\(\int (a+b x^4)^{3/4} (c+d x^4) \, dx\) [79]

Optimal result

Integrand size = 19, antiderivative size = 125 \[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=\frac {(8 b c-a d) x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}+\frac {3 a (8 b c-a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}+\frac {3 a (8 b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}} \] Output:

1/32*(-a*d+8*b*c)*x*(b*x^4+a)^(3/4)/b+1/8*d*x*(b*x^4+a)^(7/4)/b+3/64*a*(-a 
*d+8*b*c)*arctan(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(5/4)+3/64*a*(-a*d+8*b*c)*ar 
ctanh(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(5/4)
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=\frac {2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4} \left (8 b c+3 a d+4 b d x^4\right )-3 a (-8 b c+a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-3 a (-8 b c+a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}} \] Input:

Integrate[(a + b*x^4)^(3/4)*(c + d*x^4),x]
 

Output:

(2*b^(1/4)*x*(a + b*x^4)^(3/4)*(8*b*c + 3*a*d + 4*b*d*x^4) - 3*a*(-8*b*c + 
 a*d)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)] - 3*a*(-8*b*c + a*d)*ArcTanh[( 
b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(5/4))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {913, 748, 770, 756, 216, 219}

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.

\(\displaystyle \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {(8 b c-a d) \int \left (b x^4+a\right )^{3/4}dx}{8 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}\)

\(\Big \downarrow \) 748

\(\displaystyle \frac {(8 b c-a d) \left (\frac {3}{4} a \int \frac {1}{\sqrt [4]{b x^4+a}}dx+\frac {1}{4} x \left (a+b x^4\right )^{3/4}\right )}{8 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {(8 b c-a d) \left (\frac {3}{4} a \int \frac {1}{1-\frac {b x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {1}{4} x \left (a+b x^4\right )^{3/4}\right )}{8 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {(8 b c-a d) \left (\frac {3}{4} a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}+1}d\frac {x}{\sqrt [4]{b x^4+a}}\right )+\frac {1}{4} x \left (a+b x^4\right )^{3/4}\right )}{8 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {(8 b c-a d) \left (\frac {3}{4} a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )+\frac {1}{4} x \left (a+b x^4\right )^{3/4}\right )}{8 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(8 b c-a d) \left (\frac {3}{4} a \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )+\frac {1}{4} x \left (a+b x^4\right )^{3/4}\right )}{8 b}+\frac {d x \left (a+b x^4\right )^{7/4}}{8 b}\)

Input:

Int[(a + b*x^4)^(3/4)*(c + d*x^4),x]
 

Output:

(d*x*(a + b*x^4)^(7/4))/(8*b) + ((8*b*c - a*d)*((x*(a + b*x^4)^(3/4))/4 + 
(3*a*(ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(1/4)) + ArcTanh[(b^(1/4) 
*x)/(a + b*x^4)^(1/4)]/(2*b^(1/4))))/4))/(8*b)
 

Defintions of rubi rules used

rule 216

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

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 [A] (verified)

Time = 1.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (\frac {d \,x^{4}}{2}+c \right ) x \,b^{\frac {5}{4}}}{4}+\frac {3 a \left (2 \left (b \,x^{4}+a \right )^{\frac {3}{4}} x d \,b^{\frac {1}{4}}+\left (a d -8 b c \right ) \left (\arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )-\frac {\ln \left (\frac {-b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )}{2}\right )\right )}{64}}{b^{\frac {5}{4}}}\)	\(117\)

Input:

int((b*x^4+a)^(3/4)*(d*x^4+c),x,method=_RETURNVERBOSE)
 

Output:

3/64*(16/3*(b*x^4+a)^(3/4)*(1/2*d*x^4+c)*x*b^(5/4)+a*(2*(b*x^4+a)^(3/4)*x* 
d*b^(1/4)+(a*d-8*b*c)*(arctan(1/b^(1/4)/x*(b*x^4+a)^(1/4))-1/2*ln((-b^(1/4 
)*x-(b*x^4+a)^(1/4))/(b^(1/4)*x-(b*x^4+a)^(1/4))))))/b^(5/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 795, normalized size of antiderivative = 6.36 \[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx =\text {Too large to display} \] Input:

integrate((b*x^4+a)^(3/4)*(d*x^4+c),x, algorithm="fricas")
 

Output:

1/128*(3*b*((4096*a^4*b^4*c^4 - 2048*a^5*b^3*c^3*d + 384*a^6*b^2*c^2*d^2 - 
 32*a^7*b*c*d^3 + a^8*d^4)/b^5)^(1/4)*log(-27*(b^4*x*((4096*a^4*b^4*c^4 - 
2048*a^5*b^3*c^3*d + 384*a^6*b^2*c^2*d^2 - 32*a^7*b*c*d^3 + a^8*d^4)/b^5)^ 
(3/4) + (512*a^3*b^3*c^3 - 192*a^4*b^2*c^2*d + 24*a^5*b*c*d^2 - a^6*d^3)*( 
b*x^4 + a)^(1/4))/x) - 3*b*((4096*a^4*b^4*c^4 - 2048*a^5*b^3*c^3*d + 384*a 
^6*b^2*c^2*d^2 - 32*a^7*b*c*d^3 + a^8*d^4)/b^5)^(1/4)*log(27*(b^4*x*((4096 
*a^4*b^4*c^4 - 2048*a^5*b^3*c^3*d + 384*a^6*b^2*c^2*d^2 - 32*a^7*b*c*d^3 + 
 a^8*d^4)/b^5)^(3/4) - (512*a^3*b^3*c^3 - 192*a^4*b^2*c^2*d + 24*a^5*b*c*d 
^2 - a^6*d^3)*(b*x^4 + a)^(1/4))/x) - 3*I*b*((4096*a^4*b^4*c^4 - 2048*a^5* 
b^3*c^3*d + 384*a^6*b^2*c^2*d^2 - 32*a^7*b*c*d^3 + a^8*d^4)/b^5)^(1/4)*log 
(-27*(I*b^4*x*((4096*a^4*b^4*c^4 - 2048*a^5*b^3*c^3*d + 384*a^6*b^2*c^2*d^ 
2 - 32*a^7*b*c*d^3 + a^8*d^4)/b^5)^(3/4) + (512*a^3*b^3*c^3 - 192*a^4*b^2* 
c^2*d + 24*a^5*b*c*d^2 - a^6*d^3)*(b*x^4 + a)^(1/4))/x) + 3*I*b*((4096*a^4 
*b^4*c^4 - 2048*a^5*b^3*c^3*d + 384*a^6*b^2*c^2*d^2 - 32*a^7*b*c*d^3 + a^8 
*d^4)/b^5)^(1/4)*log(-27*(-I*b^4*x*((4096*a^4*b^4*c^4 - 2048*a^5*b^3*c^3*d 
 + 384*a^6*b^2*c^2*d^2 - 32*a^7*b*c*d^3 + a^8*d^4)/b^5)^(3/4) + (512*a^3*b 
^3*c^3 - 192*a^4*b^2*c^2*d + 24*a^5*b*c*d^2 - a^6*d^3)*(b*x^4 + a)^(1/4))/ 
x) + 4*(4*b*d*x^5 + (8*b*c + 3*a*d)*x)*(b*x^4 + a)^(3/4))/b
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=\frac {a^{\frac {3}{4}} c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {3}{4}} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((b*x**4+a)**(3/4)*(d*x**4+c),x)
 

Output:

a**(3/4)*c*x*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/ 
a)/(4*gamma(5/4)) + a**(3/4)*d*x**5*gamma(5/4)*hyper((-3/4, 5/4), (9/4,), 
b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (101) = 202\).

Time = 0.11 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.06 \[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=-\frac {1}{16} \, {\left (3 \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {1}{4}}}\right )} + \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a}{{\left (b - \frac {b x^{4} + a}{x^{4}}\right )} x^{3}}\right )} c + \frac {1}{128} \, {\left (\frac {3 \, a^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {1}{4}}}\right )}}{b} + \frac {4 \, {\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{2} b}{x^{3}} + \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{x^{7}}\right )}}{b^{3} - \frac {2 \, {\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac {{\left (b x^{4} + a\right )}^{2} b}{x^{8}}}\right )} d \] Input:

integrate((b*x^4+a)^(3/4)*(d*x^4+c),x, algorithm="maxima")
 

Output:

-1/16*(3*a*(2*arctan((b*x^4 + a)^(1/4)/(b^(1/4)*x))/b^(1/4) + log(-(b^(1/4 
) - (b*x^4 + a)^(1/4)/x)/(b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(1/4)) + 4*(b* 
x^4 + a)^(3/4)*a/((b - (b*x^4 + a)/x^4)*x^3))*c + 1/128*(3*a^2*(2*arctan(( 
b*x^4 + a)^(1/4)/(b^(1/4)*x))/b^(1/4) + log(-(b^(1/4) - (b*x^4 + a)^(1/4)/ 
x)/(b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(1/4))/b + 4*((b*x^4 + a)^(3/4)*a^2* 
b/x^3 + 3*(b*x^4 + a)^(7/4)*a^2/x^7)/(b^3 - 2*(b*x^4 + a)*b^2/x^4 + (b*x^4 
 + a)^2*b/x^8))*d
 

Giac [F]

\[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{4}} {\left (d x^{4} + c\right )} \,d x } \] Input:

integrate((b*x^4+a)^(3/4)*(d*x^4+c),x, algorithm="giac")
 

Output:

integrate((b*x^4 + a)^(3/4)*(d*x^4 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=\int {\left (b\,x^4+a\right )}^{3/4}\,\left (d\,x^4+c\right ) \,d x \] Input:

int((a + b*x^4)^(3/4)*(c + d*x^4),x)
 

Output:

int((a + b*x^4)^(3/4)*(c + d*x^4), x)
 

Reduce [F]

\[ \int \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) \, dx=\frac {3 \left (b \,x^{4}+a \right )^{\frac {3}{4}} a d x +8 \left (b \,x^{4}+a \right )^{\frac {3}{4}} b c x +4 \left (b \,x^{4}+a \right )^{\frac {3}{4}} b d \,x^{5}-3 \left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x \right ) a^{2} d +24 \left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x \right ) a b c}{32 b} \] Input:

int((b*x^4+a)^(3/4)*(d*x^4+c),x)
 

Output:

(3*(a + b*x**4)**(3/4)*a*d*x + 8*(a + b*x**4)**(3/4)*b*c*x + 4*(a + b*x**4 
)**(3/4)*b*d*x**5 - 3*int((a + b*x**4)**(3/4)/(a + b*x**4),x)*a**2*d + 24* 
int((a + b*x**4)**(3/4)/(a + b*x**4),x)*a*b*c)/(32*b)