\(\int (e x)^m (c+d x)^{5/2} (a+b x^2) \, dx\) [130]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int (e x)^m (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=-\frac {4 b c (2+m) (e x)^{1+m} (c+d x)^{7/2}}{d^2 e (9+2 m) (11+2 m)}+\frac {2 b (e x)^{2+m} (c+d x)^{7/2}}{d e^2 (11+2 m)}+\frac {2 \left (4 b c^2 (1+m) (2+m)+a d^2 (9+2 m) (11+2 m)\right ) \left (-\frac {d x}{c}\right )^{-m} (e x)^m (c+d x)^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m,\frac {9}{2},1+\frac {d x}{c}\right )}{7 d^3 (9+2 m) (11+2 m)} \] Output:

-4*b*c*(2+m)*(e*x)^(1+m)*(d*x+c)^(7/2)/d^2/e/(9+2*m)/(11+2*m)+2*b*(e*x)^(2 
+m)*(d*x+c)^(7/2)/d/e^2/(11+2*m)+2/7*(4*b*c^2*(1+m)*(2+m)+a*d^2*(9+2*m)*(1 
1+2*m))*(e*x)^m*(d*x+c)^(7/2)*hypergeom([7/2, -m],[9/2],1+d*x/c)/d^3/(9+2* 
m)/(11+2*m)/((-d*x/c)^m)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.71 \[ \int (e x)^m (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \left (-\frac {d x}{c}\right )^{-m} (e x)^m (c+d x)^{7/2} \left (99 \left (b c^2+a d^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m,\frac {9}{2},1+\frac {d x}{c}\right )-7 b (c+d x) \left (22 c \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-m,\frac {11}{2},1+\frac {d x}{c}\right )-9 (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {11}{2},-m,\frac {13}{2},1+\frac {d x}{c}\right )\right )\right )}{693 d^3} \] Input:

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

Output:

(2*(e*x)^m*(c + d*x)^(7/2)*(99*(b*c^2 + a*d^2)*Hypergeometric2F1[7/2, -m, 
9/2, 1 + (d*x)/c] - 7*b*(c + d*x)*(22*c*Hypergeometric2F1[9/2, -m, 11/2, 1 
 + (d*x)/c] - 9*(c + d*x)*Hypergeometric2F1[11/2, -m, 13/2, 1 + (d*x)/c])) 
)/(693*d^3*(-((d*x)/c))^m)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {521, 27, 90, 77, 75}

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.

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

\(\Big \downarrow \) 521

\(\displaystyle \frac {2 \int \frac {1}{2} e^2 (e x)^m (c+d x)^{5/2} (a d (2 m+11)-2 b c (m+2) x)dx}{d e^2 (2 m+11)}+\frac {2 b (c+d x)^{7/2} (e x)^{m+2}}{d e^2 (2 m+11)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^m (c+d x)^{5/2} (a d (2 m+11)-2 b c (m+2) x)dx}{d (2 m+11)}+\frac {2 b (c+d x)^{7/2} (e x)^{m+2}}{d e^2 (2 m+11)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\left (a d^2 (2 m+9) (2 m+11)+4 b c^2 (m+1) (m+2)\right ) \int (e x)^m (c+d x)^{5/2}dx}{d (2 m+9)}-\frac {4 b c (m+2) (c+d x)^{7/2} (e x)^{m+1}}{d e (2 m+9)}}{d (2 m+11)}+\frac {2 b (c+d x)^{7/2} (e x)^{m+2}}{d e^2 (2 m+11)}\)

\(\Big \downarrow \) 77

\(\displaystyle \frac {\frac {(e x)^m \left (-\frac {d x}{c}\right )^{-m} \left (a d^2 (2 m+9) (2 m+11)+4 b c^2 (m+1) (m+2)\right ) \int \left (-\frac {d x}{c}\right )^m (c+d x)^{5/2}dx}{d (2 m+9)}-\frac {4 b c (m+2) (c+d x)^{7/2} (e x)^{m+1}}{d e (2 m+9)}}{d (2 m+11)}+\frac {2 b (c+d x)^{7/2} (e x)^{m+2}}{d e^2 (2 m+11)}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {\frac {2 (c+d x)^{7/2} (e x)^m \left (-\frac {d x}{c}\right )^{-m} \left (a d^2 (2 m+9) (2 m+11)+4 b c^2 (m+1) (m+2)\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m,\frac {9}{2},\frac {d x}{c}+1\right )}{7 d^2 (2 m+9)}-\frac {4 b c (m+2) (c+d x)^{7/2} (e x)^{m+1}}{d e (2 m+9)}}{d (2 m+11)}+\frac {2 b (c+d x)^{7/2} (e x)^{m+2}}{d e^2 (2 m+11)}\)

Input:

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

Output:

(2*b*(e*x)^(2 + m)*(c + d*x)^(7/2))/(d*e^2*(11 + 2*m)) + ((-4*b*c*(2 + m)* 
(e*x)^(1 + m)*(c + d*x)^(7/2))/(d*e*(9 + 2*m)) + (2*(4*b*c^2*(1 + m)*(2 + 
m) + a*d^2*(9 + 2*m)*(11 + 2*m))*(e*x)^m*(c + d*x)^(7/2)*Hypergeometric2F1 
[7/2, -m, 9/2, 1 + (d*x)/c])/(7*d^2*(9 + 2*m)*(-((d*x)/c))^m))/(d*(11 + 2* 
m))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 77

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-b)*(c/ 
d))^IntPart[m]*((b*x)^FracPart[m]/((-d)*(x/c))^FracPart[m])   Int[((-d)*(x/ 
c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && 
 !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 521

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> Simp[b^p*(e*x)^(m + 2*p)*((c + d*x)^(n + 1)/(d*e^(2*p)*(m + n 
 + 2*p + 1))), x] + Simp[1/(d*e^(2*p)*(m + n + 2*p + 1))   Int[(e*x)^m*(c + 
 d*x)^n*ExpandToSum[d*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x^2)^p - b^p*(e*x)^ 
(2*p)) - b^p*(e*c)*(m + 2*p)*(e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] &&  !IntegerQ[m] &&  !I 
ntegerQ[n]
 

Maple [F]

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

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]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.07 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.65 \[ \int (e x)^m (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {a c^{\frac {5}{2}} e^{m} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 a c^{\frac {3}{2}} d e^{m} x^{m + 2} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac {a \sqrt {c} d^{2} e^{m} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} + \frac {b c^{\frac {5}{2}} e^{m} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} + \frac {2 b c^{\frac {3}{2}} d e^{m} x^{m + 4} \Gamma \left (m + 4\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 4 \\ m + 5 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 5\right )} + \frac {b \sqrt {c} d^{2} e^{m} x^{m + 5} \Gamma \left (m + 5\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 5 \\ m + 6 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 6\right )} \] Input:

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

Output:

a*c**(5/2)*e**m*x**(m + 1)*gamma(m + 1)*hyper((-1/2, m + 1), (m + 2,), d*x 
*exp_polar(I*pi)/c)/gamma(m + 2) + 2*a*c**(3/2)*d*e**m*x**(m + 2)*gamma(m 
+ 2)*hyper((-1/2, m + 2), (m + 3,), d*x*exp_polar(I*pi)/c)/gamma(m + 3) + 
a*sqrt(c)*d**2*e**m*x**(m + 3)*gamma(m + 3)*hyper((-1/2, m + 3), (m + 4,), 
 d*x*exp_polar(I*pi)/c)/gamma(m + 4) + b*c**(5/2)*e**m*x**(m + 3)*gamma(m 
+ 3)*hyper((-1/2, m + 3), (m + 4,), d*x*exp_polar(I*pi)/c)/gamma(m + 4) + 
2*b*c**(3/2)*d*e**m*x**(m + 4)*gamma(m + 4)*hyper((-1/2, m + 4), (m + 5,), 
 d*x*exp_polar(I*pi)/c)/gamma(m + 5) + b*sqrt(c)*d**2*e**m*x**(m + 5)*gamm 
a(m + 5)*hyper((-1/2, m + 5), (m + 6,), d*x*exp_polar(I*pi)/c)/gamma(m + 6 
)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*e**m*(60*x**m*sqrt(c + d*x)*a*c**3*d**2*m**2 + 600*x**m*sqrt(c + d*x)*a 
*c**3*d**2*m + 1485*x**m*sqrt(c + d*x)*a*c**3*d**2 + 32*x**m*sqrt(c + d*x) 
*a*c**2*d**3*m**5*x + 544*x**m*sqrt(c + d*x)*a*c**2*d**3*m**4*x + 3496*x** 
m*sqrt(c + d*x)*a*c**2*d**3*m**3*x + 10364*x**m*sqrt(c + d*x)*a*c**2*d**3* 
m**2*x + 13284*x**m*sqrt(c + d*x)*a*c**2*d**3*m*x + 4455*x**m*sqrt(c + d*x 
)*a*c**2*d**3*x + 64*x**m*sqrt(c + d*x)*a*c*d**4*m**5*x**2 + 1008*x**m*sqr 
t(c + d*x)*a*c*d**4*m**4*x**2 + 5792*x**m*sqrt(c + d*x)*a*c*d**4*m**3*x**2 
 + 14568*x**m*sqrt(c + d*x)*a*c*d**4*m**2*x**2 + 14868*x**m*sqrt(c + d*x)* 
a*c*d**4*m*x**2 + 4455*x**m*sqrt(c + d*x)*a*c*d**4*x**2 + 32*x**m*sqrt(c + 
 d*x)*a*d**5*m**5*x**3 + 464*x**m*sqrt(c + d*x)*a*d**5*m**4*x**3 + 2416*x* 
*m*sqrt(c + d*x)*a*d**5*m**3*x**3 + 5464*x**m*sqrt(c + d*x)*a*d**5*m**2*x* 
*3 + 5154*x**m*sqrt(c + d*x)*a*d**5*m*x**3 + 1485*x**m*sqrt(c + d*x)*a*d** 
5*x**3 + 60*x**m*sqrt(c + d*x)*b*c**5*m**2 + 180*x**m*sqrt(c + d*x)*b*c**5 
*m + 120*x**m*sqrt(c + d*x)*b*c**5 - 60*x**m*sqrt(c + d*x)*b*c**4*d*m**2*x 
 - 150*x**m*sqrt(c + d*x)*b*c**4*d*m*x - 60*x**m*sqrt(c + d*x)*b*c**4*d*x 
+ 60*x**m*sqrt(c + d*x)*b*c**3*d**2*m**2*x**2 + 120*x**m*sqrt(c + d*x)*b*c 
**3*d**2*m*x**2 + 45*x**m*sqrt(c + d*x)*b*c**3*d**2*x**2 + 32*x**m*sqrt(c 
+ d*x)*b*c**2*d**3*m**5*x**3 + 480*x**m*sqrt(c + d*x)*b*c**2*d**3*m**4*x** 
3 + 2600*x**m*sqrt(c + d*x)*b*c**2*d**3*m**3*x**3 + 6060*x**m*sqrt(c + d*x 
)*b*c**2*d**3*m**2*x**3 + 5828*x**m*sqrt(c + d*x)*b*c**2*d**3*m*x**3 + ...