Integrand size = 21, antiderivative size = 71 \[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\frac {c (d x)^{1+m} \sqrt {\frac {c}{\left (a+b x^n\right )^2}} \left (1+\frac {b x^n}{a}\right ) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2 d (1+m)} \] Output:
c*(d*x)^(1+m)*(c/(a+b*x^n)^2)^(1/2)*(1+b*x^n/a)*hypergeom([3, (1+m)/n],[(1 +m+n)/n],-b*x^n/a)/a^2/d/(1+m)
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\frac {x (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \left (a+b x^n\right )^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{n},1+\frac {1+m}{n},-\frac {b x^n}{a}\right )}{a^3 (1+m)} \] Input:
Integrate[(d*x)^m*(c/(a + b*x^n)^2)^(3/2),x]
Output:
(x*(d*x)^m*(c/(a + b*x^n)^2)^(3/2)*(a + b*x^n)^3*Hypergeometric2F1[3, (1 + m)/n, 1 + (1 + m)/n, -((b*x^n)/a)])/(a^3*(1 + m))
Time = 0.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2045, 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.
\(\displaystyle \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \frac {c \left (\frac {b x^n}{a}+1\right ) \sqrt {\frac {c}{\left (a+b x^n\right )^2}} \int \frac {(d x)^m}{\left (\frac {b x^n}{a}+1\right )^3}dx}{a^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {c (d x)^{m+1} \left (\frac {b x^n}{a}+1\right ) \sqrt {\frac {c}{\left (a+b x^n\right )^2}} \operatorname {Hypergeometric2F1}\left (3,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a^2 d (m+1)}\) |
Input:
Int[(d*x)^m*(c/(a + b*x^n)^2)^(3/2),x]
Output:
(c*(d*x)^(1 + m)*Sqrt[c/(a + b*x^n)^2]*(1 + (b*x^n)/a)*Hypergeometric2F1[3 , (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^2*d*(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[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
\[\int \left (x d \right )^{m} {\left (\frac {c}{\left (a +b \,x^{n}\right )^{2}}\right )}^{\frac {3}{2}}d x\]
Input:
int((x*d)^m*(c/(a+b*x^n)^2)^(3/2),x)
Output:
int((x*d)^m*(c/(a+b*x^n)^2)^(3/2),x)
\[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\int { \left (d x\right )^{m} \left (\frac {c}{{\left (b x^{n} + a\right )}^{2}}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x)^m*(c/(a+b*x^n)^2)^(3/2),x, algorithm="fricas")
Output:
integral((d*x)^m*c*sqrt(c/(b^2*x^(2*n) + 2*a*b*x^n + a^2))/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)
Timed out. \[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((d*x)**m*(c/(a+b*x**n)**2)**(3/2),x)
Output:
Timed out
\[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\int { \left (d x\right )^{m} \left (\frac {c}{{\left (b x^{n} + a\right )}^{2}}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x)^m*(c/(a+b*x^n)^2)^(3/2),x, algorithm="maxima")
Output:
(m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*c^(3/2)*d^m*integrate(1/2*x^m/(a^2*b *n^2*x^n + a^3*n^2), x) - 1/2*(a*c^(3/2)*d^m*(m - 3*n + 1)*x*x^m + b*c^(3/ 2)*d^m*(m - 2*n + 1)*x*e^(m*log(x) + n*log(x)))/(a^2*b^2*n^2*x^(2*n) + 2*a ^3*b*n^2*x^n + a^4*n^2)
\[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\int { \left (d x\right )^{m} \left (\frac {c}{{\left (b x^{n} + a\right )}^{2}}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x)^m*(c/(a+b*x^n)^2)^(3/2),x, algorithm="giac")
Output:
integrate((d*x)^m*(c/(b*x^n + a)^2)^(3/2), x)
Timed out. \[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=\int {\left (d\,x\right )}^m\,{\left (\frac {c}{{\left (a+b\,x^n\right )}^2}\right )}^{3/2} \,d x \] Input:
int((d*x)^m*(c/(a + b*x^n)^2)^(3/2),x)
Output:
int((d*x)^m*(c/(a + b*x^n)^2)^(3/2), x)
\[ \int (d x)^m \left (\frac {c}{\left (a+b x^n\right )^2}\right )^{3/2} \, dx=d^{m} \sqrt {c}\, \left (\int \frac {x^{m}}{x^{3 n} b^{3}+3 x^{2 n} a \,b^{2}+3 x^{n} a^{2} b +a^{3}}d x \right ) c \] Input:
int((d*x)^m*(c/(a+b*x^n)^2)^(3/2),x)
Output:
d**m*sqrt(c)*int(x**m/(x**(3*n)*b**3 + 3*x**(2*n)*a*b**2 + 3*x**n*a**2*b + a**3),x)*c