Integrand size = 20, antiderivative size = 48 \[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=-\frac {b^2 x (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{a^3 (1+n) \sqrt {c x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 67} \[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=-\frac {b^2 x (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (3,n+1,n+2,\frac {b x}{a}+1\right )}{a^3 (n+1) \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 67
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(a+b x)^n}{x^3} \, dx}{\sqrt {c x^2}} \\ & = -\frac {b^2 x (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n) \sqrt {c x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=-\frac {b^2 c x^3 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{a^3 (1+n) \left (c x^2\right )^{3/2}} \]
[In]
[Out]
\[\int \frac {\left (b x +a \right )^{n}}{x^{2} \sqrt {c \,x^{2}}}d x\]
[In]
[Out]
\[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{\sqrt {c x^{2}} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=\int \frac {\left (a + b x\right )^{n}}{x^{2} \sqrt {c x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{\sqrt {c x^{2}} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{\sqrt {c x^{2}} x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^n}{x^2 \sqrt {c x^2}} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^2\,\sqrt {c\,x^2}} \,d x \]
[In]
[Out]