\(\int \frac {(e x)^m (a+b x^4)^2}{\sqrt {c+d x^4}} \, dx\) [279]

Optimal result

Integrand size = 26, antiderivative size = 188 \[ \int \frac {(e x)^m \left (a+b x^4\right )^2}{\sqrt {c+d x^4}} \, dx=-\frac {b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m) (7+m)}+\frac {b^2 (e x)^{5+m} \sqrt {c+d x^4}}{d e^5 (7+m)}+\frac {\left (\frac {a^2}{1+m}+\frac {b c (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) (e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{4},\frac {5+m}{4},-\frac {d x^4}{c}\right )}{e \sqrt {c+d x^4}} \] Output:

-b*(b*c*(5+m)-2*a*d*(7+m))*(e*x)^(1+m)*(d*x^4+c)^(1/2)/d^2/e/(3+m)/(7+m)+b 
^2*(e*x)^(5+m)*(d*x^4+c)^(1/2)/d/e^5/(7+m)+(a^2/(1+m)+b*c*(b*c*(5+m)-2*a*d 
*(7+m))/d^2/(3+m)/(7+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 = 9.44 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.87 \[ \int \frac {(e x)^m \left (a+b x^4\right )^2}{\sqrt {c+d x^4}} \, dx=\frac {x (e x)^m \sqrt {1+\frac {d x^4}{c}} \left (a^2 \left (45+14 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{4},\frac {5+m}{4},-\frac {d x^4}{c}\right )+b (1+m) x^4 \left (2 a (9+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{4},\frac {9+m}{4},-\frac {d x^4}{c}\right )+b (5+m) x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9+m}{4},\frac {13+m}{4},-\frac {d x^4}{c}\right )\right )\right )}{(1+m) (5+m) (9+m) \sqrt {c+d x^4}} \] Input:

Integrate[((e*x)^m*(a + b*x^4)^2)/Sqrt[c + d*x^4],x]
 

Output:

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a^2*(45 + 14*m + m^2)*Hypergeometric2F1[1/ 
2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*(2*a*(9 + m)*Hyperg 
eometric2F1[1/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)] + b*(5 + m)*x^4*Hyper 
geometric2F1[1/2, (9 + m)/4, (13 + m)/4, -((d*x^4)/c)])))/((1 + m)*(5 + m) 
*(9 + m)*Sqrt[c + d*x^4])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {964, 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.

\(\displaystyle \int \frac {\left (a+b x^4\right )^2 (e x)^m}{\sqrt {c+d x^4}} \, dx\)

\(\Big \downarrow \) 964

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {\left (a^2 d (m+7)+\frac {b c (m+1) (b c (m+5)-2 a d (m+7))}{d (m+3)}\right ) \int \frac {(e x)^m}{\sqrt {d x^4+c}}dx-\frac {b \sqrt {c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d e (m+3)}}{d (m+7)}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+5}}{d e^5 (m+7)}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {\frac {\sqrt {\frac {d x^4}{c}+1} \left (a^2 d (m+7)+\frac {b c (m+1) (b c (m+5)-2 a d (m+7))}{d (m+3)}\right ) \int \frac {(e x)^m}{\sqrt {\frac {d x^4}{c}+1}}dx}{\sqrt {c+d x^4}}-\frac {b \sqrt {c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d e (m+3)}}{d (m+7)}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+5}}{d e^5 (m+7)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {\frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (a^2 d (m+7)+\frac {b c (m+1) (b c (m+5)-2 a d (m+7))}{d (m+3)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{4},\frac {m+5}{4},-\frac {d x^4}{c}\right )}{e (m+1) \sqrt {c+d x^4}}-\frac {b \sqrt {c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d e (m+3)}}{d (m+7)}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+5}}{d e^5 (m+7)}\)

Input:

Int[((e*x)^m*(a + b*x^4)^2)/Sqrt[c + d*x^4],x]
 

Output:

(b^2*(e*x)^(5 + m)*Sqrt[c + d*x^4])/(d*e^5*(7 + m)) + (-((b*(b*c*(5 + m) - 
 2*a*d*(7 + m))*(e*x)^(1 + m)*Sqrt[c + d*x^4])/(d*e*(3 + m))) + ((a^2*d*(7 
 + m) + (b*c*(1 + m)*(b*c*(5 + m) - 2*a*d*(7 + 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]))/(d*(7 + m))
 

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]
 

rule 964

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

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (b \,x^{4}+a \right )^{2}}{\sqrt {d \,x^{4}+c}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*x^8 + 2*a*b*x^4 + a^2)*(e*x)^m/sqrt(d*x^4 + c), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.47 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.97 \[ \int \frac {(e x)^m \left (a+b x^4\right )^2}{\sqrt {c+d x^4}} \, dx=\frac {a^{2} e^{m} x^{m + 1} \Gamma \left (\frac {m}{4} + \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {1}{4} \\ \frac {m}{4} + \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right )} + \frac {a b e^{m} x^{m + 5} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {5}{4} \\ \frac {m}{4} + \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {9}{4}\right )} + \frac {b^{2} e^{m} x^{m + 9} \Gamma \left (\frac {m}{4} + \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {9}{4} \\ \frac {m}{4} + \frac {13}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {13}{4}\right )} \] Input:

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

Output:

a**2*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)) + a*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_pol 
ar(I*pi)/c)/(2*sqrt(c)*gamma(m/4 + 9/4)) + b**2*e**m*x**(m + 9)*gamma(m/4 
+ 9/4)*hyper((1/2, m/4 + 9/4), (m/4 + 13/4,), d*x**4*exp_polar(I*pi)/c)/(4 
*sqrt(c)*gamma(m/4 + 13/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m \left (a+b x^4\right )^2}{\sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

(e**m*(2*x**m*sqrt(c + d*x**4)*a*b*d*m*x + 14*x**m*sqrt(c + d*x**4)*a*b*d* 
x - x**m*sqrt(c + d*x**4)*b**2*c*m*x - 5*x**m*sqrt(c + d*x**4)*b**2*c*x + 
x**m*sqrt(c + d*x**4)*b**2*d*m*x**5 + 3*x**m*sqrt(c + d*x**4)*b**2*d*x**5 
+ int((x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 21*c + d*m**2*x**4 + 10*d 
*m*x**4 + 21*d*x**4),x)*a**2*d**2*m**4 + 20*int((x**m*sqrt(c + d*x**4))/(c 
*m**2 + 10*c*m + 21*c + d*m**2*x**4 + 10*d*m*x**4 + 21*d*x**4),x)*a**2*d** 
2*m**3 + 142*int((x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 21*c + d*m**2* 
x**4 + 10*d*m*x**4 + 21*d*x**4),x)*a**2*d**2*m**2 + 420*int((x**m*sqrt(c + 
 d*x**4))/(c*m**2 + 10*c*m + 21*c + d*m**2*x**4 + 10*d*m*x**4 + 21*d*x**4) 
,x)*a**2*d**2*m + 441*int((x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 21*c 
+ d*m**2*x**4 + 10*d*m*x**4 + 21*d*x**4),x)*a**2*d**2 - 2*int((x**m*sqrt(c 
 + d*x**4))/(c*m**2 + 10*c*m + 21*c + d*m**2*x**4 + 10*d*m*x**4 + 21*d*x** 
4),x)*a*b*c*d*m**4 - 36*int((x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 21* 
c + d*m**2*x**4 + 10*d*m*x**4 + 21*d*x**4),x)*a*b*c*d*m**3 - 216*int((x**m 
*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 21*c + d*m**2*x**4 + 10*d*m*x**4 + 2 
1*d*x**4),x)*a*b*c*d*m**2 - 476*int((x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c 
*m + 21*c + d*m**2*x**4 + 10*d*m*x**4 + 21*d*x**4),x)*a*b*c*d*m - 294*int( 
(x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 21*c + d*m**2*x**4 + 10*d*m*x** 
4 + 21*d*x**4),x)*a*b*c*d + int((x**m*sqrt(c + d*x**4))/(c*m**2 + 10*c*m + 
 21*c + d*m**2*x**4 + 10*d*m*x**4 + 21*d*x**4),x)*b**2*c**2*m**4 + 16*i...