Optimal. Leaf size=205 \[ -\frac {(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (a e^2 (m+1)-c d^2 (m+2 p+3)\right ) \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {c x^2}{a}\right )}{c g (m+1) (m+2 p+3)}+\frac {2 d e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};-\frac {c x^2}{a}\right )}{g^2 (m+2)}+\frac {e^2 (g x)^{m+1} \left (a+c x^2\right )^{p+1}}{c g (m+2 p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 194, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1809, 808, 365, 364} \[ \frac {(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (\frac {d^2}{m+1}-\frac {a e^2}{c (m+2 p+3)}\right ) \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {c x^2}{a}\right )}{g}+\frac {2 d e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};-\frac {c x^2}{a}\right )}{g^2 (m+2)}+\frac {e^2 (g x)^{m+1} \left (a+c x^2\right )^{p+1}}{c g (m+2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rule 808
Rule 1809
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^2 \left (a+c x^2\right )^p \, dx &=\frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {\int (g x)^m \left (-a e^2 (1+m)+c d^2 (3+m+2 p)+2 c d e (3+m+2 p) x\right ) \left (a+c x^2\right )^p \, dx}{c (3+m+2 p)}\\ &=\frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {(2 d e) \int (g x)^{1+m} \left (a+c x^2\right )^p \, dx}{g}+\left (d^2-\frac {a e^2 (1+m)}{c (3+m+2 p)}\right ) \int (g x)^m \left (a+c x^2\right )^p \, dx\\ &=\frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {\left (2 d e \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^{1+m} \left (1+\frac {c x^2}{a}\right )^p \, dx}{g}+\left (\left (d^2-\frac {a e^2 (1+m)}{c (3+m+2 p)}\right ) \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^m \left (1+\frac {c x^2}{a}\right )^p \, dx\\ &=\frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {\left (d^2-\frac {a e^2 (1+m)}{c (3+m+2 p)}\right ) (g x)^{1+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};-\frac {c x^2}{a}\right )}{g (1+m)}+\frac {2 d e (g x)^{2+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};-\frac {c x^2}{a}\right )}{g^2 (2+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 158, normalized size = 0.77 \[ \frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d^2 \left (m^2+5 m+6\right ) \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {c x^2}{a}\right )+e (m+1) x \left (2 d (m+3) \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};-\frac {c x^2}{a}\right )+e (m+2) x \, _2F_1\left (\frac {m+3}{2},-p;\frac {m+5}{2};-\frac {c x^2}{a}\right )\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{2} \left (g x \right )^{m} \left (c \,x^{2}+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 170.87, size = 172, normalized size = 0.84 \[ \frac {a^{p} d^{2} g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{p} d e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} + \frac {a^{p} e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________