\(\int \frac {x}{(c+d x) \sqrt [4]{a+b x^2}} \, dx\) [1646]

Optimal result

Integrand size = 20, antiderivative size = 360 \[ \int \frac {x}{(c+d x) \sqrt [4]{a+b x^2}} \, dx=\frac {2 x}{d \sqrt [4]{a+b x^2}}-\frac {c \arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{b c^2+a d^2}}\right )}{d^{3/2} \sqrt [4]{b c^2+a d^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{b c^2+a d^2}}\right )}{d^{3/2} \sqrt [4]{b c^2+a d^2}}-\frac {2 \sqrt {a} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} c^2 \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \sqrt {b c^2+a d^2} x}-\frac {\sqrt [4]{a} c^2 \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \sqrt {b c^2+a d^2} x} \] Output:

2*x/d/(b*x^2+a)^(1/4)-c*arctan(d^(1/2)*(b*x^2+a)^(1/4)/(a*d^2+b*c^2)^(1/4) 
)/d^(3/2)/(a*d^2+b*c^2)^(1/4)+c*arctanh(d^(1/2)*(b*x^2+a)^(1/4)/(a*d^2+b*c 
^2)^(1/4))/d^(3/2)/(a*d^2+b*c^2)^(1/4)-2*a^(1/2)*(1+b*x^2/a)^(1/4)*Ellipti 
cE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/2))/b^(1/2)/d/(b*x^2+a)^(1/4)+a 
^(1/4)*c^2*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)/(a 
*d^2+b*c^2)^(1/2)*d,I)/d^2/(a*d^2+b*c^2)^(1/2)/x-a^(1/4)*c^2*(-b*x^2/a)^(1 
/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)/(a*d^2+b*c^2)^(1/2)*d,I)/d^ 
2/(a*d^2+b*c^2)^(1/2)/x
                                                                                    
                                                                                    
 

Mathematica [F]

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

Integrate[x/((c + d*x)*(a + b*x^2)^(1/4)),x]
 

Output:

Integrate[x/((c + d*x)*(a + b*x^2)^(1/4)), x]
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {605, 227, 225, 212, 504, 310, 353, 73, 27, 827, 218, 221, 993, 1542}

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.

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

\(\Big \downarrow \) 605

\(\displaystyle \frac {\int \frac {1}{\sqrt [4]{b x^2+a}}dx}{d}-\frac {c \int \frac {1}{(c+d x) \sqrt [4]{b x^2+a}}dx}{d}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{d \sqrt [4]{a+b x^2}}-\frac {c \int \frac {1}{(c+d x) \sqrt [4]{b x^2+a}}dx}{d}\)

\(\Big \downarrow \) 225

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

\(\Big \downarrow \) 212

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

\(\Big \downarrow \) 504

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

\(\Big \downarrow \) 310

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1542

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

Input:

Int[x/((c + d*x)*(a + b*x^2)^(1/4)),x]
 

Output:

((1 + (b*x^2)/a)^(1/4)*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE 
[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/Sqrt[b]))/(d*(a + b*x^2)^(1/4)) - (c*( 
-2*d*(-1/2*ArcTan[(Sqrt[d]*(a + b*x^2)^(1/4))/(b*c^2 + a*d^2)^(1/4)]/(d^(3 
/2)*(b*c^2 + a*d^2)^(1/4)) + ArcTanh[(Sqrt[d]*(a + b*x^2)^(1/4))/(b*c^2 + 
a*d^2)^(1/4)]/(2*d^(3/2)*(b*c^2 + a*d^2)^(1/4))) + (2*c*Sqrt[-((b*x^2)/a)] 
*(-1/2*(a^(1/4)*EllipticPi[-((Sqrt[a]*d)/Sqrt[b*c^2 + a*d^2]), ArcSin[(a + 
 b*x^2)^(1/4)/a^(1/4)], -1])/(d*Sqrt[b*c^2 + a*d^2]) + (a^(1/4)*EllipticPi 
[(Sqrt[a]*d)/Sqrt[b*c^2 + a*d^2], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/ 
(2*d*Sqrt[b*c^2 + a*d^2])))/x))/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 218

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

rule 221

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

rule 225

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

rule 227

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

rule 310

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[2*(Sqrt[(-b)*(x^2/a)]/x)   Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* 
x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 353

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

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 605

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

rule 827

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

rule 993

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

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

Maple [F]

\[\int \frac {x}{\left (d x +c \right ) \left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {x}{(c+d x) \sqrt [4]{a+b x^2}} \, dx=\int { \frac {x}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x}{(c+d x) \sqrt [4]{a+b x^2}} \, dx=\int { \frac {x}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(c+d x) \sqrt [4]{a+b x^2}} \, dx=\int \frac {x}{{\left (b\,x^2+a\right )}^{1/4}\,\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x}{(c+d x) \sqrt [4]{a+b x^2}} \, dx=\int \frac {x}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} c +\left (b \,x^{2}+a \right )^{\frac {1}{4}} d x}d x \] Input:

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

Output:

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