\(\int \frac {1}{x \sqrt {a+b (c+d x)^4}} \, dx\) [127]

Optimal result

Integrand size = 19, antiderivative size = 413 \[ \int \frac {1}{x \sqrt {a+b (c+d x)^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b c^4} (c+d x)}{c \sqrt {a+b (c+d x)^4}}\right )}{2 \sqrt {a+b c^4}}-\frac {\text {arctanh}\left (\frac {a+b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{2 \sqrt {a+b c^4}}-\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}}-\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {a}+\sqrt {b} c^2\right )^2}{4 \sqrt {a} \sqrt {b} c^2},2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.53 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \sqrt {a+b (c+d x)^4}} \, dx=-\frac {(-1)^{3/4} \left (\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)\right )^2 \sqrt {\frac {(1-i) \left ((-1)^{3/4} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \sqrt {-\frac {(1+i) \left ((-1)^{3/4} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \left (\left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right ),-1\right )-2 \sqrt [4]{-1} \sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {(-1)^{3/4} \sqrt [4]{a}-i \sqrt [4]{b} c}{\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} c},\arcsin \left (\sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right ),-1\right )\right )}{\sqrt [4]{a} \left (\sqrt {a}+i \sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {896, 25, 2263, 1541, 27, 761, 1577, 488, 219, 2223}

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.

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

\(\Big \downarrow \) 896

\(\displaystyle \int \frac {1}{d x \sqrt {a+b (c+d x)^4}}d(c+d x)\)

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2263

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{2 \sqrt {a+b c^4}}-c \left (\frac {\int \frac {\sqrt {b} (c+d x)^2+\sqrt {a}}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {a}+\sqrt {b} c^2}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}}\right )\)

\(\Big \downarrow \) 2223

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 488

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

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 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1541

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

rule 1577

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, c, d, e, p, q}, x]
 

rule 2223

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* 
(d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] 
)), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 
2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc 
Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 
] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[c*(d/e) + a*(e 
/d)]
 

rule 2263

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[d 
  Int[1/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x], x] - Simp[e   Int[x/((d^2 - 
e^2*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.07 (sec) , antiderivative size = 1582, normalized size of antiderivative = 3.83

method	result	size
default	\(\text {Expression too large to display}\)	\(1582\)
elliptic	\(\text {Expression too large to display}\)	\(1582\)

Input:

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

Output:

2*(-(-I/b*(-a*b^3)^(1/4)-c)/d+(1/b*(-a*b^3)^(1/4)-c)/d)*(((-I/b*(-a*b^3)^( 
1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)*(x-(1/b*(-a*b^3)^(1/4)-c)/d)/((-I/b*(- 
a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))^ 
(1/2)*(x-(I/b*(-a*b^3)^(1/4)-c)/d)^2*(((I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b 
^3)^(1/4)-c)/d)*(x-(-1/b*(-a*b^3)^(1/4)-c)/d)/((-1/b*(-a*b^3)^(1/4)-c)/d-( 
1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))^(1/2)*(((I/b*(-a*b^ 
3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-I/b*(-a*b^3)^(1/4)-c)/d)/((-I 
/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c) 
/d))^(1/2)/((-I/b*(-a*b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)/((I/b*(-a* 
b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(d^4*b*(x-(1/b*(-a*b^3)^(1/4)-c) 
/d)*(x-(I/b*(-a*b^3)^(1/4)-c)/d)*(x-(-1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-I/b*(- 
a*b^3)^(1/4)-c)/d))^(1/2)/(I/b*(-a*b^3)^(1/4)-c)*d*(EllipticF((((-I/b*(-a* 
b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)*(x-(1/b*(-a*b^3)^(1/4)-c)/d)/((- 
I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c 
)/d))^(1/2),(((I/b*(-a*b^3)^(1/4)-c)/d-(-1/b*(-a*b^3)^(1/4)-c)/d)*(-(-I/b* 
(-a*b^3)^(1/4)-c)/d+(1/b*(-a*b^3)^(1/4)-c)/d)/((1/b*(-a*b^3)^(1/4)-c)/d-(- 
1/b*(-a*b^3)^(1/4)-c)/d)/((I/b*(-a*b^3)^(1/4)-c)/d-(-I/b*(-a*b^3)^(1/4)-c) 
/d))^(1/2))+((I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(1/b*(-a*b 
^3)^(1/4)-c)*d*EllipticPi((((-I/b*(-a*b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)- 
c)/d)*(x-(1/b*(-a*b^3)^(1/4)-c)/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*...
 

Fricas [F]

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

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

Output:

integral(sqrt(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + 
b*c^4 + a)/(b*d^4*x^5 + 4*b*c*d^3*x^4 + 6*b*c^2*d^2*x^3 + 4*b*c^3*d*x^2 + 
(b*c^4 + a)*x), x)
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {a+b (c+d x)^4}} \, dx=\int \frac {1}{x \sqrt {a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4}}}\, dx \] Input:

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

Output:

Integral(1/(x*sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c* 
d**3*x**3 + b*d**4*x**4)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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