Integrand size = 20, antiderivative size = 135 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\frac {c (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e (1+m)}+\frac {d (e x)^{2+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},-\frac {b x^2}{a}\right )}{e^2 (2+m)} \] Output:
c*(e*x)^(1+m)*(b*x^2+a)^p*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/ e/(1+m)/((1+b*x^2/a)^p)+d*(e*x)^(2+m)*(b*x^2+a)^p*hypergeom([-p, 1+1/2*m], [2+1/2*m],-b*x^2/a)/e^2/(2+m)/((1+b*x^2/a)^p)
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.79 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\frac {x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )+d (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},-\frac {b x^2}{a}\right )\right )}{(1+m) (2+m)} \] Input:
Integrate[(e*x)^m*(c + d*x)*(a + b*x^2)^p,x]
Output:
(x*(e*x)^m*(a + b*x^2)^p*(c*(2 + m)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)] + d*(1 + m)*x*Hypergeometric2F1[(2 + m)/2, -p, (4 + m) /2, -((b*x^2)/a)]))/((1 + m)*(2 + m)*(1 + (b*x^2)/a)^p)
Time = 0.24 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {557, 279, 278}
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.
| \(\displaystyle \int (c+d x) (e x)^m \left (a+b x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle c \int (e x)^m \left (b x^2+a\right )^pdx+\frac {d \int (e x)^{m+1} \left (b x^2+a\right )^pdx}{e}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^m \left (\frac {b x^2}{a}+1\right )^pdx+\frac {d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{m+1} \left (\frac {b x^2}{a}+1\right )^pdx}{e}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {c (e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right )}{e (m+1)}+\frac {d (e x)^{m+2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-p,\frac {m+4}{2},-\frac {b x^2}{a}\right )}{e^2 (m+2)}\) |
Input:
Int[(e*x)^m*(c + d*x)*(a + b*x^2)^p,x]
Output:
(c*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(e*(1 + m)*(1 + (b*x^2)/a)^p) + (d*(e*x)^(2 + m)*(a + b*x^ 2)^p*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, -((b*x^2)/a)])/(e^2*(2 + m)*(1 + (b*x^2)/a)^p)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
\[\int \left (e x \right )^{m} \left (d x +c \right ) \left (b \,x^{2}+a \right )^{p}d x\]
Input:
int((e*x)^m*(d*x+c)*(b*x^2+a)^p,x)
Output:
int((e*x)^m*(d*x+c)*(b*x^2+a)^p,x)
\[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a)^p,x, algorithm="fricas")
Output:
integral((d*x + c)*(b*x^2 + a)^p*(e*x)^m, x)
Result contains complex when optimal does not.
Time = 37.86 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} c e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{p} d e^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} \] Input:
integrate((e*x)**m*(d*x+c)*(b*x**2+a)**p,x)
Output:
a**p*c*e**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2, ), b*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 3/2)) + a**p*d*e**m*x**(m + 2) *gamma(m/2 + 1)*hyper((-p, m/2 + 1), (m/2 + 2,), b*x**2*exp_polar(I*pi)/a) /(2*gamma(m/2 + 2))
\[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a)^p,x, algorithm="maxima")
Output:
integrate((d*x + c)*(b*x^2 + a)^p*(e*x)^m, x)
\[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a)^p,x, algorithm="giac")
Output:
integrate((d*x + c)*(b*x^2 + a)^p*(e*x)^m, x)
Timed out. \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\int {\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,\left (c+d\,x\right ) \,d x \] Input:
int((e*x)^m*(a + b*x^2)^p*(c + d*x),x)
Output:
int((e*x)^m*(a + b*x^2)^p*(c + d*x), x)
\[ \int (e x)^m (c+d x) \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:
int((e*x)^m*(d*x+c)*(b*x^2+a)^p,x)
Output:
(e**m*(2*x**m*(a + b*x**2)**p*a*d*m*p + 4*x**m*(a + b*x**2)**p*a*d*p**2 + 2*x**m*(a + b*x**2)**p*a*d*p + x**m*(a + b*x**2)**p*b*c*m**2*x + 4*x**m*(a + b*x**2)**p*b*c*m*p*x + 2*x**m*(a + b*x**2)**p*b*c*m*x + 4*x**m*(a + b*x **2)**p*b*c*p**2*x + 4*x**m*(a + b*x**2)**p*b*c*p*x + x**m*(a + b*x**2)**p *b*d*m**2*x**2 + 4*x**m*(a + b*x**2)**p*b*d*m*p*x**2 + x**m*(a + b*x**2)** p*b*d*m*x**2 + 4*x**m*(a + b*x**2)**p*b*d*p**2*x**2 + 2*x**m*(a + b*x**2)* *p*b*d*p*x**2 - 2*int((x**m*(a + b*x**2)**p)/(a*m**3*x + 6*a*m**2*p*x + 3* a*m**2*x + 12*a*m*p**2*x + 12*a*m*p*x + 2*a*m*x + 8*a*p**3*x + 12*a*p**2*x + 4*a*p*x + b*m**3*x**3 + 6*b*m**2*p*x**3 + 3*b*m**2*x**3 + 12*b*m*p**2*x **3 + 12*b*m*p*x**3 + 2*b*m*x**3 + 8*b*p**3*x**3 + 12*b*p**2*x**3 + 4*b*p* x**3),x)*a**2*d*m**5*p - 16*int((x**m*(a + b*x**2)**p)/(a*m**3*x + 6*a*m** 2*p*x + 3*a*m**2*x + 12*a*m*p**2*x + 12*a*m*p*x + 2*a*m*x + 8*a*p**3*x + 1 2*a*p**2*x + 4*a*p*x + b*m**3*x**3 + 6*b*m**2*p*x**3 + 3*b*m**2*x**3 + 12* b*m*p**2*x**3 + 12*b*m*p*x**3 + 2*b*m*x**3 + 8*b*p**3*x**3 + 12*b*p**2*x** 3 + 4*b*p*x**3),x)*a**2*d*m**4*p**2 - 8*int((x**m*(a + b*x**2)**p)/(a*m**3 *x + 6*a*m**2*p*x + 3*a*m**2*x + 12*a*m*p**2*x + 12*a*m*p*x + 2*a*m*x + 8* a*p**3*x + 12*a*p**2*x + 4*a*p*x + b*m**3*x**3 + 6*b*m**2*p*x**3 + 3*b*m** 2*x**3 + 12*b*m*p**2*x**3 + 12*b*m*p*x**3 + 2*b*m*x**3 + 8*b*p**3*x**3 + 1 2*b*p**2*x**3 + 4*b*p*x**3),x)*a**2*d*m**4*p - 48*int((x**m*(a + b*x**2)** p)/(a*m**3*x + 6*a*m**2*p*x + 3*a*m**2*x + 12*a*m*p**2*x + 12*a*m*p*x +...