\(\int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx\) [479]

Optimal result

Integrand size = 19, antiderivative size = 119 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{5 \sqrt {d} (b c-a d)^{3/2} \sqrt [4]{c+d x}} \] Output:

4/5*(b*x+a)^(1/2)/(-a*d+b*c)/(d*x+c)^(5/4)+12/5*b*(b*(d*x+c)/(-a*d+b*c))^( 
1/4)*EllipticE(sin(1/2*arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))),2^( 
1/2))/d^(1/2)/(-a*d+b*c)^(3/2)/(d*x+c)^(1/4)
 

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{9/4}} \] Input:

Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(9/4)),x]
 

Output:

(2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(9/4)*Hypergeometric2F1[1/2, 
9/4, 3/2, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(c + d*x)^(9/4))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(287\) vs. \(2(119)=238\).

Time = 0.47 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.41, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {61, 61, 73, 836, 765, 762, 1390, 1388, 327}

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.

\(\displaystyle \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {3 b \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}}dx}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\)

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\)

Input:

Int[1/(Sqrt[a + b*x]*(c + d*x)^(9/4)),x]
 

Output:

(4*Sqrt[a + b*x])/(5*(b*c - a*d)*(c + d*x)^(5/4)) + (3*b*((4*Sqrt[a + b*x] 
)/((b*c - a*d)*(c + d*x)^(1/4)) - (4*b*(((b*c - a*d)^(3/4)*Sqrt[1 - (b*(c 
+ d*x))/(b*c - a*d)]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d 
)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) - ((b*c - a*d 
)^(3/4)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4)*(c + 
 d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + 
d*x))/d])))/(d*(b*c - a*d))))/(5*(b*c - a*d))
 

Defintions of rubi rules used

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, 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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

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

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [F]

\[\int \frac {1}{\sqrt {b x +a}\, \left (x d +c \right )^{\frac {9}{4}}}d x\]

Input:

int(1/(b*x+a)^(1/2)/(d*x+c)^(9/4),x)
 

Output:

int(1/(b*x+a)^(1/2)/(d*x+c)^(9/4),x)
 

Fricas [F]

\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(9/4),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {9}{4}}}\, dx \] Input:

integrate(1/(b*x+a)**(1/2)/(d*x+c)**(9/4),x)
 

Output:

Integral(1/(sqrt(a + b*x)*(c + d*x)**(9/4)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(9/4),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(b*x + a)*(d*x + c)^(9/4)), x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(9/4),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(b*x + a)*(d*x + c)^(9/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{9/4}} \,d x \] Input:

int(1/((a + b*x)^(1/2)*(c + d*x)^(9/4)),x)
 

Output:

int(1/((a + b*x)^(1/2)*(c + d*x)^(9/4)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\frac {-4 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}-3 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{4} x^{4}-2 b^{2} c \,d^{3} x^{4}+3 a^{2} d^{4} x^{3}+7 a b c \,d^{3} x^{3}-6 b^{2} c^{2} d^{2} x^{3}+9 a^{2} c \,d^{3} x^{2}+3 a b \,c^{2} d^{2} x^{2}-6 b^{2} c^{3} d \,x^{2}+9 a^{2} c^{2} d^{2} x -3 a b \,c^{3} d x -2 b^{2} c^{4} x +3 a^{2} c^{3} d -2 a b \,c^{4}}d x \right ) a b c \,d^{2}-3 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{4} x^{4}-2 b^{2} c \,d^{3} x^{4}+3 a^{2} d^{4} x^{3}+7 a b c \,d^{3} x^{3}-6 b^{2} c^{2} d^{2} x^{3}+9 a^{2} c \,d^{3} x^{2}+3 a b \,c^{2} d^{2} x^{2}-6 b^{2} c^{3} d \,x^{2}+9 a^{2} c^{2} d^{2} x -3 a b \,c^{3} d x -2 b^{2} c^{4} x +3 a^{2} c^{3} d -2 a b \,c^{4}}d x \right ) a b \,d^{3} x +2 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{4} x^{4}-2 b^{2} c \,d^{3} x^{4}+3 a^{2} d^{4} x^{3}+7 a b c \,d^{3} x^{3}-6 b^{2} c^{2} d^{2} x^{3}+9 a^{2} c \,d^{3} x^{2}+3 a b \,c^{2} d^{2} x^{2}-6 b^{2} c^{3} d \,x^{2}+9 a^{2} c^{2} d^{2} x -3 a b \,c^{3} d x -2 b^{2} c^{4} x +3 a^{2} c^{3} d -2 a b \,c^{4}}d x \right ) b^{2} c^{2} d +2 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{4} x^{4}-2 b^{2} c \,d^{3} x^{4}+3 a^{2} d^{4} x^{3}+7 a b c \,d^{3} x^{3}-6 b^{2} c^{2} d^{2} x^{3}+9 a^{2} c \,d^{3} x^{2}+3 a b \,c^{2} d^{2} x^{2}-6 b^{2} c^{3} d \,x^{2}+9 a^{2} c^{2} d^{2} x -3 a b \,c^{3} d x -2 b^{2} c^{4} x +3 a^{2} c^{3} d -2 a b \,c^{4}}d x \right ) b^{2} c \,d^{2} x}{\sqrt {d x +c}\, \left (3 a \,d^{2} x -2 b c d x +3 a c d -2 b \,c^{2}\right )} \] Input:

int(1/(b*x+a)^(1/2)/(d*x+c)^(9/4),x)
 

Output:

( - 4*(c + d*x)**(1/4)*sqrt(a + b*x) - 3*sqrt(c + d*x)*int(((c + d*x)**(3/ 
4)*sqrt(a + b*x)*x)/(3*a**2*c**3*d + 9*a**2*c**2*d**2*x + 9*a**2*c*d**3*x* 
*2 + 3*a**2*d**4*x**3 - 2*a*b*c**4 - 3*a*b*c**3*d*x + 3*a*b*c**2*d**2*x**2 
 + 7*a*b*c*d**3*x**3 + 3*a*b*d**4*x**4 - 2*b**2*c**4*x - 6*b**2*c**3*d*x** 
2 - 6*b**2*c**2*d**2*x**3 - 2*b**2*c*d**3*x**4),x)*a*b*c*d**2 - 3*sqrt(c + 
 d*x)*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**2*c**3*d + 9*a**2*c**2* 
d**2*x + 9*a**2*c*d**3*x**2 + 3*a**2*d**4*x**3 - 2*a*b*c**4 - 3*a*b*c**3*d 
*x + 3*a*b*c**2*d**2*x**2 + 7*a*b*c*d**3*x**3 + 3*a*b*d**4*x**4 - 2*b**2*c 
**4*x - 6*b**2*c**3*d*x**2 - 6*b**2*c**2*d**2*x**3 - 2*b**2*c*d**3*x**4),x 
)*a*b*d**3*x + 2*sqrt(c + d*x)*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a 
**2*c**3*d + 9*a**2*c**2*d**2*x + 9*a**2*c*d**3*x**2 + 3*a**2*d**4*x**3 - 
2*a*b*c**4 - 3*a*b*c**3*d*x + 3*a*b*c**2*d**2*x**2 + 7*a*b*c*d**3*x**3 + 3 
*a*b*d**4*x**4 - 2*b**2*c**4*x - 6*b**2*c**3*d*x**2 - 6*b**2*c**2*d**2*x** 
3 - 2*b**2*c*d**3*x**4),x)*b**2*c**2*d + 2*sqrt(c + d*x)*int(((c + d*x)**( 
3/4)*sqrt(a + b*x)*x)/(3*a**2*c**3*d + 9*a**2*c**2*d**2*x + 9*a**2*c*d**3* 
x**2 + 3*a**2*d**4*x**3 - 2*a*b*c**4 - 3*a*b*c**3*d*x + 3*a*b*c**2*d**2*x* 
*2 + 7*a*b*c*d**3*x**3 + 3*a*b*d**4*x**4 - 2*b**2*c**4*x - 6*b**2*c**3*d*x 
**2 - 6*b**2*c**2*d**2*x**3 - 2*b**2*c*d**3*x**4),x)*b**2*c*d**2*x)/(sqrt( 
c + d*x)*(3*a*c*d + 3*a*d**2*x - 2*b*c**2 - 2*b*c*d*x))