Integrand size = 20, antiderivative size = 91 \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {c (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2 e (1+m)}+\frac {d (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a^2 e^2 (2+m)} \] Output:
c*(e*x)^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^2/e/(1+m)+d *(e*x)^(2+m)*hypergeom([2, 1+1/2*m],[2+1/2*m],-b*x^2/a)/a^2/e^2/(2+m)
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {x (e x)^m \left (c (2+m) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+d (1+m) x \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )\right )}{a^2 (1+m) (2+m)} \] Input:
Integrate[((e*x)^m*(c + d*x))/(a + b*x^2)^2,x]
Output:
(x*(e*x)^m*(c*(2 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2) /a)] + d*(1 + m)*x*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a) ]))/(a^2*(1 + m)*(2 + m))
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {557, 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.
| \(\displaystyle \int \frac {(c+d x) (e x)^m}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle c \int \frac {(e x)^m}{\left (b x^2+a\right )^2}dx+\frac {d \int \frac {(e x)^{m+1}}{\left (b x^2+a\right )^2}dx}{e}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {c (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a^2 e (m+1)}+\frac {d (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},-\frac {b x^2}{a}\right )}{a^2 e^2 (m+2)}\) |
Input:
Int[((e*x)^m*(c + d*x))/(a + b*x^2)^2,x]
Output:
(c*(e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]) /(a^2*e*(1 + m)) + (d*(e*x)^(2 + m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m )/2, -((b*x^2)/a)])/(a^2*e^2*(2 + m))
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[((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]
\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )}{\left (b \,x^{2}+a \right )^{2}}d x\]
Input:
int((e*x)^m*(d*x+c)/(b*x^2+a)^2,x)
Output:
int((e*x)^m*(d*x+c)/(b*x^2+a)^2,x)
\[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
integral((d*x + c)*(e*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2), x)
Result contains complex when optimal does not.
Time = 15.83 (sec) , antiderivative size = 770, normalized size of antiderivative = 8.46 \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x)**m*(d*x+c)/(b*x**2+a)**2,x)
Output:
c*(-a*e**m*m**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2 )*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/ 2)) + 2*a*e**m*m*x**(m + 1)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8* a**2*b*x**2*gamma(m/2 + 3/2)) + a*e**m*x**(m + 1)*lerchphi(b*x**2*exp_pola r(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a** 2*b*x**2*gamma(m/2 + 3/2)) + 2*a*e**m*x**(m + 1)*gamma(m/2 + 1/2)/(8*a**3* gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) - b*e**m*m**2*x**2*x**( m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/( 8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + b*e**m*x**2*x* *(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2) /(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2))) + d*(-a*e**m* m**2*x**(m + 2)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*b*x**2*gamma(m/2 + 2)) - 2*a*e**m*m*x* *(m + 2)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(8* a**3*gamma(m/2 + 2) + 8*a**2*b*x**2*gamma(m/2 + 2)) + 2*a*e**m*m*x**(m + 2 )*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*b*x**2*gamma(m/2 + 2)) + 4*a*e**m*x**(m + 2)*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*b*x**2* gamma(m/2 + 2)) - b*e**m*m**2*x**2*x**(m + 2)*lerchphi(b*x**2*exp_polar(I* pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*b*x**2*g amma(m/2 + 2)) - 2*b*e**m*m*x**2*x**(m + 2)*lerchphi(b*x**2*exp_polar(I...
\[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
integrate((d*x + c)*(e*x)^m/(b*x^2 + a)^2, x)
\[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
Output:
integrate((d*x + c)*(e*x)^m/(b*x^2 + a)^2, x)
Timed out. \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int(((e*x)^m*(c + d*x))/(a + b*x^2)^2,x)
Output:
int(((e*x)^m*(c + d*x))/(a + b*x^2)^2, x)
\[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {e^{m} \left (x^{m} d +\left (\int \frac {x^{m}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c m -2 \left (\int \frac {x^{m}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c +\left (\int \frac {x^{m}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b^{2} c m \,x^{2}-2 \left (\int \frac {x^{m}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b^{2} c \,x^{2}-\left (\int \frac {x^{m}}{b^{2} m \,x^{5}-2 b^{2} x^{5}+2 a b m \,x^{3}-4 a b \,x^{3}+a^{2} m x -2 a^{2} x}d x \right ) a^{2} d \,m^{2}+2 \left (\int \frac {x^{m}}{b^{2} m \,x^{5}-2 b^{2} x^{5}+2 a b m \,x^{3}-4 a b \,x^{3}+a^{2} m x -2 a^{2} x}d x \right ) a^{2} d m -\left (\int \frac {x^{m}}{b^{2} m \,x^{5}-2 b^{2} x^{5}+2 a b m \,x^{3}-4 a b \,x^{3}+a^{2} m x -2 a^{2} x}d x \right ) a b d \,m^{2} x^{2}+2 \left (\int \frac {x^{m}}{b^{2} m \,x^{5}-2 b^{2} x^{5}+2 a b m \,x^{3}-4 a b \,x^{3}+a^{2} m x -2 a^{2} x}d x \right ) a b d m \,x^{2}\right )}{b \left (b m \,x^{2}-2 b \,x^{2}+a m -2 a \right )} \] Input:
int((e*x)^m*(d*x+c)/(b*x^2+a)^2,x)
Output:
(e**m*(x**m*d + int(x**m/(a**2 + 2*a*b*x**2 + b**2*x**4),x)*a*b*c*m - 2*in t(x**m/(a**2 + 2*a*b*x**2 + b**2*x**4),x)*a*b*c + int(x**m/(a**2 + 2*a*b*x **2 + b**2*x**4),x)*b**2*c*m*x**2 - 2*int(x**m/(a**2 + 2*a*b*x**2 + b**2*x **4),x)*b**2*c*x**2 - int(x**m/(a**2*m*x - 2*a**2*x + 2*a*b*m*x**3 - 4*a*b *x**3 + b**2*m*x**5 - 2*b**2*x**5),x)*a**2*d*m**2 + 2*int(x**m/(a**2*m*x - 2*a**2*x + 2*a*b*m*x**3 - 4*a*b*x**3 + b**2*m*x**5 - 2*b**2*x**5),x)*a**2 *d*m - int(x**m/(a**2*m*x - 2*a**2*x + 2*a*b*m*x**3 - 4*a*b*x**3 + b**2*m* x**5 - 2*b**2*x**5),x)*a*b*d*m**2*x**2 + 2*int(x**m/(a**2*m*x - 2*a**2*x + 2*a*b*m*x**3 - 4*a*b*x**3 + b**2*m*x**5 - 2*b**2*x**5),x)*a*b*d*m*x**2))/ (b*(a*m - 2*a + b*m*x**2 - 2*b*x**2))