\(\int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx\) [228]

Optimal result

Integrand size = 34, antiderivative size = 100 \[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\frac {(c+d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )}{a e (1+m)}+\frac {b (c-d) (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\frac {b^2 x^2}{a^2}\right )}{a^2 e^2 (2+m)} \] Output:

(c+d)*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],b^2*x^2/a^2)/a/e/(1 
+m)+b*(c-d)*(e*x)^(2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],b^2*x^2/a^2)/a^2/ 
e^2/(2+m)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.86 \[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\frac {x (e x)^m \left (\frac {b (c-d) x \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},\frac {b^2 x^2}{a^2}\right )}{2+m}+\frac {a (c+d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )}{1+m}\right )}{a^2} \] Input:

Integrate[((e*x)^m*(a*(c + d) + b*(c - d)*x))/(a^2 - b^2*x^2),x]
 

Output:

(x*(e*x)^m*((b*(c - d)*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, (b^2*x^2)/ 
a^2])/(2 + m) + (a*(c + d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b^2 
*x^2)/a^2])/(1 + m)))/a^2
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.

\(\displaystyle \int \frac {(e x)^m (a (c+d)+b x (c-d))}{a^2-b^2 x^2} \, dx\)

\(\Big \downarrow \) 557

\(\displaystyle a (c+d) \int \frac {(e x)^m}{a^2-b^2 x^2}dx+\frac {b (c-d) \int \frac {(e x)^{m+1}}{a^2-b^2 x^2}dx}{e}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {b (c-d) (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},\frac {b^2 x^2}{a^2}\right )}{a^2 e^2 (m+2)}+\frac {(c+d) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\frac {b^2 x^2}{a^2}\right )}{a e (m+1)}\)

Input:

Int[((e*x)^m*(a*(c + d) + b*(c - d)*x))/(a^2 - b^2*x^2),x]
 

Output:

((c + d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b^2*x^2 
)/a^2])/(a*e*(1 + m)) + (b*(c - d)*(e*x)^(2 + m)*Hypergeometric2F1[1, (2 + 
 m)/2, (4 + m)/2, (b^2*x^2)/a^2])/(a^2*e^2*(2 + m))
 

Defintions of rubi rules used

rule 278

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])
 

rule 557

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]
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (a \left (c +d \right )+b \left (c -d \right ) x \right )}{-b^{2} x^{2}+a^{2}}d x\]

Input:

int((e*x)^m*(a*(c+d)+b*(c-d)*x)/(-b^2*x^2+a^2),x)
 

Output:

int((e*x)^m*(a*(c+d)+b*(c-d)*x)/(-b^2*x^2+a^2),x)
 

Fricas [F]

\[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\int { -\frac {{\left (b {\left (c - d\right )} x + a {\left (c + d\right )}\right )} \left (e x\right )^{m}}{b^{2} x^{2} - a^{2}} \,d x } \] Input:

integrate((e*x)^m*(a*(c+d)+b*(c-d)*x)/(-b^2*x^2+a^2),x, algorithm="fricas" 
)
 

Output:

integral(-(a*c + a*d + (b*c - b*d)*x)*(e*x)^m/(b^2*x^2 - a^2), x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (78) = 156\).

Time = 2.77 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.79 \[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\frac {c e^{m} m x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c e^{m} x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d e^{m} m x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d e^{m} x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b c e^{m} m x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} + \frac {b c e^{m} x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} - \frac {b d e^{m} m x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} - \frac {b d e^{m} x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} \] Input:

integrate((e*x)**m*(a*(c+d)+b*(c-d)*x)/(-b**2*x**2+a**2),x)
 

Output:

c*e**m*m*x**(m + 1)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1/2)*gamma(m/2 + 1/2 
)/(4*a*gamma(m/2 + 3/2)) + c*e**m*x**(m + 1)*lerchphi(b**2*x**2/a**2, 1, m 
/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)) + d*e**m*m*x**(m + 1)*le 
rchphi(b**2*x**2/a**2, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2 
)) + d*e**m*x**(m + 1)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1/2)*gamma(m/2 + 
1/2)/(4*a*gamma(m/2 + 3/2)) + b*c*e**m*m*x**(m + 2)*lerchphi(b**2*x**2/a** 
2, 1, m/2 + 1)*gamma(m/2 + 1)/(4*a**2*gamma(m/2 + 2)) + b*c*e**m*x**(m + 2 
)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a**2*gamma(m/2 + 
2)) - b*d*e**m*m*x**(m + 2)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1)*gamma(m/2 
 + 1)/(4*a**2*gamma(m/2 + 2)) - b*d*e**m*x**(m + 2)*lerchphi(b**2*x**2/a** 
2, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a**2*gamma(m/2 + 2))
 

Maxima [F]

\[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\int { -\frac {{\left (b {\left (c - d\right )} x + a {\left (c + d\right )}\right )} \left (e x\right )^{m}}{b^{2} x^{2} - a^{2}} \,d x } \] Input:

integrate((e*x)^m*(a*(c+d)+b*(c-d)*x)/(-b^2*x^2+a^2),x, algorithm="maxima" 
)
 

Output:

-integrate((b*(c - d)*x + a*(c + d))*(e*x)^m/(b^2*x^2 - a^2), x)
 

Giac [F]

\[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\int { -\frac {{\left (b {\left (c - d\right )} x + a {\left (c + d\right )}\right )} \left (e x\right )^{m}}{b^{2} x^{2} - a^{2}} \,d x } \] Input:

integrate((e*x)^m*(a*(c+d)+b*(c-d)*x)/(-b^2*x^2+a^2),x, algorithm="giac")
 

Output:

integrate(-(b*(c - d)*x + a*(c + d))*(e*x)^m/(b^2*x^2 - a^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\int \frac {\left (a\,\left (c+d\right )+b\,x\,\left (c-d\right )\right )\,{\left (e\,x\right )}^m}{a^2-b^2\,x^2} \,d x \] Input:

int(((a*(c + d) + b*x*(c - d))*(e*x)^m)/(a^2 - b^2*x^2),x)
 

Output:

int(((a*(c + d) + b*x*(c - d))*(e*x)^m)/(a^2 - b^2*x^2), x)
 

Reduce [F]

\[ \int \frac {(e x)^m (a (c+d)+b (c-d) x)}{a^2-b^2 x^2} \, dx=\frac {e^{m} \left (-x^{m} c +x^{m} d +\left (\int \frac {x^{m}}{-b^{2} x^{2}+a^{2}}d x \right ) a b c m +\left (\int \frac {x^{m}}{-b^{2} x^{2}+a^{2}}d x \right ) a b d m +\left (\int \frac {x^{m}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2} c m -\left (\int \frac {x^{m}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2} d m \right )}{b m} \] Input:

int((e*x)^m*(a*(c+d)+b*(c-d)*x)/(-b^2*x^2+a^2),x)
 

Output:

(e**m*( - x**m*c + x**m*d + int(x**m/(a**2 - b**2*x**2),x)*a*b*c*m + int(x 
**m/(a**2 - b**2*x**2),x)*a*b*d*m + int(x**m/(a**2*x - b**2*x**3),x)*a**2* 
c*m - int(x**m/(a**2*x - b**2*x**3),x)*a**2*d*m))/(b*m)