Integrand size = 20, antiderivative size = 198 \[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=-\frac {b (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 a \left (\sqrt {-a} \sqrt {b} c+a d\right ) (1+n)}-\frac {b (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 a^2 \left (\frac {\sqrt {b} c}{\sqrt {-a}}+d\right ) (1+n)}+\frac {d (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )}{a c^2 (1+n)} \] Output:
-1/2*b*(d*x+c)^(1+n)*hypergeom([1, 1+n],[2+n],b^(1/2)*(d*x+c)/(b^(1/2)*c-( -a)^(1/2)*d))/a/((-a)^(1/2)*b^(1/2)*c+a*d)/(1+n)-1/2*b*(d*x+c)^(1+n)*hyper geom([1, 1+n],[2+n],b^(1/2)*(d*x+c)/(b^(1/2)*c+(-a)^(1/2)*d))/a^2/(b^(1/2) *c/(-a)^(1/2)+d)/(1+n)+d*(d*x+c)^(1+n)*hypergeom([2, 1+n],[2+n],1+d*x/c)/a /c^2/(1+n)
Time = 0.18 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\frac {(c+d x)^{1+n} \left (-\frac {b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{\sqrt {-a} \sqrt {b} c+a d}+\frac {b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{\sqrt {-a} \sqrt {b} c-a d}+\frac {2 d \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )}{c^2}\right )}{2 a (1+n)} \] Input:
Integrate[(c + d*x)^n/(x^2*(a + b*x^2)),x]
Output:
((c + d*x)^(1 + n)*(-((b*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c - Sqrt[-a]*d)])/(Sqrt[-a]*Sqrt[b]*c + a*d)) + (b*Hypergeo metric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)])/ (Sqrt[-a]*Sqrt[b]*c - a*d) + (2*d*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + ( d*x)/c])/c^2))/(2*a*(1 + n))
Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
\(\displaystyle \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \int \left (\frac {(c+d x)^n}{a x^2}-\frac {b (c+d x)^n}{a \left (a+b x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt {b} c-\sqrt {-a} d\right )}-\frac {b (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}+\frac {d (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {d x}{c}+1\right )}{a c^2 (n+1)}\) |
Input:
Int[(c + d*x)^n/(x^2*(a + b*x^2)),x]
Output:
(b*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x) )/(Sqrt[b]*c - Sqrt[-a]*d)])/(2*(-a)^(3/2)*(Sqrt[b]*c - Sqrt[-a]*d)*(1 + n )) - (b*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)])/(2*(-a)^(3/2)*(Sqrt[b]*c + Sqrt[-a]*d)*( 1 + n)) + (d*(c + d*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (d*x )/c])/(a*c^2*(1 + 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]
\[\int \frac {\left (d x +c \right )^{n}}{x^{2} \left (b \,x^{2}+a \right )}d x\]
Input:
int((d*x+c)^n/x^2/(b*x^2+a),x)
Output:
int((d*x+c)^n/x^2/(b*x^2+a),x)
\[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \] Input:
integrate((d*x+c)^n/x^2/(b*x^2+a),x, algorithm="fricas")
Output:
integral((d*x + c)^n/(b*x^4 + a*x^2), x)
Exception generated. \[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((d*x+c)**n/x**2/(b*x**2+a),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \] Input:
integrate((d*x+c)^n/x^2/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^n/((b*x^2 + a)*x^2), x)
\[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \] Input:
integrate((d*x+c)^n/x^2/(b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x + c)^n/((b*x^2 + a)*x^2), x)
Timed out. \[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^n}{x^2\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^n/(x^2*(a + b*x^2)),x)
Output:
int((c + d*x)^n/(x^2*(a + b*x^2)), x)
\[ \int \frac {(c+d x)^n}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {\left (d x +c \right )^{n}}{b \,x^{4}+a \,x^{2}}d x \] Input:
int((d*x+c)^n/x^2/(b*x^2+a),x)
Output:
int((c + d*x)**n/(a*x**2 + b*x**4),x)