\(\int (a+b x)^{3/4} (c+d x)^{5/4} \, dx\) [573]

Optimal result

Integrand size = 19, antiderivative size = 205 \[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\frac {5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac {5 (b c-a d)^3 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac {5 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}} \] Output:

5/96*(-a*d+b*c)^2*(b*x+a)^(3/4)*(d*x+c)^(1/4)/b^2/d+5/24*(-a*d+b*c)*(b*x+a 
)^(7/4)*(d*x+c)^(1/4)/b^2+1/3*(b*x+a)^(7/4)*(d*x+c)^(5/4)/b+5/64*(-a*d+b*c 
)^3*arctan(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(9/4)/d^(7/4)-5/ 
64*(-a*d+b*c)^3*arctanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(9/ 
4)/d^(7/4)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.85 \[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\frac {2 \sqrt [4]{b} d^{3/4} (a+b x)^{3/4} \sqrt [4]{c+d x} \left (-15 a^2 d^2+6 a b d (7 c+2 d x)+b^2 \left (5 c^2+52 c d x+32 d^2 x^2\right )\right )-15 (b c-a d)^3 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )-15 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{192 b^{9/4} d^{7/4}} \] Input:

Integrate[(a + b*x)^(3/4)*(c + d*x)^(5/4),x]
 

Output:

(2*b^(1/4)*d^(3/4)*(a + b*x)^(3/4)*(c + d*x)^(1/4)*(-15*a^2*d^2 + 6*a*b*d* 
(7*c + 2*d*x) + b^2*(5*c^2 + 52*c*d*x + 32*d^2*x^2)) - 15*(b*c - a*d)^3*Ar 
cTan[(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))] - 15*(b*c - a*d) 
^3*ArcTanh[(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))])/(192*b^(9 
/4)*d^(7/4))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {60, 60, 60, 73, 854, 27, 827, 218, 221}

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.

\(\displaystyle \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \int (a+b x)^{3/4} \sqrt [4]{c+d x}dx}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}}dx}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}}dx}{4 d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{3/4}}d\sqrt [4]{a+b x}}{b d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {b \sqrt {a+b x}}{b-d (a+b x)}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{b-d (a+b x)}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt {d}}-\frac {\int \frac {1}{\sqrt {b}+\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt {d}}\right )}{d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}\right )}{d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}\right )}{d}\right )}{8 b}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 b}\right )}{12 b}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}\)

Input:

Int[(a + b*x)^(3/4)*(c + d*x)^(5/4),x]
 

Output:

((a + b*x)^(7/4)*(c + d*x)^(5/4))/(3*b) + (5*(b*c - a*d)*(((a + b*x)^(7/4) 
*(c + d*x)^(1/4))/(2*b) + ((b*c - a*d)*(((a + b*x)^(3/4)*(c + d*x)^(1/4))/ 
d - (3*(b*c - a*d)*(-1/2*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c - (a 
*d)/b + (d*(a + b*x))/b)^(1/4))]/(b^(1/4)*d^(3/4)) + ArcTanh[(d^(1/4)*(a + 
 b*x)^(1/4))/(b^(1/4)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))]/(2*b^(1/4)*d 
^(3/4))))/d))/(8*b)))/(12*b)
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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 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 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 854

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

Maple [F]

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

Input:

int((b*x+a)^(3/4)*(d*x+c)^(5/4),x)
 

Output:

int((b*x+a)^(3/4)*(d*x+c)^(5/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1786, normalized size of antiderivative = 8.71 \[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/4)*(d*x+c)^(5/4),x, algorithm="fricas")
 

Output:

-1/384*(15*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 2 
20*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b 
^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d 
^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4) 
*log(-5*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x + a)^(3/ 
4)*(d*x + c)^(1/4) + (b^3*d^2*x + a*b^2*d^2)*((b^12*c^12 - 12*a*b^11*c^11* 
d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792 
*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4 
*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + 
 a^12*d^12)/(b^9*d^7))^(1/4))/(b*x + a)) - 15*b^2*d*((b^12*c^12 - 12*a*b^1 
1*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^ 
4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495* 
a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c 
*d^11 + a^12*d^12)/(b^9*d^7))^(1/4)*log(-5*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a 
^2*b*c*d^2 - a^3*d^3)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b^3*d^2*x + a*b^2 
*d^2)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9* 
c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 
- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^1 
0*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4))/(b*x + a) 
) + 15*I*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - ...
 

Sympy [F]

\[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\int \left (a + b x\right )^{\frac {3}{4}} \left (c + d x\right )^{\frac {5}{4}}\, dx \] Input:

integrate((b*x+a)**(3/4)*(d*x+c)**(5/4),x)
 

Output:

Integral((a + b*x)**(3/4)*(c + d*x)**(5/4), x)
 

Maxima [F]

\[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {5}{4}} \,d x } \] Input:

integrate((b*x+a)^(3/4)*(d*x+c)^(5/4),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(3/4)*(d*x + c)^(5/4), x)
 

Giac [F(-1)]

Timed out. \[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)^(3/4)*(d*x+c)^(5/4),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\int {\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{5/4} \,d x \] Input:

int((a + b*x)^(3/4)*(c + d*x)^(5/4),x)
 

Output:

int((a + b*x)^(3/4)*(c + d*x)^(5/4), x)
 

Reduce [F]

\[ \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx=\frac {-48 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a^{2} c d +12 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a^{2} d^{2} x +176 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a b \,c^{2}+88 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a b c d x +32 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a b \,d^{2} x^{2}+156 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} b^{2} c^{2} x +96 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} b^{2} c d \,x^{2}-15 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} x}{a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+a^{2} d^{2} x +4 a b c d x +3 b^{2} c^{2} x +a^{2} c d +3 a b \,c^{2}}d x \right ) a^{4} d^{4}+90 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} x}{a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+a^{2} d^{2} x +4 a b c d x +3 b^{2} c^{2} x +a^{2} c d +3 a b \,c^{2}}d x \right ) a^{2} b^{2} c^{2} d^{2}-120 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} x}{a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+a^{2} d^{2} x +4 a b c d x +3 b^{2} c^{2} x +a^{2} c d +3 a b \,c^{2}}d x \right ) a \,b^{3} c^{3} d +45 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} x}{a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+a^{2} d^{2} x +4 a b c d x +3 b^{2} c^{2} x +a^{2} c d +3 a b \,c^{2}}d x \right ) b^{4} c^{4}}{96 b \left (a d +3 b c \right )} \] Input:

int((b*x+a)^(3/4)*(d*x+c)^(5/4),x)
 

Output:

( - 48*(c + d*x)**(1/4)*(a + b*x)**(3/4)*a**2*c*d + 12*(c + d*x)**(1/4)*(a 
 + b*x)**(3/4)*a**2*d**2*x + 176*(c + d*x)**(1/4)*(a + b*x)**(3/4)*a*b*c** 
2 + 88*(c + d*x)**(1/4)*(a + b*x)**(3/4)*a*b*c*d*x + 32*(c + d*x)**(1/4)*( 
a + b*x)**(3/4)*a*b*d**2*x**2 + 156*(c + d*x)**(1/4)*(a + b*x)**(3/4)*b**2 
*c**2*x + 96*(c + d*x)**(1/4)*(a + b*x)**(3/4)*b**2*c*d*x**2 - 15*int(((c 
+ d*x)**(1/4)*(a + b*x)**(3/4)*x)/(a**2*c*d + a**2*d**2*x + 3*a*b*c**2 + 4 
*a*b*c*d*x + a*b*d**2*x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x**2),x)*a**4*d**4 
 + 90*int(((c + d*x)**(1/4)*(a + b*x)**(3/4)*x)/(a**2*c*d + a**2*d**2*x + 
3*a*b*c**2 + 4*a*b*c*d*x + a*b*d**2*x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x**2 
),x)*a**2*b**2*c**2*d**2 - 120*int(((c + d*x)**(1/4)*(a + b*x)**(3/4)*x)/( 
a**2*c*d + a**2*d**2*x + 3*a*b*c**2 + 4*a*b*c*d*x + a*b*d**2*x**2 + 3*b**2 
*c**2*x + 3*b**2*c*d*x**2),x)*a*b**3*c**3*d + 45*int(((c + d*x)**(1/4)*(a 
+ b*x)**(3/4)*x)/(a**2*c*d + a**2*d**2*x + 3*a*b*c**2 + 4*a*b*c*d*x + a*b* 
d**2*x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x**2),x)*b**4*c**4)/(96*b*(a*d + 3* 
b*c))