Optimal. Leaf size=206 \[ -\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}+\frac {2 d^2 (2+m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (3+m+2 p)}+\frac {2 d e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1823, 822, 372,
371} \begin {gather*} \frac {2 d e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2)}+\frac {2 d^2 (m+p+2) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)}-\frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 372
Rule 822
Rule 1823
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {\int (g x)^m \left (-2 d^2 e^2 (2+m+p)-2 d e^3 (3+m+2 p) x\right ) \left (d^2-e^2 x^2\right )^p \, dx}{e^2 (3+m+2 p)}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}+\frac {(2 d e) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \, dx}{g}+\frac {\left (2 d^2 (2+m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^p \, dx}{3+m+2 p}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}+\frac {\left (2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{g}+\frac {\left (2 d^2 (2+m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{3+m+2 p}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}+\frac {2 d^2 (2+m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (3+m+2 p)}+\frac {2 d e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 169, normalized size = 0.82 \begin {gather*} \frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2 \left (6+5 m+m^2\right ) \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )+e (1+m) x \left (2 d (3+m) \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )+e (2+m) x \, _2F_1\left (\frac {3+m}{2},-p;\frac {5+m}{2};\frac {e^2 x^2}{d^2}\right )\right )\right )}{(1+m) (2+m) (3+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{m} \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 7.16, size = 192, normalized size = 0.93 \begin {gather*} \frac {d^{2} d^{2 p} 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 {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d d^{2 p} 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 {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} + \frac {d^{2 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 {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________