\(\int \frac {(e x)^m}{\sqrt {c+d x} (a+b x^2)^2} \, dx\) [152]

Optimal result

Integrand size = 24, antiderivative size = 377 \[ \int \frac {(e x)^m}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=\frac {b (e x)^{1+m} (c-d x) \sqrt {c+d x}}{2 a \left (b c^2+a d^2\right ) e \left (a+b x^2\right )}-\frac {\left (\sqrt {-a} \sqrt {b} c d-a d^2 (3-2 m)-2 b c^2 (1-m)\right ) (e x)^{1+m} \sqrt {1+\frac {d x}{c}} \operatorname {AppellF1}\left (1+m,\frac {1}{2},1,2+m,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{8 a^2 \left (b c^2+a d^2\right ) e (1+m) \sqrt {c+d x}}+\frac {\left (\sqrt {-a} \sqrt {b} c d+a d^2 (3-2 m)+2 b c^2 (1-m)\right ) (e x)^{1+m} \sqrt {1+\frac {d x}{c}} \operatorname {AppellF1}\left (1+m,\frac {1}{2},1,2+m,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{8 a^2 \left (b c^2+a d^2\right ) e (1+m) \sqrt {c+d x}}+\frac {d (1+2 m) \left (-\frac {d x}{c}\right )^{-m} (e x)^m \sqrt {c+d x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+\frac {d x}{c}\right )}{2 a \left (b c^2+a d^2\right )} \] Output:

1/2*b*(e*x)^(1+m)*(-d*x+c)*(d*x+c)^(1/2)/a/(a*d^2+b*c^2)/e/(b*x^2+a)-1/8*( 
(-a)^(1/2)*b^(1/2)*c*d-a*d^2*(3-2*m)-2*b*c^2*(1-m))*(e*x)^(1+m)*(1+d*x/c)^ 
(1/2)*AppellF1(1+m,1,1/2,2+m,-b^(1/2)*x/(-a)^(1/2),-d*x/c)/a^2/(a*d^2+b*c^ 
2)/e/(1+m)/(d*x+c)^(1/2)+1/8*((-a)^(1/2)*b^(1/2)*c*d+a*d^2*(3-2*m)+2*b*c^2 
*(1-m))*(e*x)^(1+m)*(1+d*x/c)^(1/2)*AppellF1(1+m,1,1/2,2+m,b^(1/2)*x/(-a)^ 
(1/2),-d*x/c)/a^2/(a*d^2+b*c^2)/e/(1+m)/(d*x+c)^(1/2)+1/2*d*(1+2*m)*(e*x)^ 
m*(d*x+c)^(1/2)*hypergeom([1/2, -m],[3/2],1+d*x/c)/a/(a*d^2+b*c^2)/((-d*x/ 
c)^m)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b (e x)^m}{2 a \left (-a b-b^2 x^2\right ) \sqrt {c+d x}}-\frac {b (e x)^m}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2 \sqrt {c+d x}}-\frac {b (e x)^m}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2 \sqrt {c+d x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\frac {d x}{c}+1} (e x)^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2},1,m+2,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1) \sqrt {c+d x}}+\frac {\sqrt {\frac {d x}{c}+1} (e x)^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2},1,m+2,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1) \sqrt {c+d x}}+\frac {\sqrt {\frac {d x}{c}+1} (e x)^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2},2,m+2,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1) \sqrt {c+d x}}+\frac {\sqrt {\frac {d x}{c}+1} (e x)^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2},2,m+2,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1) \sqrt {c+d x}}\)

Input:

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

Output:

((e*x)^(1 + m)*Sqrt[1 + (d*x)/c]*AppellF1[1 + m, 1/2, 1, 2 + m, -((d*x)/c) 
, -((Sqrt[b]*x)/Sqrt[-a])])/(4*a^2*e*(1 + m)*Sqrt[c + d*x]) + ((e*x)^(1 + 
m)*Sqrt[1 + (d*x)/c]*AppellF1[1 + m, 1/2, 1, 2 + m, -((d*x)/c), (Sqrt[b]*x 
)/Sqrt[-a]])/(4*a^2*e*(1 + m)*Sqrt[c + d*x]) + ((e*x)^(1 + m)*Sqrt[1 + (d* 
x)/c]*AppellF1[1 + m, 1/2, 2, 2 + m, -((d*x)/c), -((Sqrt[b]*x)/Sqrt[-a])]) 
/(4*a^2*e*(1 + m)*Sqrt[c + d*x]) + ((e*x)^(1 + m)*Sqrt[1 + (d*x)/c]*Appell 
F1[1 + m, 1/2, 2, 2 + m, -((d*x)/c), (Sqrt[b]*x)/Sqrt[-a]])/(4*a^2*e*(1 + 
m)*Sqrt[c + d*x])
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

int((e*x)^m/(d*x+c)^(1/2)/(b*x^2+a)^2,x)
 

Output:

int((e*x)^m/(d*x+c)^(1/2)/(b*x^2+a)^2,x)
 

Fricas [F]

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

integrate((e*x)^m/(d*x+c)^(1/2)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

integral(sqrt(d*x + c)*(e*x)^m/(b^2*d*x^5 + b^2*c*x^4 + 2*a*b*d*x^3 + 2*a* 
b*c*x^2 + a^2*d*x + a^2*c), x)
 

Sympy [F(-1)]

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

integrate((e*x)**m/(d*x+c)**(1/2)/(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^m/(d*x+c)^(1/2)/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^m/(d*x+c)^(1/2)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((e*x)^m/((a + b*x^2)^2*(c + d*x)^(1/2)),x)
 

Output:

int((e*x)^m/((a + b*x^2)^2*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^m}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=e^{m} \left (\int \frac {x^{m} \sqrt {d x +c}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}+2 a b d \,x^{3}+2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) \] Input:

int((e*x)^m/(d*x+c)^(1/2)/(b*x^2+a)^2,x)
 

Output:

e**m*int((x**m*sqrt(c + d*x))/(a**2*c + a**2*d*x + 2*a*b*c*x**2 + 2*a*b*d* 
x**3 + b**2*c*x**4 + b**2*d*x**5),x)