\(\int \sqrt {a+b x^4} (c+d x^4)^2 \, dx\) [28]

Optimal result

Integrand size = 21, antiderivative size = 203 \[ \int \sqrt {a+b x^4} \left (c+d x^4\right )^2 \, dx=\frac {1}{231} \left (77 c^2-\frac {a d (22 b c-5 a d)}{b^2}\right ) x \sqrt {a+b x^4}+\frac {d (22 b c-5 a d) x \left (a+b x^4\right )^{3/2}}{77 b^2}+\frac {d^2 x^5 \left (a+b x^4\right )^{3/2}}{11 b}+\frac {a^{3/4} \left (77 b^2 c^2-22 a b c d+5 a^2 d^2\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^4}} \] Output:

1/231*(77*c^2-a*d*(-5*a*d+22*b*c)/b^2)*x*(b*x^4+a)^(1/2)+1/77*d*(-5*a*d+22 
*b*c)*x*(b*x^4+a)^(3/2)/b^2+1/11*d^2*x^5*(b*x^4+a)^(3/2)/b+1/231*a^(3/4)*( 
5*a^2*d^2-22*a*b*c*d+77*b^2*c^2)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2) 
+b^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^( 
1/2))/b^(9/4)/(b*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 13.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.79 \[ \int \sqrt {a+b x^4} \left (c+d x^4\right )^2 \, dx=\frac {x \sqrt {a+b x^4} \left (13 a \left (45 c^2+18 c d x^4+5 d^2 x^8\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {13}{4},-\frac {b x^4}{a}\right )+4 b x^4 \left (7 c^2+10 c d x^4+3 d^2 x^8\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {17}{4},-\frac {b x^4}{a}\right )+8 b x^4 \left (c+d x^4\right )^2 \, _3F_2\left (\frac {1}{2},\frac {5}{4},2;1,\frac {17}{4};-\frac {b x^4}{a}\right )\right )}{585 a \sqrt {1+\frac {b x^4}{a}}} \] Input:

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

Output:

(x*Sqrt[a + b*x^4]*(13*a*(45*c^2 + 18*c*d*x^4 + 5*d^2*x^8)*Hypergeometric2 
F1[-1/2, 1/4, 13/4, -((b*x^4)/a)] + 4*b*x^4*(7*c^2 + 10*c*d*x^4 + 3*d^2*x^ 
8)*Hypergeometric2F1[1/2, 5/4, 17/4, -((b*x^4)/a)] + 8*b*x^4*(c + d*x^4)^2 
*HypergeometricPFQ[{1/2, 5/4, 2}, {1, 17/4}, -((b*x^4)/a)]))/(585*a*Sqrt[1 
 + (b*x^4)/a])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {933, 913, 748, 761}

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.

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

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int \sqrt {b x^4+a} \left (5 d (3 b c-a d) x^4+c (11 b c-a d)\right )dx}{11 b}+\frac {d x \left (a+b x^4\right )^{3/2} \left (c+d x^4\right )}{11 b}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {\frac {\left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) \int \sqrt {b x^4+a}dx}{7 b}+\frac {5 d x \left (a+b x^4\right )^{3/2} (3 b c-a d)}{7 b}}{11 b}+\frac {d x \left (a+b x^4\right )^{3/2} \left (c+d x^4\right )}{11 b}\)

\(\Big \downarrow \) 748

\(\displaystyle \frac {\frac {\left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) \left (\frac {2}{3} a \int \frac {1}{\sqrt {b x^4+a}}dx+\frac {1}{3} x \sqrt {a+b x^4}\right )}{7 b}+\frac {5 d x \left (a+b x^4\right )^{3/2} (3 b c-a d)}{7 b}}{11 b}+\frac {d x \left (a+b x^4\right )^{3/2} \left (c+d x^4\right )}{11 b}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {\left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) \left (\frac {a^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {1}{3} x \sqrt {a+b x^4}\right )}{7 b}+\frac {5 d x \left (a+b x^4\right )^{3/2} (3 b c-a d)}{7 b}}{11 b}+\frac {d x \left (a+b x^4\right )^{3/2} \left (c+d x^4\right )}{11 b}\)

Input:

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

Output:

(d*x*(a + b*x^4)^(3/2)*(c + d*x^4))/(11*b) + ((5*d*(3*b*c - a*d)*x*(a + b* 
x^4)^(3/2))/(7*b) + ((77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*((x*Sqrt[a + b* 
x^4])/3 + (a^(3/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqr 
t[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(3*b^(1/4)*Sqr 
t[a + b*x^4])))/(7*b))/(11*b)
 

Defintions of rubi rules used

rule 748

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p 
+ 1)), x] + Simp[a*n*(p/(n*p + 1))   Int[(a + b*x^n)^(p - 1), x], x] /; Fre 
eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat 
or[p + 1/n], Denominator[p]])
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 913

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

rule 933

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.79 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {x \left (-21 b^{2} d^{2} x^{8}-6 a b \,d^{2} x^{4}-66 b^{2} c d \,x^{4}+10 a^{2} d^{2}-44 a b c d -77 b^{2} c^{2}\right ) \sqrt {b \,x^{4}+a}}{231 b^{2}}+\frac {2 a \left (5 a^{2} d^{2}-22 a b c d +77 b^{2} c^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{231 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(168\)
elliptic	\(\frac {d^{2} x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {\left (\frac {2}{11} a \,d^{2}+2 b c d \right ) x^{5} \sqrt {b \,x^{4}+a}}{7 b}+\frac {\left (2 a c d +b \,c^{2}-\frac {5 a \left (\frac {2}{11} a \,d^{2}+2 b c d \right )}{7 b}\right ) x \sqrt {b \,x^{4}+a}}{3 b}+\frac {\left (a \,c^{2}-\frac {a \left (2 a c d +b \,c^{2}-\frac {5 a \left (\frac {2}{11} a \,d^{2}+2 b c d \right )}{7 b}\right )}{3 b}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(202\)
default	\(c^{2} \left (\frac {x \sqrt {b \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d^{2} \left (\frac {x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {2 a \,x^{5} \sqrt {b \,x^{4}+a}}{77 b}-\frac {10 a^{2} x \sqrt {b \,x^{4}+a}}{231 b^{2}}+\frac {10 a^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{231 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+2 c d \left (\frac {x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {2 a x \sqrt {b \,x^{4}+a}}{21 b}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(332\)

Input:

int((b*x^4+a)^(1/2)*(d*x^4+c)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/231*x*(-21*b^2*d^2*x^8-6*a*b*d^2*x^4-66*b^2*c*d*x^4+10*a^2*d^2-44*a*b*c 
*d-77*b^2*c^2)/b^2*(b*x^4+a)^(1/2)+2/231*a*(5*a^2*d^2-22*a*b*c*d+77*b^2*c^ 
2)/b^2/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I*b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1 
/2)*x^2/a^(1/2))^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/ 
2),I)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.61 \[ \int \sqrt {a+b x^4} \left (c+d x^4\right )^2 \, dx=\frac {2 \, {\left (77 \, b^{2} c^{2} - 22 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (21 \, b^{2} d^{2} x^{9} + 6 \, {\left (11 \, b^{2} c d + a b d^{2}\right )} x^{5} + {\left (77 \, b^{2} c^{2} + 44 \, a b c d - 10 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x^{4} + a}}{231 \, b^{2}} \] Input:

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

Output:

1/231*(2*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*sqrt(b)*(-a/b)^(3/4)*ellipt 
ic_f(arcsin((-a/b)^(1/4)/x), -1) + (21*b^2*d^2*x^9 + 6*(11*b^2*c*d + a*b*d 
^2)*x^5 + (77*b^2*c^2 + 44*a*b*c*d - 10*a^2*d^2)*x)*sqrt(b*x^4 + a))/b^2
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.51 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.64 \[ \int \sqrt {a+b x^4} \left (c+d x^4\right )^2 \, dx=\frac {\sqrt {a} c^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} c d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {\sqrt {a} d^{2} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \] Input:

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

Output:

sqrt(a)*c**2*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**4*exp_polar(I*pi 
)/a)/(4*gamma(5/4)) + sqrt(a)*c*d*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4, 
), b*x**4*exp_polar(I*pi)/a)/(2*gamma(9/4)) + sqrt(a)*d**2*x**9*gamma(9/4) 
*hyper((-1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(13/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b x^4} \left (c+d x^4\right )^2 \, dx=\frac {-10 \sqrt {b \,x^{4}+a}\, a^{2} d^{2} x +44 \sqrt {b \,x^{4}+a}\, a b c d x +6 \sqrt {b \,x^{4}+a}\, a b \,d^{2} x^{5}+77 \sqrt {b \,x^{4}+a}\, b^{2} c^{2} x +66 \sqrt {b \,x^{4}+a}\, b^{2} c d \,x^{5}+21 \sqrt {b \,x^{4}+a}\, b^{2} d^{2} x^{9}+10 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a^{3} d^{2}-44 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a^{2} b c d +154 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a \,b^{2} c^{2}}{231 b^{2}} \] Input:

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

Output:

( - 10*sqrt(a + b*x**4)*a**2*d**2*x + 44*sqrt(a + b*x**4)*a*b*c*d*x + 6*sq 
rt(a + b*x**4)*a*b*d**2*x**5 + 77*sqrt(a + b*x**4)*b**2*c**2*x + 66*sqrt(a 
 + b*x**4)*b**2*c*d*x**5 + 21*sqrt(a + b*x**4)*b**2*d**2*x**9 + 10*int(sqr 
t(a + b*x**4)/(a + b*x**4),x)*a**3*d**2 - 44*int(sqrt(a + b*x**4)/(a + b*x 
**4),x)*a**2*b*c*d + 154*int(sqrt(a + b*x**4)/(a + b*x**4),x)*a*b**2*c**2) 
/(231*b**2)