\(\int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx\) [131]

Optimal result

Integrand size = 49, antiderivative size = 55 \[ \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx=\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} (e (c+d x))^{3/2}}{e^{3/2} \sqrt {a+b (c+d x)^3}}\right )}{3 \sqrt {b} d} \] Output:

2/3*e^(1/2)*arctanh(b^(1/2)*(e*(d*x+c))^(3/2)/e^(3/2)/(a+b*(d*x+c)^3)^(1/2 
))/b^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx=\frac {2 \sqrt {e (c+d x)} \log \left (\sqrt {b} (c+d x)^{3/2}+\sqrt {a+b (c+d x)^3}\right )}{3 \sqrt {b} d \sqrt {c+d x}} \] Input:

Integrate[Sqrt[e*(c + d*x)]/Sqrt[a + b*c^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2 + 
 b*d^3*x^3],x]
 

Output:

(2*Sqrt[e*(c + d*x)]*Log[Sqrt[b]*(c + d*x)^(3/2) + Sqrt[a + b*(c + d*x)^3] 
])/(3*Sqrt[b]*d*Sqrt[c + d*x])
 

Rubi [A] (warning: unable to verify)

Time = 0.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {2510, 851, 807, 224, 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.

\(\displaystyle \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx\)

\(\Big \downarrow \) 2510

\(\displaystyle \frac {\int \frac {\sqrt {e (c+d x)}}{\sqrt {b (c+d x)^3+a}}d(e (c+d x))}{d e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {2 \int \frac {e^2 (c+d x)^2}{\sqrt {b e^3 (c+d x)^6+a}}d\sqrt {e (c+d x)}}{d e}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {2 \int \frac {1}{\sqrt {\frac {b (c+d x)^2}{e}+a}}d\left (e^3 (c+d x)^3\right )}{3 d e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {2 \int \frac {1}{1-\frac {b (c+d x)^2}{e}}d\frac {e^3 (c+d x)^3}{\sqrt {\frac {b (c+d x)^2}{e}+a}}}{3 d e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {e}}\right )}{3 \sqrt {b} d}\)

Input:

Int[Sqrt[e*(c + d*x)]/Sqrt[a + b*c^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2 + b*d^3 
*x^3],x]
 

Output:

(2*Sqrt[e]*ArcTanh[(Sqrt[b]*(c + d*x))/Sqrt[e]])/(3*Sqrt[b]*d)
 

Defintions of rubi rules used

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 224

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

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 851

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

rule 2510

Int[(Pn_)^(p_.)*(u_)^(m_.), x_Symbol] :> With[{Px = Pn /. x -> (x - Coeff[u 
, x, 0])/Coeff[u, x, 1]}, Simp[1/Coeff[u, x, 1]   Subst[Int[x^m*ExpandToSum 
[Px, x]^p, x], x, u], x] /; BinomialQ[Px, x]] /; FreeQ[{m, p}, x] && Linear 
Q[u, x] && PolyQ[Pn, x] && NeQ[Coeff[u, x, 0], 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 2.23 (sec) , antiderivative size = 864, normalized size of antiderivative = 15.71

method	result	size
default	\(\frac {2 \sqrt {e \left (x d +c \right )}\, \sqrt {b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}\, e \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-b d x -b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, {\left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}^{2} \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b d x -2 b c -\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{\left (i \sqrt {3}-3\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (x d +c \right ) b}{i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-b d x -b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}}}, 2 \sqrt {-\frac {i \sqrt {3}}{\left (i \sqrt {3}-3\right ) \left (1+i \sqrt {3}\right )}}\right )-i \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-b d x -b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}}}, -\frac {2}{1+i \sqrt {3}}, 2 \sqrt {-\frac {i \sqrt {3}}{\left (i \sqrt {3}-3\right ) \left (1+i \sqrt {3}\right )}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-b d x -b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}}}, 2 \sqrt {-\frac {i \sqrt {3}}{\left (i \sqrt {3}-3\right ) \left (1+i \sqrt {3}\right )}}\right )-3 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-b d x -b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}}}, -\frac {2}{1+i \sqrt {3}}, 2 \sqrt {-\frac {i \sqrt {3}}{\left (i \sqrt {3}-3\right ) \left (1+i \sqrt {3}\right )}}\right )\right )}{b^{2} d \sqrt {\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) e \left (x d +c \right )}\, \left (1+i \sqrt {3}\right ) \left (i \sqrt {3}+3\right ) \sqrt {\frac {e \left (-b d x -b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b d x +2 b c +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b d x -2 b c -\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (x d +c \right )}{b^{2}}}}\)	\(864\)
elliptic	\(\text {Expression too large to display}\)	\(2959\)

Input:

int((e*(d*x+c))^(1/2)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^(1/2), 
x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).

Time = 0.36 (sec) , antiderivative size = 379, normalized size of antiderivative = 6.89 \[ \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx=\left [\frac {\sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} d^{6} e x^{6} - 48 \, b^{2} c d^{5} e x^{5} - 120 \, b^{2} c^{2} d^{4} e x^{4} - 8 \, {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} e x^{3} - 24 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} e x^{2} - 24 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d e x - 4 \, {\left (2 \, b^{2} d^{4} x^{4} + 8 \, b^{2} c d^{3} x^{3} + 12 \, b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c^{4} + a b c + {\left (8 \, b^{2} c^{3} + a b\right )} d x\right )} \sqrt {b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} \sqrt {d e x + c e} \sqrt {\frac {e}{b}} - {\left (8 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} e\right )}{6 \, d}, -\frac {\sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} {\left (b d x + b c\right )} \sqrt {d e x + c e} \sqrt {-\frac {e}{b}}}{2 \, b d^{3} e x^{3} + 6 \, b c d^{2} e x^{2} + 6 \, b c^{2} d e x + {\left (2 \, b c^{3} + a\right )} e}\right )}{3 \, d}\right ] \] Input:

integrate((e*(d*x+c))^(1/2)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^ 
(1/2),x, algorithm="fricas")
 

Output:

[1/6*sqrt(e/b)*log(-8*b^2*d^6*e*x^6 - 48*b^2*c*d^5*e*x^5 - 120*b^2*c^2*d^4 
*e*x^4 - 8*(20*b^2*c^3 + a*b)*d^3*e*x^3 - 24*(5*b^2*c^4 + a*b*c)*d^2*e*x^2 
 - 24*(2*b^2*c^5 + a*b*c^2)*d*e*x - 4*(2*b^2*d^4*x^4 + 8*b^2*c*d^3*x^3 + 1 
2*b^2*c^2*d^2*x^2 + 2*b^2*c^4 + a*b*c + (8*b^2*c^3 + a*b)*d*x)*sqrt(b*d^3* 
x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*sqrt(d*e*x + c*e)*sqrt(e/b) 
 - (8*b^2*c^6 + 8*a*b*c^3 + a^2)*e)/d, -1/3*sqrt(-e/b)*arctan(2*sqrt(b*d^3 
*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(b*d*x + b*c)*sqrt(d*e*x + 
 c*e)*sqrt(-e/b)/(2*b*d^3*e*x^3 + 6*b*c*d^2*e*x^2 + 6*b*c^2*d*e*x + (2*b*c 
^3 + a)*e))/d]
 

Sympy [F]

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

integrate((e*(d*x+c))**(1/2)/(b*d**3*x**3+3*b*c*d**2*x**2+3*b*c**2*d*x+b*c 
**3+a)**(1/2),x)
 

Output:

Integral(sqrt(e*(c + d*x))/sqrt(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x** 
2 + b*d**3*x**3), x)
 

Maxima [F]

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

integrate((e*(d*x+c))^(1/2)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^ 
(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt((d*x + c)*e)/sqrt(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + 
 b*c^3 + a), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (41) = 82\).

Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx=-\frac {2 \, e^{4} \log \left ({\left | -2 \, \sqrt {b e} {\left (3 \, \sqrt {d e x + c e} c - \frac {3 \, \sqrt {d e x + c e} c e - {\left (d e x + c e\right )}^{\frac {3}{2}}}{e}\right )} + 2 \, \sqrt {{\left (3 \, \sqrt {d e x + c e} c - \frac {3 \, \sqrt {d e x + c e} c e - {\left (d e x + c e\right )}^{\frac {3}{2}}}{e}\right )}^{2} b e + a e^{2}} \right |}\right )}{3 \, \sqrt {b e} d {\left | e \right |}^{3}} \] Input:

integrate((e*(d*x+c))^(1/2)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^ 
(1/2),x, algorithm="giac")
 

Output:

-2/3*e^4*log(abs(-2*sqrt(b*e)*(3*sqrt(d*e*x + c*e)*c - (3*sqrt(d*e*x + c*e 
)*c*e - (d*e*x + c*e)^(3/2))/e) + 2*sqrt((3*sqrt(d*e*x + c*e)*c - (3*sqrt( 
d*e*x + c*e)*c*e - (d*e*x + c*e)^(3/2))/e)^2*b*e + a*e^2)))/(sqrt(b*e)*d*a 
bs(e)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx=\int \frac {\sqrt {e\,\left (c+d\,x\right )}}{\sqrt {b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}} \,d x \] Input:

int((e*(c + d*x))^(1/2)/(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x 
^2)^(1/2),x)
 

Output:

int((e*(c + d*x))^(1/2)/(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x 
^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt {e (c+d x)}}{\sqrt {a+b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3}} \, dx=\frac {\sqrt {e}\, \sqrt {b}\, \left (-\mathrm {log}\left (\sqrt {b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}-\sqrt {b}\, \sqrt {d x +c}\, c -\sqrt {b}\, \sqrt {d x +c}\, d x \right )+\mathrm {log}\left (\sqrt {b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}+\sqrt {b}\, \sqrt {d x +c}\, c +\sqrt {b}\, \sqrt {d x +c}\, d x \right )\right )}{3 b d} \] Input:

int((e*(d*x+c))^(1/2)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^(1/2), 
x)
 

Output:

(sqrt(e)*sqrt(b)*( - log(sqrt(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x**2 
+ b*d**3*x**3) - sqrt(b)*sqrt(c + d*x)*c - sqrt(b)*sqrt(c + d*x)*d*x) + lo 
g(sqrt(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3) + sqrt(b 
)*sqrt(c + d*x)*c + sqrt(b)*sqrt(c + d*x)*d*x)))/(3*b*d)