\(\int \frac {(c+d x)^{3/4}}{(a+b x)^{9/2}} \, dx\) [443]

Optimal result

Integrand size = 19, antiderivative size = 216 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{9/2}} \, dx=-\frac {3 d}{35 b^2 (a+b x)^{5/2} \sqrt [4]{c+d x}}+\frac {d^2}{70 b^2 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}-\frac {d^3}{20 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}-\frac {3 d^{7/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{20 b^2 (b c-a d)^{5/2} \sqrt [4]{c+d x}} \] Output:

-3/35*d/b^2/(b*x+a)^(5/2)/(d*x+c)^(1/4)+1/70*d^2/b^2/(-a*d+b*c)/(b*x+a)^(3 
/2)/(d*x+c)^(1/4)-1/20*d^3/b^2/(-a*d+b*c)^2/(b*x+a)^(1/2)/(d*x+c)^(1/4)-2/ 
7*(d*x+c)^(3/4)/b/(b*x+a)^(7/2)-3/20*d^(7/2)*(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)) 
/b^2/(-a*d+b*c)^(5/2)/(d*x+c)^(1/4)
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.34 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{9/2}} \, dx=-\frac {2 (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {3}{4},-\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{7 b (a+b x)^{7/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \] Input:

Integrate[(c + d*x)^(3/4)/(a + b*x)^(9/2),x]
 

Output:

(-2*(c + d*x)^(3/4)*Hypergeometric2F1[-7/2, -3/4, -5/2, (d*(a + b*x))/(-(b 
*c) + a*d)])/(7*b*(a + b*x)^(7/2)*((b*(c + d*x))/(b*c - a*d))^(3/4))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.69, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {57, 61, 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.

\(\displaystyle \int \frac {(c+d x)^{3/4}}{(a+b x)^{9/2}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {3 d \int \frac {1}{(a+b x)^{7/2} \sqrt [4]{c+d x}}dx}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {3 d \left (-\frac {7 d \int \frac {1}{(a+b x)^{5/2} \sqrt [4]{c+d x}}dx}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \int \frac {1}{(a+b x)^{3/2} \sqrt [4]{c+d x}}dx}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {d \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}}dx}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {3 d \left (-\frac {7 d \left (-\frac {d \left (\frac {2 \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{10 (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{14 b}-\frac {2 (c+d x)^{3/4}}{7 b (a+b x)^{7/2}}\)

Input:

Int[(c + d*x)^(3/4)/(a + b*x)^(9/2),x]
 

Output:

(-2*(c + d*x)^(3/4))/(7*b*(a + b*x)^(7/2)) + (3*d*((-2*(c + d*x)^(3/4))/(5 
*(b*c - a*d)*(a + b*x)^(5/2)) - (7*d*((-2*(c + d*x)^(3/4))/(3*(b*c - a*d)* 
(a + b*x)^(3/2)) - (d*((-2*(c + d*x)^(3/4))/((b*c - a*d)*Sqrt[a + b*x]) + 
(2*(((b*c - a*d)^(3/4)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d)]*EllipticE[ArcSi 
n[(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])))/(b*c - a*d)))/(2*(b*c - 
a*d))))/(10*(b*c - a*d))))/(14*b)
 

Defintions of rubi rules used

rule 57

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

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]

\[\int \frac {\left (x d +c \right )^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {9}{2}}}d x\]

Input:

int((d*x+c)^(3/4)/(b*x+a)^(9/2),x)
 

Output:

int((d*x+c)^(3/4)/(b*x+a)^(9/2),x)
 

Fricas [F]

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

integrate((d*x+c)^(3/4)/(b*x+a)^(9/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x + a)*(d*x + c)^(3/4)/(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3 
*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5), x)
 

Sympy [F]

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

integrate((d*x+c)**(3/4)/(b*x+a)**(9/2),x)
 

Output:

Integral((c + d*x)**(3/4)/(a + b*x)**(9/2), x)
 

Maxima [F]

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

integrate((d*x+c)^(3/4)/(b*x+a)^(9/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((d*x + c)^(3/4)/(b*x + a)^(9/2), x)
 

Giac [F]

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

integrate((d*x+c)^(3/4)/(b*x+a)^(9/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(3/4)/(b*x + a)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{9/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/4}}{{\left (a+b\,x\right )}^{9/2}} \,d x \] Input:

int((c + d*x)^(3/4)/(a + b*x)^(9/2),x)
 

Output:

int((c + d*x)^(3/4)/(a + b*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{9/2}} \, dx=\text {too large to display} \] Input:

int((d*x+c)^(3/4)/(b*x+a)^(9/2),x)
 

Output:

(4*(c + d*x)**(3/4)*sqrt(a + b*x)*c + 9*int(((c + d*x)**(3/4)*sqrt(a + b*x 
)*x)/(3*a**6*c*d + 3*a**6*d**2*x - 14*a**5*b*c**2 + a**5*b*c*d*x + 15*a**5 
*b*d**2*x**2 - 70*a**4*b**2*c**2*x - 40*a**4*b**2*c*d*x**2 + 30*a**4*b**2* 
d**2*x**3 - 140*a**3*b**3*c**2*x**2 - 110*a**3*b**3*c*d*x**3 + 30*a**3*b** 
3*d**2*x**4 - 140*a**2*b**4*c**2*x**3 - 125*a**2*b**4*c*d*x**4 + 15*a**2*b 
**4*d**2*x**5 - 70*a*b**5*c**2*x**4 - 67*a*b**5*c*d*x**5 + 3*a*b**5*d**2*x 
**6 - 14*b**6*c**2*x**5 - 14*b**6*c*d*x**6),x)*a**6*d**3 - 51*int(((c + d* 
x)**(3/4)*sqrt(a + b*x)*x)/(3*a**6*c*d + 3*a**6*d**2*x - 14*a**5*b*c**2 + 
a**5*b*c*d*x + 15*a**5*b*d**2*x**2 - 70*a**4*b**2*c**2*x - 40*a**4*b**2*c* 
d*x**2 + 30*a**4*b**2*d**2*x**3 - 140*a**3*b**3*c**2*x**2 - 110*a**3*b**3* 
c*d*x**3 + 30*a**3*b**3*d**2*x**4 - 140*a**2*b**4*c**2*x**3 - 125*a**2*b** 
4*c*d*x**4 + 15*a**2*b**4*d**2*x**5 - 70*a*b**5*c**2*x**4 - 67*a*b**5*c*d* 
x**5 + 3*a*b**5*d**2*x**6 - 14*b**6*c**2*x**5 - 14*b**6*c*d*x**6),x)*a**5* 
b*c*d**2 + 36*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**6*c*d + 3*a**6* 
d**2*x - 14*a**5*b*c**2 + a**5*b*c*d*x + 15*a**5*b*d**2*x**2 - 70*a**4*b** 
2*c**2*x - 40*a**4*b**2*c*d*x**2 + 30*a**4*b**2*d**2*x**3 - 140*a**3*b**3* 
c**2*x**2 - 110*a**3*b**3*c*d*x**3 + 30*a**3*b**3*d**2*x**4 - 140*a**2*b** 
4*c**2*x**3 - 125*a**2*b**4*c*d*x**4 + 15*a**2*b**4*d**2*x**5 - 70*a*b**5* 
c**2*x**4 - 67*a*b**5*c*d*x**5 + 3*a*b**5*d**2*x**6 - 14*b**6*c**2*x**5 - 
14*b**6*c*d*x**6),x)*a**5*b*d**3*x + 42*int(((c + d*x)**(3/4)*sqrt(a + ...