\(\int \frac {(e x)^m (a+b x^4)}{\sqrt {c+d x^4}} \, dx\) [280]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {(e x)^m \left (a+b x^4\right )}{\sqrt {c+d x^4}} \, dx=\frac {b (e x)^{1+m} \sqrt {c+d x^4}}{d e (3+m)}+\frac {\left (\frac {a}{1+m}-\frac {b c}{d (3+m)}\right ) (e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{4},\frac {5+m}{4},-\frac {d x^4}{c}\right )}{e \sqrt {c+d x^4}} \] Output:

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

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(e x)^m \left (a+b x^4\right )}{\sqrt {c+d x^4}} \, dx=\frac {x (e x)^m \sqrt {1+\frac {d x^4}{c}} \left (a (5+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{4},\frac {5+m}{4},-\frac {d x^4}{c}\right )+b (1+m) x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{4},\frac {9+m}{4},-\frac {d x^4}{c}\right )\right )}{(1+m) (5+m) \sqrt {c+d x^4}} \] Input:

Integrate[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]
 

Output:

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a*(5 + m)*Hypergeometric2F1[1/2, (1 + m)/4 
, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*Hypergeometric2F1[1/2, (5 + m)/ 
4, (9 + m)/4, -((d*x^4)/c)]))/((1 + m)*(5 + m)*Sqrt[c + d*x^4])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {959, 889, 888}

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.

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

\(\Big \downarrow \) 959

\(\displaystyle \left (a-\frac {b c (m+1)}{d (m+3)}\right ) \int \frac {(e x)^m}{\sqrt {d x^4+c}}dx+\frac {b \sqrt {c+d x^4} (e x)^{m+1}}{d e (m+3)}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {\sqrt {\frac {d x^4}{c}+1} \left (a-\frac {b c (m+1)}{d (m+3)}\right ) \int \frac {(e x)^m}{\sqrt {\frac {d x^4}{c}+1}}dx}{\sqrt {c+d x^4}}+\frac {b \sqrt {c+d x^4} (e x)^{m+1}}{d e (m+3)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (a-\frac {b c (m+1)}{d (m+3)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{4},\frac {m+5}{4},-\frac {d x^4}{c}\right )}{e (m+1) \sqrt {c+d x^4}}+\frac {b \sqrt {c+d x^4} (e x)^{m+1}}{d e (m+3)}\)

Input:

Int[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]
 

Output:

(b*(e*x)^(1 + m)*Sqrt[c + d*x^4])/(d*e*(3 + m)) + ((a - (b*c*(1 + m))/(d*( 
3 + m)))*(e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[1/2, (1 + m)/ 
4, (5 + m)/4, -((d*x^4)/c)])/(e*(1 + m)*Sqrt[c + d*x^4])
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 959

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

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (b \,x^{4}+a \right )}{\sqrt {d \,x^{4}+c}}d x\]

Input:

int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x)
 

Output:

int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x)
 

Fricas [F]

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

integrate((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {(e x)^m \left (a+b x^4\right )}{\sqrt {c+d x^4}} \, dx=\frac {a e^{m} x^{m + 1} \Gamma \left (\frac {m}{4} + \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {1}{4} \\ \frac {m}{4} + \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right )} + \frac {b e^{m} x^{m + 5} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {5}{4} \\ \frac {m}{4} + \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {9}{4}\right )} \] Input:

integrate((e*x)**m*(b*x**4+a)/(d*x**4+c)**(1/2),x)
 

Output:

a*e**m*x**(m + 1)*gamma(m/4 + 1/4)*hyper((1/2, m/4 + 1/4), (m/4 + 5/4,), d 
*x**4*exp_polar(I*pi)/c)/(4*sqrt(c)*gamma(m/4 + 5/4)) + b*e**m*x**(m + 5)* 
gamma(m/4 + 5/4)*hyper((1/2, m/4 + 5/4), (m/4 + 9/4,), d*x**4*exp_polar(I* 
pi)/c)/(4*sqrt(c)*gamma(m/4 + 9/4))
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c), x)
 

Giac [F]

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

integrate((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c), x)
 

Mupad [F(-1)]

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

int(((e*x)^m*(a + b*x^4))/(c + d*x^4)^(1/2),x)
 

Output:

int(((e*x)^m*(a + b*x^4))/(c + d*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (a+b x^4\right )}{\sqrt {c+d x^4}} \, dx=\frac {e^{m} \left (x^{m} \sqrt {d \,x^{4}+c}\, b x +\left (\int \frac {x^{m} \sqrt {d \,x^{4}+c}}{d m \,x^{4}+3 d \,x^{4}+c m +3 c}d x \right ) a d \,m^{2}+6 \left (\int \frac {x^{m} \sqrt {d \,x^{4}+c}}{d m \,x^{4}+3 d \,x^{4}+c m +3 c}d x \right ) a d m +9 \left (\int \frac {x^{m} \sqrt {d \,x^{4}+c}}{d m \,x^{4}+3 d \,x^{4}+c m +3 c}d x \right ) a d -\left (\int \frac {x^{m} \sqrt {d \,x^{4}+c}}{d m \,x^{4}+3 d \,x^{4}+c m +3 c}d x \right ) b c \,m^{2}-4 \left (\int \frac {x^{m} \sqrt {d \,x^{4}+c}}{d m \,x^{4}+3 d \,x^{4}+c m +3 c}d x \right ) b c m -3 \left (\int \frac {x^{m} \sqrt {d \,x^{4}+c}}{d m \,x^{4}+3 d \,x^{4}+c m +3 c}d x \right ) b c \right )}{d \left (m +3\right )} \] Input:

int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x)
 

Output:

(e**m*(x**m*sqrt(c + d*x**4)*b*x + int((x**m*sqrt(c + d*x**4))/(c*m + 3*c 
+ d*m*x**4 + 3*d*x**4),x)*a*d*m**2 + 6*int((x**m*sqrt(c + d*x**4))/(c*m + 
3*c + d*m*x**4 + 3*d*x**4),x)*a*d*m + 9*int((x**m*sqrt(c + d*x**4))/(c*m + 
 3*c + d*m*x**4 + 3*d*x**4),x)*a*d - int((x**m*sqrt(c + d*x**4))/(c*m + 3* 
c + d*m*x**4 + 3*d*x**4),x)*b*c*m**2 - 4*int((x**m*sqrt(c + d*x**4))/(c*m 
+ 3*c + d*m*x**4 + 3*d*x**4),x)*b*c*m - 3*int((x**m*sqrt(c + d*x**4))/(c*m 
 + 3*c + d*m*x**4 + 3*d*x**4),x)*b*c))/(d*(m + 3))