Integrand size = 22, antiderivative size = 124 \[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=-\frac {(c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},-n,1,\frac {1}{2},-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{a \sqrt {x}}-\frac {(c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},-n,1,\frac {1}{2},-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{a \sqrt {x}} \] Output:
-(d*x+c)^n*AppellF1(-1/2,1,-n,1/2,-b^(1/2)*x/(-a)^(1/2),-d*x/c)/a/x^(1/2)/ ((1+d*x/c)^n)-(d*x+c)^n*AppellF1(-1/2,1,-n,1/2,b^(1/2)*x/(-a)^(1/2),-d*x/c )/a/x^(1/2)/((1+d*x/c)^n)
\[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx \] Input:
Integrate[(c + d*x)^n/(x^(3/2)*(a + b*x^2)),x]
Output:
Integrate[(c + d*x)^n/(x^(3/2)*(a + b*x^2)), x]
Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {615, 2009}
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.
\(\displaystyle \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \int \left (\frac {\sqrt {-a} (c+d x)^n}{2 a x^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} (c+d x)^n}{2 a x^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},-\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{a \sqrt {x}}-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},\frac {\sqrt {b} x}{\sqrt {-a}},-\frac {d x}{c}\right )}{a \sqrt {x}}\) |
Input:
Int[(c + d*x)^n/(x^(3/2)*(a + b*x^2)),x]
Output:
-(((c + d*x)^n*AppellF1[-1/2, 1, -n, 1/2, -((Sqrt[b]*x)/Sqrt[-a]), -((d*x) /c)])/(a*Sqrt[x]*(1 + (d*x)/c)^n)) - ((c + d*x)^n*AppellF1[-1/2, 1, -n, 1/ 2, (Sqrt[b]*x)/Sqrt[-a], -((d*x)/c)])/(a*Sqrt[x]*(1 + (d*x)/c)^n)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
\[\int \frac {\left (d x +c \right )^{n}}{x^{\frac {3}{2}} \left (b \,x^{2}+a \right )}d x\]
Input:
int((d*x+c)^n/x^(3/2)/(b*x^2+a),x)
Output:
int((d*x+c)^n/x^(3/2)/(b*x^2+a),x)
\[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x^{2} + a\right )} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^n/x^(3/2)/(b*x^2+a),x, algorithm="fricas")
Output:
integral((d*x + c)^n*sqrt(x)/(b*x^4 + a*x^2), x)
Timed out. \[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**n/x**(3/2)/(b*x**2+a),x)
Output:
Timed out
\[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x^{2} + a\right )} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^n/x^(3/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^n/((b*x^2 + a)*x^(3/2)), x)
\[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x^{2} + a\right )} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^n/x^(3/2)/(b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x + c)^n/((b*x^2 + a)*x^(3/2)), x)
Timed out. \[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^n}{x^{3/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^n/(x^(3/2)*(a + b*x^2)),x)
Output:
int((c + d*x)^n/(x^(3/2)*(a + b*x^2)), x)
\[ \int \frac {(c+d x)^n}{x^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (d x +c \right )^{n}}{\sqrt {x}\, a x +\sqrt {x}\, b \,x^{3}}d x \] Input:
int((d*x+c)^n/x^(3/2)/(b*x^2+a),x)
Output:
int((c + d*x)**n/(sqrt(x)*a*x + sqrt(x)*b*x**3),x)