Integrand size = 24, antiderivative size = 120 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=-\frac {2 d \left (a+b x^n\right )^{1+p}}{b e (1-2 n-2 n p) \sqrt {e x}}-\frac {2 \left (c-\frac {a d}{b-2 b n (1+p)}\right ) \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2 n},-p,1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{e \sqrt {e x}} \] Output:
-2*d*(a+b*x^n)^(p+1)/b/e/(-2*n*p-2*n+1)/(e*x)^(1/2)-2*(c-a*d/(b-2*b*n*(p+1 )))*(a+b*x^n)^p*hypergeom([-p, -1/2/n],[1-1/2/n],-b*x^n/a)/e/(e*x)^(1/2)/( (1+b*x^n/a)^p)
Time = 0.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\frac {2 x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (d x^n \operatorname {Hypergeometric2F1}\left (1-\frac {1}{2 n},-p,2-\frac {1}{2 n},-\frac {b x^n}{a}\right )+c (1-2 n) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2 n},-p,1-\frac {1}{2 n},-\frac {b x^n}{a}\right )\right )}{(-1+2 n) (e x)^{3/2}} \] Input:
Integrate[((a + b*x^n)^p*(c + d*x^n))/(e*x)^(3/2),x]
Output:
(2*x*(a + b*x^n)^p*(d*x^n*Hypergeometric2F1[1 - 1/(2*n), -p, 2 - 1/(2*n), -((b*x^n)/a)] + c*(1 - 2*n)*Hypergeometric2F1[-1/2*1/n, -p, 1 - 1/(2*n), - ((b*x^n)/a)]))/((-1 + 2*n)*(e*x)^(3/2)*(1 + (b*x^n)/a)^p)
Time = 0.45 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
| \(\displaystyle \int \frac {\left (c+d x^n\right ) \left (a+b x^n\right )^p}{(e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \left (c-\frac {a d}{b-2 b n (p+1)}\right ) \int \frac {\left (b x^n+a\right )^p}{(e x)^{3/2}}dx-\frac {2 d \left (a+b x^n\right )^{p+1}}{b e \sqrt {e x} (-2 n p-2 n+1)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d}{b-2 b n (p+1)}\right ) \int \frac {\left (\frac {b x^n}{a}+1\right )^p}{(e x)^{3/2}}dx-\frac {2 d \left (a+b x^n\right )^{p+1}}{b e \sqrt {e x} (-2 n p-2 n+1)}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle -\frac {2 \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d}{b-2 b n (p+1)}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2 n},-p,1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{e \sqrt {e x}}-\frac {2 d \left (a+b x^n\right )^{p+1}}{b e \sqrt {e x} (-2 n p-2 n+1)}\) |
Input:
Int[((a + b*x^n)^p*(c + d*x^n))/(e*x)^(3/2),x]
Output:
(-2*d*(a + b*x^n)^(1 + p))/(b*e*(1 - 2*n - 2*n*p)*Sqrt[e*x]) - (2*(c - (a* d)/(b - 2*b*n*(1 + p)))*(a + b*x^n)^p*Hypergeometric2F1[-1/2*1/n, -p, 1 - 1/(2*n), -((b*x^n)/a)])/(e*Sqrt[e*x]*(1 + (b*x^n)/a)^p)
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]
\[\int \frac {\left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )}{\left (e x \right )^{\frac {3}{2}}}d x\]
Input:
int((a+b*x^n)^p*(c+d*x^n)/(e*x)^(3/2),x)
Output:
int((a+b*x^n)^p*(c+d*x^n)/(e*x)^(3/2),x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)/(e*x)^(3/2),x, algorithm="fricas")
Output:
integral((d*x^n + c)*sqrt(e*x)*(b*x^n + a)^p/(e^2*x^2), x)
Result contains complex when optimal does not.
Time = 38.60 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\frac {a^{1 - \frac {1}{2 n}} a^{p - 1 + \frac {1}{2 n}} d x^{n - \frac {1}{2}} \Gamma \left (1 - \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - \frac {1}{2 n} \\ 2 - \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{e^{\frac {3}{2}} n \Gamma \left (2 - \frac {1}{2 n}\right )} + \frac {a^{- \frac {1}{2 n}} a^{p + \frac {1}{2 n}} c \Gamma \left (- \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2 n}, - p \\ 1 - \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{e^{\frac {3}{2}} n \sqrt {x} \Gamma \left (1 - \frac {1}{2 n}\right )} \] Input:
integrate((a+b*x**n)**p*(c+d*x**n)/(e*x)**(3/2),x)
Output:
a**(1 - 1/(2*n))*a**(p - 1 + 1/(2*n))*d*x**(n - 1/2)*gamma(1 - 1/(2*n))*hy per((-p, 1 - 1/(2*n)), (2 - 1/(2*n),), b*x**n*exp_polar(I*pi)/a)/(e**(3/2) *n*gamma(2 - 1/(2*n))) + a**(p + 1/(2*n))*c*gamma(-1/(2*n))*hyper((-1/(2*n ), -p), (1 - 1/(2*n),), b*x**n*exp_polar(I*pi)/a)/(a**(1/(2*n))*e**(3/2)*n *sqrt(x)*gamma(1 - 1/(2*n)))
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)/(e*x)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^n + c)*(b*x^n + a)^p/(e*x)^(3/2), x)
Exception generated. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)/(e*x)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-2,[0,0,0,2,1,0,1,2]%%%}+%%%{-2,[0,0,0,2,1,0,0,2]%%%} / %% %{4,[0,0,
Timed out. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right )}{{\left (e\,x\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^n)^p*(c + d*x^n))/(e*x)^(3/2),x)
Output:
int(((a + b*x^n)^p*(c + d*x^n))/(e*x)^(3/2), x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{3/2}} \, dx=\text {too large to display} \] Input:
int((a+b*x^n)^p*(c+d*x^n)/(e*x)^(3/2),x)
Output:
(2*sqrt(e)*(2*x**n*(x**n*b + a)**p*b*d*n*p - x**n*(x**n*b + a)**p*b*d + 2* (x**n*b + a)**p*a*d*n*p + 2*(x**n*b + a)**p*b*c*n*p + 2*(x**n*b + a)**p*b* c*n - (x**n*b + a)**p*b*c + 4*sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/(4*x** n*b*n**2*p**2*x**2 + 4*x**n*b*n**2*p*x**2 - 4*x**n*b*n*p*x**2 - 2*x**n*b*n *x**2 + x**n*b*x**2 + 4*a*n**2*p**2*x**2 + 4*a*n**2*p*x**2 - 4*a*n*p*x**2 - 2*a*n*x**2 + a*x**2),x)*a**2*d*n**3*p**3 + 4*sqrt(x)*int((sqrt(x)*(x**n* b + a)**p)/(4*x**n*b*n**2*p**2*x**2 + 4*x**n*b*n**2*p*x**2 - 4*x**n*b*n*p* x**2 - 2*x**n*b*n*x**2 + x**n*b*x**2 + 4*a*n**2*p**2*x**2 + 4*a*n**2*p*x** 2 - 4*a*n*p*x**2 - 2*a*n*x**2 + a*x**2),x)*a**2*d*n**3*p**2 - 4*sqrt(x)*in t((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2*x**2 + 4*x**n*b*n**2*p*x** 2 - 4*x**n*b*n*p*x**2 - 2*x**n*b*n*x**2 + x**n*b*x**2 + 4*a*n**2*p**2*x**2 + 4*a*n**2*p*x**2 - 4*a*n*p*x**2 - 2*a*n*x**2 + a*x**2),x)*a**2*d*n**2*p* *2 - 2*sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2*x**2 + 4* x**n*b*n**2*p*x**2 - 4*x**n*b*n*p*x**2 - 2*x**n*b*n*x**2 + x**n*b*x**2 + 4 *a*n**2*p**2*x**2 + 4*a*n**2*p*x**2 - 4*a*n*p*x**2 - 2*a*n*x**2 + a*x**2), x)*a**2*d*n**2*p + sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p* *2*x**2 + 4*x**n*b*n**2*p*x**2 - 4*x**n*b*n*p*x**2 - 2*x**n*b*n*x**2 + x** n*b*x**2 + 4*a*n**2*p**2*x**2 + 4*a*n**2*p*x**2 - 4*a*n*p*x**2 - 2*a*n*x** 2 + a*x**2),x)*a**2*d*n*p + 8*sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/(4*x** n*b*n**2*p**2*x**2 + 4*x**n*b*n**2*p*x**2 - 4*x**n*b*n*p*x**2 - 2*x**n*...