\(\int (e x)^m (c+d x) (a+b x^2)^{3/2} \, dx\) [163]

Optimal result

Integrand size = 22, antiderivative size = 141 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^{3/2} \, dx=\frac {a c (e x)^{1+m} \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e (1+m) \sqrt {1+\frac {b x^2}{a}}}+\frac {a d (e x)^{2+m} \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{e^2 (2+m) \sqrt {1+\frac {b x^2}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^{3/2} \, dx=\frac {a x (e x)^m \sqrt {a+b x^2} \left (c (2+m) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+d (1+m) x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )\right )}{(1+m) (2+m) \sqrt {1+\frac {b x^2}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {557, 279, 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.

\(\displaystyle \int \left (a+b x^2\right )^{3/2} (c+d x) (e x)^m \, dx\)

\(\Big \downarrow \) 557

\(\displaystyle c \int (e x)^m \left (b x^2+a\right )^{3/2}dx+\frac {d \int (e x)^{m+1} \left (b x^2+a\right )^{3/2}dx}{e}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {a c \sqrt {a+b x^2} \int (e x)^m \left (\frac {b x^2}{a}+1\right )^{3/2}dx}{\sqrt {\frac {b x^2}{a}+1}}+\frac {a d \sqrt {a+b x^2} \int (e x)^{m+1} \left (\frac {b x^2}{a}+1\right )^{3/2}dx}{e \sqrt {\frac {b x^2}{a}+1}}\)

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], 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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^m*(d*x+c)*(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.55 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.65 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} c e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{\frac {3}{2}} d e^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {\sqrt {a} b c e^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {\sqrt {a} b d e^{m} x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \] Input:

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

Output:

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

Maxima [F]

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

integrate((e*x)^m*(d*x+c)*(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(d*x + c)*(e*x)^m, x)
 

Giac [F]

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

integrate((e*x)^m*(d*x+c)*(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(d*x + c)*(e*x)^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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