3.18.2 \(\int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx\) [1702]

3.18.2.1 Optimal result

Integrand size = 19, antiderivative size = 688 \[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\frac {2 \sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{\sqrt {b} \sqrt {d} (b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )}-\frac {\sqrt {2} (b c-a d)^{3/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{b^{3/4} d^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {(b c-a d)^{3/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right ),\frac {1}{2}\right )}{\sqrt {2} b^{3/4} d^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}} \]

output

2*((b*x+a)*(d*x+c))^(1/2)*((2*b*d*x+a*d+b*c)^2)^(1/2)*((a*d+b*(2*d*x+c))^2 
)^(1/2)/(-a*d+b*c)/(b*x+a)^(1/4)/(d*x+c)^(1/4)/(2*b*d*x+a*d+b*c)/b^(1/2)/d 
^(1/2)/(1+2*b^(1/2)*d^(1/2)*((b*x+a)*(d*x+c))^(1/2)/(-a*d+b*c))+1/2*(-a*d+ 
b*c)^(3/2)*((b*x+a)*(d*x+c))^(1/4)*(cos(2*arctan(b^(1/4)*d^(1/4)*((b*x+a)* 
(d*x+c))^(1/4)*2^(1/2)/(-a*d+b*c)^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*d^ 
(1/4)*((b*x+a)*(d*x+c))^(1/4)*2^(1/2)/(-a*d+b*c)^(1/2)))*EllipticF(sin(2*a 
rctan(b^(1/4)*d^(1/4)*((b*x+a)*(d*x+c))^(1/4)*2^(1/2)/(-a*d+b*c)^(1/2))),1 
/2*2^(1/2))*(1+2*b^(1/2)*d^(1/2)*((b*x+a)*(d*x+c))^(1/2)/(-a*d+b*c))*((2*b 
*d*x+a*d+b*c)^2)^(1/2)*((a*d+b*(2*d*x+c))^2/(-a*d+b*c)^2/(1+2*b^(1/2)*d^(1 
/2)*((b*x+a)*(d*x+c))^(1/2)/(-a*d+b*c))^2)^(1/2)/b^(3/4)/d^(3/4)/(b*x+a)^( 
1/4)/(d*x+c)^(1/4)/(2*b*d*x+a*d+b*c)*2^(1/2)/((a*d+b*(2*d*x+c))^2)^(1/2)-( 
-a*d+b*c)^(3/2)*((b*x+a)*(d*x+c))^(1/4)*(cos(2*arctan(b^(1/4)*d^(1/4)*((b* 
x+a)*(d*x+c))^(1/4)*2^(1/2)/(-a*d+b*c)^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/ 
4)*d^(1/4)*((b*x+a)*(d*x+c))^(1/4)*2^(1/2)/(-a*d+b*c)^(1/2)))*EllipticE(si 
n(2*arctan(b^(1/4)*d^(1/4)*((b*x+a)*(d*x+c))^(1/4)*2^(1/2)/(-a*d+b*c)^(1/2 
))),1/2*2^(1/2))*2^(1/2)*(1+2*b^(1/2)*d^(1/2)*((b*x+a)*(d*x+c))^(1/2)/(-a* 
d+b*c))*((2*b*d*x+a*d+b*c)^2)^(1/2)*((a*d+b*(2*d*x+c))^2/(-a*d+b*c)^2/(1+2 
*b^(1/2)*d^(1/2)*((b*x+a)*(d*x+c))^(1/2)/(-a*d+b*c))^2)^(1/2)/b^(3/4)/d^(3 
/4)/(b*x+a)^(1/4)/(d*x+c)^(1/4)/(2*b*d*x+a*d+b*c)/((a*d+b*(2*d*x+c))^2)^(1 
/2)
 

3.18.2.2 Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\frac {4 (a+b x)^{3/4} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},\frac {d (a+b x)}{-b c+a d}\right )}{3 b \sqrt [4]{c+d x}} \]

input

Integrate[1/((a + b*x)^(1/4)*(c + d*x)^(1/4)),x]
 

output

(4*(a + b*x)^(3/4)*((b*(c + d*x))/(b*c - a*d))^(1/4)*Hypergeometric2F1[1/4 
, 3/4, 7/4, (d*(a + b*x))/(-(b*c) + a*d)])/(3*b*(c + d*x)^(1/4))
 

3.18.2.3 Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {73, 839, 813, 858, 807, 212}

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.

\(\displaystyle \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 \int \frac {\sqrt {a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [4]{a+b x}}{b}\)

\(\Big \downarrow \) 839

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

\(\Big \downarrow \) 813

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

\(\Big \downarrow \) 858

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 212

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

input

Int[1/((a + b*x)^(1/4)*(c + d*x)^(1/4)),x]
 

output

(4*((a + b*x)^(3/4)/(2*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4)) + (b*(c - (a 
*d)/b)*(a + b*x)^(1/4)*(1 + (b*c - a*d)/(d*(a + b*x)))^(1/4)*EllipticE[Arc 
Tan[(Sqrt[b*c - a*d]*Sqrt[a + b*x])/Sqrt[d]]/2, 2])/(2*Sqrt[d]*Sqrt[b*c - 
a*d]*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))))/b
 

3.18.2.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

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 813

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) 
)^(1/4)/(b*(a + b*x^4)^(1/4)))   Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] 
/; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 839

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 
)^(1/4)), x] - Simp[a/2   Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} 
, x] && PosQ[b/a]
 

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]
 

3.18.2.4 Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]

input

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

output

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

3.18.2.5 Fricas [F]

\[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

integral((b*x + a)^(3/4)*(d*x + c)^(3/4)/(b*d*x^2 + a*c + (b*c + a*d)*x), 
x)
 

3.18.2.6 Sympy [F]

\[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{\sqrt [4]{a + b x} \sqrt [4]{c + d x}}\, dx \]

input

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

output

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

3.18.2.7 Maxima [F]

\[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.18.2.8 Giac [F]

\[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.18.2.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{1/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]

input

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

output

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

3.18.2.10 Reduce [F]

\[ \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}}}d x \]

input

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

output

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