Integrand size = 22, antiderivative size = 64 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\frac {(d x)^{2+m} \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (2+m,-n,3+m,-\frac {b x}{a}\right )}{d^2 (2+m) x} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 16, 68, 66} \[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\frac {\sqrt {c x^2} (d x)^{m+2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (m+2,-n,m+3,-\frac {b x}{a}\right )}{d^2 (m+2) x} \]
[In]
[Out]
Rule 15
Rule 16
Rule 66
Rule 68
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int x (d x)^m (a+b x)^n \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int (d x)^{1+m} (a+b x)^n \, dx}{d x} \\ & = \frac {\left (\sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{1+m} \left (1+\frac {b x}{a}\right )^n \, dx}{d x} \\ & = \frac {(d x)^{2+m} \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (2+m,-n;3+m;-\frac {b x}{a}\right )}{d^2 (2+m) x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\frac {x (d x)^m \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (2+m,-n,3+m,-\frac {b x}{a}\right )}{2+m} \]
[In]
[Out]
\[\int \left (d x \right )^{m} \sqrt {c \,x^{2}}\, \left (b x +a \right )^{n}d x\]
[In]
[Out]
\[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\int { \sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\int \sqrt {c x^{2}} \left (d x\right )^{m} \left (a + b x\right )^{n}\, dx \]
[In]
[Out]
\[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\int { \sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\int { \sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx=\int {\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^n \,d x \]
[In]
[Out]