\(\int \frac {(e x)^m \sqrt {a+b x^2}}{(c+d x)^2} \, dx\) [160]

Optimal result

Integrand size = 24, antiderivative size = 252 \[ \int \frac {(e x)^m \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {(e x)^{1+m} \sqrt {a+b x^2} \operatorname {AppellF1}\left (\frac {1+m}{2},-\frac {1}{2},2,\frac {3+m}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 e (1+m) \sqrt {1+\frac {b x^2}{a}}}-\frac {2 d (e x)^{2+m} \sqrt {a+b x^2} \operatorname {AppellF1}\left (\frac {2+m}{2},-\frac {1}{2},2,\frac {4+m}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^3 e^2 (2+m) \sqrt {1+\frac {b x^2}{a}}}+\frac {d^2 (e x)^{3+m} \sqrt {a+b x^2} \operatorname {AppellF1}\left (\frac {3+m}{2},-\frac {1}{2},2,\frac {5+m}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^4 e^3 (3+m) \sqrt {1+\frac {b x^2}{a}}} \] Output:

(e*x)^(1+m)*(b*x^2+a)^(1/2)*AppellF1(1/2+1/2*m,2,-1/2,3/2+1/2*m,d^2*x^2/c^ 
2,-b*x^2/a)/c^2/e/(1+m)/(1+b*x^2/a)^(1/2)-2*d*(e*x)^(2+m)*(b*x^2+a)^(1/2)* 
AppellF1(1+1/2*m,2,-1/2,2+1/2*m,d^2*x^2/c^2,-b*x^2/a)/c^3/e^2/(2+m)/(1+b*x 
^2/a)^(1/2)+d^2*(e*x)^(3+m)*(b*x^2+a)^(1/2)*AppellF1(3/2+1/2*m,2,-1/2,5/2+ 
1/2*m,d^2*x^2/c^2,-b*x^2/a)/c^4/e^3/(3+m)/(1+b*x^2/a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {623, 622, 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.

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

\(\Big \downarrow \) 623

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

\(\Big \downarrow \) 622

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

\(\Big \downarrow \) 2009

\(\displaystyle x^{-m} (e x)^m \left (\frac {x^{m+1} \sqrt {a+b x^2} \operatorname {AppellF1}\left (\frac {m+1}{2},-\frac {1}{2},2,\frac {m+3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 (m+1) \sqrt {\frac {b x^2}{a}+1}}+\frac {d^2 x^{m+3} \sqrt {a+b x^2} \operatorname {AppellF1}\left (\frac {m+3}{2},-\frac {1}{2},2,\frac {m+5}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^4 (m+3) \sqrt {\frac {b x^2}{a}+1}}-\frac {2 d x^{m+2} \sqrt {a+b x^2} \operatorname {AppellF1}\left (\frac {m+2}{2},-\frac {1}{2},2,\frac {m+4}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^3 (m+2) \sqrt {\frac {b x^2}{a}+1}}\right )\)

Input:

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

Output:

((e*x)^m*((x^(1 + m)*Sqrt[a + b*x^2]*AppellF1[(1 + m)/2, -1/2, 2, (3 + m)/ 
2, -((b*x^2)/a), (d^2*x^2)/c^2])/(c^2*(1 + m)*Sqrt[1 + (b*x^2)/a]) - (2*d* 
x^(2 + m)*Sqrt[a + b*x^2]*AppellF1[(2 + m)/2, -1/2, 2, (4 + m)/2, -((b*x^2 
)/a), (d^2*x^2)/c^2])/(c^3*(2 + m)*Sqrt[1 + (b*x^2)/a]) + (d^2*x^(3 + m)*S 
qrt[a + b*x^2]*AppellF1[(3 + m)/2, -1/2, 2, (5 + m)/2, -((b*x^2)/a), (d^2* 
x^2)/c^2])/(c^4*(3 + m)*Sqrt[1 + (b*x^2)/a])))/x^m
 

Defintions of rubi rules used

rule 622

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[x^m*(a + b*x^2)^p, (c/(c^2 - d^2*x^2) - d*(x/(c^2 
 - d^2*x^2)))^(-n), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[n, -1]
 

rule 623

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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