\(\int \frac {(e x)^m (c+d x)^{5/2}}{a+b x^2} \, dx\) [142]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {(e x)^m (c+d x)^{5/2}}{a+b x^2} \, dx=\frac {c^2 (e x)^{1+m} \sqrt {c+d x} \operatorname {AppellF1}\left (1+m,-\frac {5}{2},1,2+m,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{2 a e (1+m) \sqrt {1+\frac {d x}{c}}}+\frac {c^2 (e x)^{1+m} \sqrt {c+d x} \operatorname {AppellF1}\left (1+m,-\frac {5}{2},1,2+m,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{2 a e (1+m) \sqrt {1+\frac {d x}{c}}} \] Output:

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

Mathematica [F]

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

Integrate[((e*x)^m*(c + d*x)^(5/2))/(a + b*x^2),x]
 

Output:

Integrate[((e*x)^m*(c + d*x)^(5/2))/(a + b*x^2), x]
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.

\(\displaystyle \int \frac {(c+d x)^{5/2} (e x)^m}{a+b x^2} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {\sqrt {-a} (c+d x)^{5/2} (e x)^m}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} (c+d x)^{5/2} (e x)^m}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[((e*x)^m*(c + d*x)^(5/2))/(a + b*x^2),x]
 

Output:

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

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]

\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )^{\frac {5}{2}}}{b \,x^{2}+a}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^m*(d*x+c)^(5/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((e*x)**m*(d*x+c)**(5/2)/(b*x**2+a),x)
 

Output:

Integral((e*x)**m*(c + d*x)**(5/2)/(a + b*x**2), x)
 

Maxima [F]

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

integrate((e*x)^m*(d*x+c)^(5/2)/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(5/2)*(e*x)^m/(b*x^2 + a), x)
 

Giac [F]

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

integrate((e*x)^m*(d*x+c)^(5/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(5/2)*(e*x)^m/(b*x^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m (c+d x)^{5/2}}{a+b x^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (c+d\,x\right )}^{5/2}}{b\,x^2+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m (c+d x)^{5/2}}{a+b x^2} \, dx=\text {too large to display} \] Input:

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

Output:

(e**m*( - 4*x**m*sqrt(c + d*x)*a*d**3*m - 6*x**m*sqrt(c + d*x)*a*d**3 + 12 
*x**m*sqrt(c + d*x)*b*c**2*d*m + 18*x**m*sqrt(c + d*x)*b*c**2*d + 4*x**m*s 
qrt(c + d*x)*b*c*d**2*m*x + 8*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3*a 
*c + 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3*b* 
d*x**3),x)*a*b*d**4*m**3 + 28*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3*a 
*c + 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3*b* 
d*x**3),x)*a*b*d**4*m**2 + 30*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3*a 
*c + 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3*b* 
d*x**3),x)*a*b*d**4*m + 9*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3*a*c + 
 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3*b*d*x* 
*3),x)*a*b*d**4 - 8*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3*a*c + 2*a*d 
*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3*b*d*x**3),x) 
*b**2*c**2*d**2*m**3 - 32*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3*a*c + 
 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3*b*d*x* 
*3),x)*b**2*c**2*d**2*m**2 - 48*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m + 3 
*a*c + 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 3* 
b*d*x**3),x)*b**2*c**2*d**2*m - 27*int((x**m*sqrt(c + d*x)*x**2)/(2*a*c*m 
+ 3*a*c + 2*a*d*m*x + 3*a*d*x + 2*b*c*m*x**2 + 3*b*c*x**2 + 2*b*d*m*x**3 + 
 3*b*d*x**3),x)*b**2*c**2*d**2 + 8*int((x**m*sqrt(c + d*x))/(2*a*c*m*x + 3 
*a*c*x + 2*a*d*m*x**2 + 3*a*d*x**2 + 2*b*c*m*x**3 + 3*b*c*x**3 + 2*b*d*...