt {e x}}{\left (b x^n+a\right )^{3/2}}dx+\frac {2 B (e x)^{3/2}}{b e (3-n) \sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \left (A-\frac {3 a B}{b (3-n)}\right ) \int \frac {\sqrt {e x}}{\left (\frac {b x^n}{a}+1\right )^{3/2}}dx}{a \sqrt {a+b x^n}}+\frac {2 B (e x)^{3/2}}{b e (3-n) \sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 (e x)^{3/2} \sqrt {\frac {b x^n}{a}+1} \left (A-\frac {3 a B}{b (3-n)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 a e \sqrt {a+b x^n}}+\frac {2 B (e x)^{3/2}}{b e (3-n) \sqrt {a+b x^n}}\) |
Input:
Int[(Sqrt[e*x]*(A + B*x^n))/(a + b*x^n)^(3/2),x]
Output:
(2*B*(e*x)^(3/2))/(b*e*(3 - n)*Sqrt[a + b*x^n]) + (2*(A - (3*a*B)/(b*(3 - n)))*(e*x)^(3/2)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[3/2, 3/(2*n), 1 + 3 /(2*n), -((b*x^n)/a)])/(3*a*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]
\[\int \frac {\sqrt {e x}\, \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{\frac {3}{2}}}d x\]
Input:
int((e*x)^(1/2)*(A+B*x^n)/(a+b*x^n)^(3/2),x)
Output:
int((e*x)^(1/2)*(A+B*x^n)/(a+b*x^n)^(3/2),x)
Exception generated. \[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^(1/2)*(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 = 5.49 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {A a^{\frac {3}{2 n}} a^{- \frac {3}{2} - \frac {3}{2 n}} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {3}{2 n} \\ 1 + \frac {3}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {3}{2 n}\right )} + \frac {B a^{- \frac {5}{2} - \frac {3}{2 n}} a^{1 + \frac {3}{2 n}} \sqrt {e} x^{n + \frac {3}{2}} \Gamma \left (1 + \frac {3}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, 1 + \frac {3}{2 n} \\ 2 + \frac {3}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {3}{2 n}\right )} \] Input:
integrate((e*x)**(1/2)*(A+B*x**n)/(a+b*x**n)**(3/2),x)
Output:
A*a**(3/(2*n))*a**(-3/2 - 3/(2*n))*sqrt(e)*x**(3/2)*gamma(3/(2*n))*hyper(( 3/2, 3/(2*n)), (1 + 3/(2*n),), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 3/(2 *n))) + B*a**(-5/2 - 3/(2*n))*a**(1 + 3/(2*n))*sqrt(e)*x**(n + 3/2)*gamma( 1 + 3/(2*n))*hyper((3/2, 1 + 3/(2*n)), (2 + 3/(2*n),), b*x**n*exp_polar(I* pi)/a)/(n*gamma(2 + 3/(2*n)))
\[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {e x}}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(1/2)*(A+B*x^n)/(a+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^n + A)*sqrt(e*x)/(b*x^n + a)^(3/2), x)
\[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {e x}}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(1/2)*(A+B*x^n)/(a+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)*sqrt(e*x)/(b*x^n + a)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x}\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^{3/2}} \,d x \] Input:
int(((e*x)^(1/2)*(A + B*x^n))/(a + b*x^n)^(3/2),x)
Output:
int(((e*x)^(1/2)*(A + B*x^n))/(a + b*x^n)^(3/2), x)
\[ \int \frac {\sqrt {e x} \left (A+B x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b +a}d x \right ) \] Input:
int((e*x)^(1/2)*(A+B*x^n)/(a+b*x^n)^(3/2),x)
Output:
sqrt(e)*int((sqrt(x)*sqrt(x**n*b + a))/(x**n*b + a),x)