Integrand size = 17, antiderivative size = 155 \[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 \left (\sqrt {-a} \sqrt {b} c+a d\right ) (1+n)}+\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 a \left (\frac {\sqrt {b} c}{\sqrt {-a}}+d\right ) (1+n)} \] Output:
1/2*(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)^(1/2)*b^(1/2)*c+a*d)/(1+n)+1/2*(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/(b^(1/2)*c/(-a)^( 1/2)+d)/(1+n)
Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\frac {(c+d x)^{1+n} \left (\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{\sqrt {b} c-\sqrt {-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 \sqrt {-a} (1+n)} \] Input:
Integrate[(c + d*x)^n/(a + b*x^2),x]
Output:
((c + d*x)^(1 + n)*(Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x)) /(Sqrt[b]*c - Sqrt[-a]*d)]/(Sqrt[b]*c - Sqrt[-a]*d) - Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)]/(Sqrt[b]*c + Sqrt[-a]*d)))/(2*Sqrt[-a]*(1 + n))
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {485, 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.
\(\displaystyle \int \frac {(c+d x)^n}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 485 |
\(\displaystyle \int \left (\frac {\sqrt {-a} (c+d x)^n}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} (c+d x)^n}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 \sqrt {-a} (n+1) \left (\sqrt {b} c-\sqrt {-a} d\right )}-\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 \sqrt {-a} (n+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}\) |
Input:
Int[(c + d*x)^n/(a + b*x^2),x]
Output:
((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/ (Sqrt[b]*c - Sqrt[-a]*d)])/(2*Sqrt[-a]*(Sqrt[b]*c - Sqrt[-a]*d)*(1 + n)) - ((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x)) /(Sqrt[b]*c + Sqrt[-a]*d)])/(2*Sqrt[-a]*(Sqrt[b]*c + Sqrt[-a]*d)*(1 + n))
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand Integrand[(c + d*x)^n, 1/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, n}, x] & & !IntegerQ[2*n]
\[\int \frac {\left (d x +c \right )^{n}}{b \,x^{2}+a}d x\]
Input:
int((d*x+c)^n/(b*x^2+a),x)
Output:
int((d*x+c)^n/(b*x^2+a),x)
\[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{b x^{2} + a} \,d x } \] Input:
integrate((d*x+c)^n/(b*x^2+a),x, algorithm="fricas")
Output:
integral((d*x + c)^n/(b*x^2 + a), x)
\[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\int \frac {\left (c + d x\right )^{n}}{a + b x^{2}}\, dx \] Input:
integrate((d*x+c)**n/(b*x**2+a),x)
Output:
Integral((c + d*x)**n/(a + b*x**2), x)
\[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{b x^{2} + a} \,d x } \] Input:
integrate((d*x+c)^n/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^n/(b*x^2 + a), x)
\[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{b x^{2} + a} \,d x } \] Input:
integrate((d*x+c)^n/(b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x + c)^n/(b*x^2 + a), x)
Timed out. \[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\int \frac {{\left (c+d\,x\right )}^n}{b\,x^2+a} \,d x \] Input:
int((c + d*x)^n/(a + b*x^2),x)
Output:
int((c + d*x)^n/(a + b*x^2), x)
\[ \int \frac {(c+d x)^n}{a+b x^2} \, dx=\int \frac {\left (d x +c \right )^{n}}{b \,x^{2}+a}d x \] Input:
int((d*x+c)^n/(b*x^2+a),x)
Output:
int((c + d*x)**n/(a + b*x**2),x)