\(\int (a+b x^4)^{5/4} (c+d x^4) \, dx\) [85]

Optimal result

Integrand size = 19, antiderivative size = 150 \[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\frac {a (10 b c-a d) x \sqrt [4]{a+b x^4}}{24 b}+\frac {(10 b c-a d) x \left (a+b x^4\right )^{5/4}}{60 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}-\frac {a^{3/2} (10 b c-a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{24 \sqrt {b} \left (a+b x^4\right )^{3/4}} \] Output:

1/24*a*(-a*d+10*b*c)*x*(b*x^4+a)^(1/4)/b+1/60*(-a*d+10*b*c)*x*(b*x^4+a)^(5 
/4)/b+1/10*d*x*(b*x^4+a)^(9/4)/b-1/24*a^(3/2)*(-a*d+10*b*c)*(1+a/b/x^4)^(3 
/4)*x^3*InverseJacobiAM(1/2*arccot(b^(1/2)*x^2/a^(1/2)),2^(1/2))/b^(1/2)/( 
b*x^4+a)^(3/4)
 

Mathematica [C] (verified)

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

Time = 10.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.50 \[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\frac {x \sqrt [4]{a+b x^4} \left (d \left (a+b x^4\right )^2-\frac {a (-10 b c+a d) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt [4]{1+\frac {b x^4}{a}}}\right )}{10 b} \] Input:

Integrate[(a + b*x^4)^(5/4)*(c + d*x^4),x]
 

Output:

(x*(a + b*x^4)^(1/4)*(d*(a + b*x^4)^2 - (a*(-10*b*c + a*d)*Hypergeometric2 
F1[-5/4, 1/4, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^(1/4)))/(10*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {913, 748, 748, 768, 858, 807, 229}

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.

\(\displaystyle \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {(10 b c-a d) \int \left (b x^4+a\right )^{5/4}dx}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

\(\Big \downarrow \) 748

\(\displaystyle \frac {(10 b c-a d) \left (\frac {5}{6} a \int \sqrt [4]{b x^4+a}dx+\frac {1}{6} x \left (a+b x^4\right )^{5/4}\right )}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

\(\Big \downarrow \) 748

\(\displaystyle \frac {(10 b c-a d) \left (\frac {5}{6} a \left (\frac {1}{2} a \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx+\frac {1}{2} x \sqrt [4]{a+b x^4}\right )+\frac {1}{6} x \left (a+b x^4\right )^{5/4}\right )}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

\(\Big \downarrow \) 768

\(\displaystyle \frac {(10 b c-a d) \left (\frac {5}{6} a \left (\frac {a x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x^3}dx}{2 \left (a+b x^4\right )^{3/4}}+\frac {1}{2} x \sqrt [4]{a+b x^4}\right )+\frac {1}{6} x \left (a+b x^4\right )^{5/4}\right )}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

\(\Big \downarrow \) 858

\(\displaystyle \frac {(10 b c-a d) \left (\frac {5}{6} a \left (\frac {1}{2} x \sqrt [4]{a+b x^4}-\frac {a x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x}d\frac {1}{x}}{2 \left (a+b x^4\right )^{3/4}}\right )+\frac {1}{6} x \left (a+b x^4\right )^{5/4}\right )}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(10 b c-a d) \left (\frac {5}{6} a \left (\frac {1}{2} x \sqrt [4]{a+b x^4}-\frac {a x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4}}d\frac {1}{x^2}}{4 \left (a+b x^4\right )^{3/4}}\right )+\frac {1}{6} x \left (a+b x^4\right )^{5/4}\right )}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

\(\Big \downarrow \) 229

\(\displaystyle \frac {(10 b c-a d) \left (\frac {5}{6} a \left (\frac {1}{2} x \sqrt [4]{a+b x^4}-\frac {\sqrt {a} \sqrt {b} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{2 \left (a+b x^4\right )^{3/4}}\right )+\frac {1}{6} x \left (a+b x^4\right )^{5/4}\right )}{10 b}+\frac {d x \left (a+b x^4\right )^{9/4}}{10 b}\)

Input:

Int[(a + b*x^4)^(5/4)*(c + d*x^4),x]
 

Output:

(d*x*(a + b*x^4)^(9/4))/(10*b) + ((10*b*c - a*d)*((x*(a + b*x^4)^(5/4))/6 
+ (5*a*((x*(a + b*x^4)^(1/4))/2 - (Sqrt[a]*Sqrt[b]*(1 + a/(b*x^4))^(3/4)*x 
^3*EllipticF[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(2*(a + b*x^4)^(3/4))))/ 
6))/(10*b)
 

Defintions of rubi rules used

rule 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

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 768

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 
/4)/(a + b*x^4)^(3/4))   Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ 
[{a, b}, x]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 858

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + 
b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int 
egerQ[m]
 

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]
 

Maple [F]

\[\int \left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (d \,x^{4}+c \right )d x\]

Input:

int((b*x^4+a)^(5/4)*(d*x^4+c),x)
 

Output:

int((b*x^4+a)^(5/4)*(d*x^4+c),x)
 

Fricas [F]

\[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )} \,d x } \] Input:

integrate((b*x^4+a)^(5/4)*(d*x^4+c),x, algorithm="fricas")
 

Output:

integral((b*d*x^8 + (b*c + a*d)*x^4 + a*c)*(b*x^4 + a)^(1/4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.13 \[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\frac {a^{\frac {5}{4}} c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {5}{4}} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {\sqrt [4]{a} b c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {\sqrt [4]{a} b d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \] Input:

integrate((b*x**4+a)**(5/4)*(d*x**4+c),x)
 

Output:

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

Maxima [F]

\[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )} \,d x } \] Input:

integrate((b*x^4+a)^(5/4)*(d*x^4+c),x, algorithm="maxima")
 

Output:

integrate((b*x^4 + a)^(5/4)*(d*x^4 + c), x)
 

Giac [F]

\[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )} \,d x } \] Input:

integrate((b*x^4+a)^(5/4)*(d*x^4+c),x, algorithm="giac")
 

Output:

integrate((b*x^4 + a)^(5/4)*(d*x^4 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\int {\left (b\,x^4+a\right )}^{5/4}\,\left (d\,x^4+c\right ) \,d x \] Input:

int((a + b*x^4)^(5/4)*(c + d*x^4),x)
 

Output:

int((a + b*x^4)^(5/4)*(c + d*x^4), x)
 

Reduce [F]

\[ \int \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) \, dx=\frac {5 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} d x +70 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b c x +22 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b d \,x^{5}+20 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} c \,x^{5}+12 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} d \,x^{9}-5 \left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x \right ) a^{3} d +50 \left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x \right ) a^{2} b c}{120 b} \] Input:

int((b*x^4+a)^(5/4)*(d*x^4+c),x)
 

Output:

(5*(a + b*x**4)**(1/4)*a**2*d*x + 70*(a + b*x**4)**(1/4)*a*b*c*x + 22*(a + 
 b*x**4)**(1/4)*a*b*d*x**5 + 20*(a + b*x**4)**(1/4)*b**2*c*x**5 + 12*(a + 
b*x**4)**(1/4)*b**2*d*x**9 - 5*int((a + b*x**4)**(1/4)/(a + b*x**4),x)*a** 
3*d + 50*int((a + b*x**4)**(1/4)/(a + b*x**4),x)*a**2*b*c)/(120*b)