Integrand size = 17, antiderivative size = 133 \[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {d e (2+p) \left (a+b x^2\right )^{1+p}}{b (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\frac {\left (a e^2-b d^2 (3+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{b (3+2 p)} \]
d*e*(2+p)*(b*x^2+a)^(p+1)/b/(2*p^2+5*p+3)+e*(e*x+d)*(b*x^2+a)^(p+1)/b/(3+2 *p)-(a*e^2-b*d^2*(3+2*p))*x*(b*x^2+a)^p*hypergeom([1/2, -p],[3/2],-b*x^2/a )/b/(3+2*p)/((1+b*x^2/a)^p)
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (3 b d^2 (1+p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )+e \left (3 d \left (b x^2 \left (1+\frac {b x^2}{a}\right )^p+a \left (-1+\left (1+\frac {b x^2}{a}\right )^p\right )\right )+b e (1+p) x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right )\right )\right )}{3 b (1+p)} \]
((a + b*x^2)^p*(3*b*d^2*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2 )/a)] + e*(3*d*(b*x^2*(1 + (b*x^2)/a)^p + a*(-1 + (1 + (b*x^2)/a)^p)) + b* e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])))/(3*b*(1 + p )*(1 + (b*x^2)/a)^p)
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {497, 25, 455, 238, 237}
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.
\(\displaystyle \int (d+e x)^2 \left (a+b x^2\right )^p \, dx\) |
\(\Big \downarrow \) 497 |
\(\displaystyle \frac {\int -\left (\left (-b (2 p+3) d^2-2 b e (p+2) x d+a e^2\right ) \left (b x^2+a\right )^p\right )dx}{b (2 p+3)}+\frac {e (d+e x) \left (a+b x^2\right )^{p+1}}{b (2 p+3)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {e (d+e x) \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac {\int \left (-b (2 p+3) d^2-2 b e (p+2) x d+a e^2\right ) \left (b x^2+a\right )^pdx}{b (2 p+3)}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {e (d+e x) \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac {\left (a e^2-b d^2 (2 p+3)\right ) \int \left (b x^2+a\right )^pdx-\frac {d e (p+2) \left (a+b x^2\right )^{p+1}}{p+1}}{b (2 p+3)}\) |
\(\Big \downarrow \) 238 |
\(\displaystyle \frac {e (d+e x) \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2-b d^2 (2 p+3)\right ) \int \left (\frac {b x^2}{a}+1\right )^pdx-\frac {d e (p+2) \left (a+b x^2\right )^{p+1}}{p+1}}{b (2 p+3)}\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \frac {e (d+e x) \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2-b d^2 (2 p+3)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )-\frac {d e (p+2) \left (a+b x^2\right )^{p+1}}{p+1}}{b (2 p+3)}\) |
(e*(d + e*x)*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) - (-((d*e*(2 + p)*(a + b*x ^2)^(1 + p))/(1 + p)) + ((a*e^2 - b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*Hyperge ometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p)/(b*(3 + 2*p))
3.4.95.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) ^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(1 + b*(x^2/a))^p, x], x] / ; FreeQ[{a, b, p}, x] && !IntegerQ[2*p] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
\[\int \left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}d x\]
\[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} \,d x } \]
Time = 4.57 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73 \[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=a^{p} d^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {a^{p} e^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + 2 d e \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) \]
a**p*d**2*x*hyper((1/2, -p), (3/2,), b*x**2*exp_polar(I*pi)/a) + a**p*e**2 *x**3*hyper((3/2, -p), (5/2,), b*x**2*exp_polar(I*pi)/a)/3 + 2*d*e*Piecewi se((a**p*x**2/2, Eq(b, 0)), (Piecewise(((a + b*x**2)**(p + 1)/(p + 1), Ne( p, -1)), (log(a + b*x**2), True))/(2*b), True))
\[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} \,d x } \]
\[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} \,d x } \]
Timed out. \[ \int (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int {\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]