\(\int \frac {\sqrt {e x}}{(c+d x)^3 (a+b x^2)^3} \, dx\) [459]

Optimal result

Integrand size = 24, antiderivative size = 886 \[ \int \frac {\sqrt {e x}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {d^3 \left (b c^2-3 a d^2\right ) \sqrt {e x}}{4 a \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {d^3 \left (3 b^2 c^4-28 a b c^2 d^2+a^2 d^4\right ) \sqrt {e x}}{4 a c \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {\sqrt {e x} (a d+b c x)}{4 a \left (b c^2+a d^2\right ) (c+d x)^2 \left (a+b x^2\right )^2}-\frac {b \sqrt {e x} \left (11 a^3 d^5-5 b^3 c^5 x-a^2 b c d^3 (58 c-43 d x)-a b^2 c^3 d (5 c+26 d x)\right )}{16 a^2 \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}-\frac {d^{9/2} \left (143 b^2 c^4-50 a b c^2 d^2-a^2 d^4\right ) \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 c^{3/2} \left (b c^2+a d^2\right )^5}-\frac {b^{3/4} \left (5 b^{7/2} c^7-9 \sqrt {a} b^3 c^6 d+43 a b^{5/2} c^5 d^2-95 a^{3/2} b^2 c^4 d^3+455 a^2 b^{3/2} c^3 d^4+605 a^{5/2} b c^2 d^5-351 a^3 \sqrt {b} c d^6-77 a^{7/2} d^7\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^5}+\frac {b^{3/4} \left (5 b^{7/2} c^7-9 \sqrt {a} b^3 c^6 d+43 a b^{5/2} c^5 d^2-95 a^{3/2} b^2 c^4 d^3+455 a^2 b^{3/2} c^3 d^4+605 a^{5/2} b c^2 d^5-351 a^3 \sqrt {b} c d^6-77 a^{7/2} d^7\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^5}-\frac {b^{3/4} \left (5 b^{7/2} c^7+9 \sqrt {a} b^3 c^6 d+43 a b^{5/2} c^5 d^2+95 a^{3/2} b^2 c^4 d^3+455 a^2 b^{3/2} c^3 d^4-605 a^{5/2} b c^2 d^5-351 a^3 \sqrt {b} c d^6+77 a^{7/2} d^7\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^5} \] Output:

1/4*d^3*(-3*a*d^2+b*c^2)*(e*x)^(1/2)/a/(a*d^2+b*c^2)^3/(d*x+c)^2+1/4*d^3*( 
a^2*d^4-28*a*b*c^2*d^2+3*b^2*c^4)*(e*x)^(1/2)/a/c/(a*d^2+b*c^2)^4/(d*x+c)+ 
1/4*(e*x)^(1/2)*(b*c*x+a*d)/a/(a*d^2+b*c^2)/(d*x+c)^2/(b*x^2+a)^2-1/16*b*( 
e*x)^(1/2)*(11*a^3*d^5-5*b^3*c^5*x-a^2*b*c*d^3*(-43*d*x+58*c)-a*b^2*c^3*d* 
(26*d*x+5*c))/a^2/(a*d^2+b*c^2)^4/(b*x^2+a)-1/4*d^(9/2)*(-a^2*d^4-50*a*b*c 
^2*d^2+143*b^2*c^4)*e^(1/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^ 
(3/2)/(a*d^2+b*c^2)^5-1/64*b^(3/4)*(5*b^(7/2)*c^7-9*a^(1/2)*b^3*c^6*d+43*a 
*b^(5/2)*c^5*d^2-95*a^(3/2)*b^2*c^4*d^3+455*a^2*b^(3/2)*c^3*d^4+605*a^(5/2 
)*b*c^2*d^5-351*a^3*b^(1/2)*c*d^6-77*a^(7/2)*d^7)*e^(1/2)*arctan(1-2^(1/2) 
*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(9/4)/(a*d^2+b*c^2)^5+1/64 
*b^(3/4)*(5*b^(7/2)*c^7-9*a^(1/2)*b^3*c^6*d+43*a*b^(5/2)*c^5*d^2-95*a^(3/2 
)*b^2*c^4*d^3+455*a^2*b^(3/2)*c^3*d^4+605*a^(5/2)*b*c^2*d^5-351*a^3*b^(1/2 
)*c*d^6-77*a^(7/2)*d^7)*e^(1/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/ 
4)/e^(1/2))*2^(1/2)/a^(9/4)/(a*d^2+b*c^2)^5-1/64*b^(3/4)*(5*b^(7/2)*c^7+9* 
a^(1/2)*b^3*c^6*d+43*a*b^(5/2)*c^5*d^2+95*a^(3/2)*b^2*c^4*d^3+455*a^2*b^(3 
/2)*c^3*d^4-605*a^(5/2)*b*c^2*d^5-351*a^3*b^(1/2)*c*d^6+77*a^(7/2)*d^7)*e^ 
(1/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2) 
*x))*2^(1/2)/a^(9/4)/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 4.25 (sec) , antiderivative size = 686, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {e x}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {\sqrt {e x} \left (\frac {4 \left (b c^2+a d^2\right ) \left (4 a^5 d^7 (-c+d x)+5 b^5 c^6 x^3 (c+d x)^2+a b^4 c^4 x (c+d x)^2 \left (9 c^2-3 c d x+38 d^2 x^2\right )+a^4 b d^5 \left (-119 c^3-130 c^2 d x-27 c d^2 x^2+8 d^3 x^3\right )+a^2 b^3 c^2 d \left (9 c^5+48 c^4 d x+147 c^3 d^2 x^2+123 c^2 d^3 x^3-148 c d^4 x^4-155 d^5 x^5\right )+a^3 b^2 d^3 \left (86 c^5+97 c^4 d x-279 c^3 d^2 x^2-289 c^2 d^3 x^3-19 c d^4 x^4+4 d^5 x^5\right )\right )}{a^2 c (c+d x)^2 \left (a+b x^2\right )^2}+\frac {16 d^{9/2} \left (-143 b^2 c^4+50 a b c^2 d^2+a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{3/2} \sqrt {x}}+\frac {\sqrt {2} b^{3/4} \left (-5 b^{7/2} c^7+9 \sqrt {a} b^3 c^6 d-43 a b^{5/2} c^5 d^2+95 a^{3/2} b^2 c^4 d^3-455 a^2 b^{3/2} c^3 d^4-605 a^{5/2} b c^2 d^5+351 a^3 \sqrt {b} c d^6+77 a^{7/2} d^7\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} \sqrt {x}}-\frac {\sqrt {2} b^{3/4} \left (5 b^{7/2} c^7+9 \sqrt {a} b^3 c^6 d+43 a b^{5/2} c^5 d^2+95 a^{3/2} b^2 c^4 d^3+455 a^2 b^{3/2} c^3 d^4-605 a^{5/2} b c^2 d^5-351 a^3 \sqrt {b} c d^6+77 a^{7/2} d^7\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} \sqrt {x}}\right )}{64 \left (b c^2+a d^2\right )^5} \] Input:

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

Output:

(Sqrt[e*x]*((4*(b*c^2 + a*d^2)*(4*a^5*d^7*(-c + d*x) + 5*b^5*c^6*x^3*(c + 
d*x)^2 + a*b^4*c^4*x*(c + d*x)^2*(9*c^2 - 3*c*d*x + 38*d^2*x^2) + a^4*b*d^ 
5*(-119*c^3 - 130*c^2*d*x - 27*c*d^2*x^2 + 8*d^3*x^3) + a^2*b^3*c^2*d*(9*c 
^5 + 48*c^4*d*x + 147*c^3*d^2*x^2 + 123*c^2*d^3*x^3 - 148*c*d^4*x^4 - 155* 
d^5*x^5) + a^3*b^2*d^3*(86*c^5 + 97*c^4*d*x - 279*c^3*d^2*x^2 - 289*c^2*d^ 
3*x^3 - 19*c*d^4*x^4 + 4*d^5*x^5)))/(a^2*c*(c + d*x)^2*(a + b*x^2)^2) + (1 
6*d^(9/2)*(-143*b^2*c^4 + 50*a*b*c^2*d^2 + a^2*d^4)*ArcTan[(Sqrt[d]*Sqrt[x 
])/Sqrt[c]])/(c^(3/2)*Sqrt[x]) + (Sqrt[2]*b^(3/4)*(-5*b^(7/2)*c^7 + 9*Sqrt 
[a]*b^3*c^6*d - 43*a*b^(5/2)*c^5*d^2 + 95*a^(3/2)*b^2*c^4*d^3 - 455*a^2*b^ 
(3/2)*c^3*d^4 - 605*a^(5/2)*b*c^2*d^5 + 351*a^3*Sqrt[b]*c*d^6 + 77*a^(7/2) 
*d^7)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^ 
(9/4)*Sqrt[x]) - (Sqrt[2]*b^(3/4)*(5*b^(7/2)*c^7 + 9*Sqrt[a]*b^3*c^6*d + 4 
3*a*b^(5/2)*c^5*d^2 + 95*a^(3/2)*b^2*c^4*d^3 + 455*a^2*b^(3/2)*c^3*d^4 - 6 
05*a^(5/2)*b*c^2*d^5 - 351*a^3*Sqrt[b]*c*d^6 + 77*a^(7/2)*d^7)*ArcTanh[(Sq 
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(9/4)*Sqrt[x]))) 
/(64*(b*c^2 + a*d^2)^5)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1977\) vs. \(2(886)=1772\).

Time = 4.74 (sec) , antiderivative size = 1977, normalized size of antiderivative = 2.23, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {2 b^2 d^2 \sqrt {e x} \left (3 c \left (b c^2-a d^2\right )-d x \left (5 b c^2-a d^2\right )\right )}{\left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^4}+\frac {b^2 \sqrt {e x} \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{\left (a+b x^2\right )^3 \left (a d^2+b c^2\right )^3}+\frac {3 b^2 d^4 \sqrt {e x} \left (c \left (5 b c^2-3 a d^2\right )-d x \left (7 b c^2-a d^2\right )\right )}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )^5}+\frac {3 b d^6 \sqrt {e x} \left (7 b c^2-a d^2\right )}{(c+d x) \left (a d^2+b c^2\right )^5}+\frac {6 b c d^6 \sqrt {e x}}{(c+d x)^2 \left (a d^2+b c^2\right )^4}+\frac {d^6 \sqrt {e x}}{(c+d x)^3 \left (a d^2+b c^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {e x} d^5}{4 c \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {6 b c \sqrt {e x} d^5}{\left (b c^2+a d^2\right )^4 (c+d x)}-\frac {\sqrt {e x} d^5}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{9/2}}{4 c^{3/2} \left (b c^2+a d^2\right )^3}+\frac {6 b \sqrt {c} \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{9/2}}{\left (b c^2+a d^2\right )^4}-\frac {6 b \sqrt {c} \left (7 b c^2-a d^2\right ) \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{9/2}}{\left (b c^2+a d^2\right )^5}-\frac {3 b^{3/4} \left (5 b^{3/2} c^3+7 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c-a^{3/2} d^3\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} \sqrt [4]{a} \left (b c^2+a d^2\right )^5}+\frac {3 b^{3/4} \left (5 b^{3/2} c^3+7 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c-a^{3/2} d^3\right ) \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} \sqrt [4]{a} \left (b c^2+a d^2\right )^5}+\frac {3 b^{3/4} \left (5 b^{3/2} c^3-7 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c+a^{3/2} d^3\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^4}{2 \sqrt {2} \sqrt [4]{a} \left (b c^2+a d^2\right )^5}-\frac {3 b^{3/4} \left (5 b^{3/2} c^3-7 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c+a^{3/2} d^3\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^4}{2 \sqrt {2} \sqrt [4]{a} \left (b c^2+a d^2\right )^5}-\frac {b^{3/4} \left (3 b^{3/2} c^3-5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c+a^{3/2} d^3\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{2 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^4}+\frac {b^{3/4} \left (3 b^{3/2} c^3-5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c+a^{3/2} d^3\right ) \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{2 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^4}+\frac {b^{3/4} \left (3 b^{3/2} c^3+5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c-a^{3/2} d^3\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{4 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^4}-\frac {b^{3/4} \left (3 b^{3/2} c^3+5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c-a^{3/2} d^3\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{4 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^4}+\frac {b \sqrt {e x} \left (a d \left (5 b c^2-a d^2\right )+3 b c \left (b c^2-a d^2\right ) x\right ) d^2}{a \left (b c^2+a d^2\right )^4 \left (b x^2+a\right )}-\frac {b^{3/4} \left (5 b^{3/2} c^3-9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c+3 a^{3/2} d^3\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3}+\frac {b^{3/4} \left (5 b^{3/2} c^3-9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c+3 a^{3/2} d^3\right ) \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3}+\frac {b^{3/4} \left (5 b^{3/2} c^3+9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c-3 a^{3/2} d^3\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{64 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3}-\frac {b^{3/4} \left (5 b^{3/2} c^3+9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c-3 a^{3/2} d^3\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{64 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3}-\frac {b \sqrt {e x} \left (a d \left (3 b c^2-a d^2\right )-5 b c \left (b c^2-3 a d^2\right ) x\right )}{16 a^2 \left (b c^2+a d^2\right )^3 \left (b x^2+a\right )}+\frac {b \sqrt {e x} \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{4 a \left (b c^2+a d^2\right )^3 \left (b x^2+a\right )^2}\)

Input:

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

Output:

-1/2*(d^5*Sqrt[e*x])/((b*c^2 + a*d^2)^3*(c + d*x)^2) - (6*b*c*d^5*Sqrt[e*x 
])/((b*c^2 + a*d^2)^4*(c + d*x)) + (d^5*Sqrt[e*x])/(4*c*(b*c^2 + a*d^2)^3* 
(c + d*x)) + (b*Sqrt[e*x]*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x 
))/(4*a*(b*c^2 + a*d^2)^3*(a + b*x^2)^2) - (b*Sqrt[e*x]*(a*d*(3*b*c^2 - a* 
d^2) - 5*b*c*(b*c^2 - 3*a*d^2)*x))/(16*a^2*(b*c^2 + a*d^2)^3*(a + b*x^2)) 
+ (b*d^2*Sqrt[e*x]*(a*d*(5*b*c^2 - a*d^2) + 3*b*c*(b*c^2 - a*d^2)*x))/(a*( 
b*c^2 + a*d^2)^4*(a + b*x^2)) - (6*b*Sqrt[c]*d^(9/2)*(7*b*c^2 - a*d^2)*Sqr 
t[e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^2)^5 + (6 
*b*Sqrt[c]*d^(9/2)*Sqrt[e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/ 
(b*c^2 + a*d^2)^4 + (d^(9/2)*Sqrt[e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*S 
qrt[e])])/(4*c^(3/2)*(b*c^2 + a*d^2)^3) - (3*b^(3/4)*d^4*(5*b^(3/2)*c^3 + 
7*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 - a^(3/2)*d^3)*Sqrt[e]*ArcTan[1 - (S 
qrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(1/4)*(b*c^2 + a* 
d^2)^5) - (b^(3/4)*d^2*(3*b^(3/2)*c^3 - 5*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c* 
d^2 + a^(3/2)*d^3)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4) 
*Sqrt[e])])/(2*Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^4) - (b^(3/4)*(5*b^(3/2)*c^ 
3 - 9*Sqrt[a]*b*c^2*d - 15*a*Sqrt[b]*c*d^2 + 3*a^(3/2)*d^3)*Sqrt[e]*ArcTan 
[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sqrt[2]*a^(9/4)*( 
b*c^2 + a*d^2)^3) + (3*b^(3/4)*d^4*(5*b^(3/2)*c^3 + 7*Sqrt[a]*b*c^2*d - 3* 
a*Sqrt[b]*c*d^2 - a^(3/2)*d^3)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt...
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 803, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(2 e^{8} \left (-\frac {b \left (\frac {\frac {b^{2} c \left (63 a^{3} d^{6}+25 a^{2} b \,c^{2} d^{4}-43 a \,b^{2} c^{4} d^{2}-5 b^{3} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a^{2}}+\frac {3 b d e \left (5 a^{3} d^{6}-21 a^{2} b \,c^{2} d^{4}-25 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a}+\frac {3 b c \,e^{2} \left (25 a^{3} d^{6}+15 a^{2} b \,c^{2} d^{4}-13 a \,b^{2} c^{4} d^{2}-3 b^{3} c^{6}\right ) \left (e x \right )^{\frac {3}{2}}}{32 a}+\left (\frac {19}{32} e^{3} a^{3} d^{7}-\frac {67}{32} b \,c^{2} d^{5} e^{3} a^{2}-\frac {95}{32} a \,b^{2} c^{4} d^{3} e^{3}-\frac {9}{32} b^{3} c^{6} d \,e^{3}\right ) \sqrt {e x}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (77 a^{4} d^{7} e -605 a^{3} b \,c^{2} d^{5} e +95 a^{2} b^{2} c^{4} d^{3} e +9 a \,b^{3} c^{6} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (351 d^{6} c \,a^{3} b -455 a^{2} c^{3} d^{4} b^{2}-43 a \,c^{5} d^{2} b^{3}-5 c^{7} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{32 a^{2}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {d^{5} \left (\frac {\frac {d \left (a^{2} d^{4}-22 b \,c^{2} d^{2} a -23 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{8 c}-\frac {e \left (a^{2} d^{4}+26 b \,c^{2} d^{2} a +25 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (a^{2} d^{4}+50 b \,c^{2} d^{2} a -143 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 c \sqrt {d e c}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\right )\)	\(803\)
default	\(2 e^{8} \left (-\frac {b \left (\frac {\frac {b^{2} c \left (63 a^{3} d^{6}+25 a^{2} b \,c^{2} d^{4}-43 a \,b^{2} c^{4} d^{2}-5 b^{3} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a^{2}}+\frac {3 b d e \left (5 a^{3} d^{6}-21 a^{2} b \,c^{2} d^{4}-25 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a}+\frac {3 b c \,e^{2} \left (25 a^{3} d^{6}+15 a^{2} b \,c^{2} d^{4}-13 a \,b^{2} c^{4} d^{2}-3 b^{3} c^{6}\right ) \left (e x \right )^{\frac {3}{2}}}{32 a}+\left (\frac {19}{32} e^{3} a^{3} d^{7}-\frac {67}{32} b \,c^{2} d^{5} e^{3} a^{2}-\frac {95}{32} a \,b^{2} c^{4} d^{3} e^{3}-\frac {9}{32} b^{3} c^{6} d \,e^{3}\right ) \sqrt {e x}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (77 a^{4} d^{7} e -605 a^{3} b \,c^{2} d^{5} e +95 a^{2} b^{2} c^{4} d^{3} e +9 a \,b^{3} c^{6} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (351 d^{6} c \,a^{3} b -455 a^{2} c^{3} d^{4} b^{2}-43 a \,c^{5} d^{2} b^{3}-5 c^{7} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{32 a^{2}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {d^{5} \left (\frac {\frac {d \left (a^{2} d^{4}-22 b \,c^{2} d^{2} a -23 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{8 c}-\frac {e \left (a^{2} d^{4}+26 b \,c^{2} d^{2} a +25 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (a^{2} d^{4}+50 b \,c^{2} d^{2} a -143 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 c \sqrt {d e c}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\right )\)	\(803\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(958\)

Input:

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

Output:

2*e^8*(-b/e^7/(a*d^2+b*c^2)^5*((1/32*b^2*c*(63*a^3*d^6+25*a^2*b*c^2*d^4-43 
*a*b^2*c^4*d^2-5*b^3*c^6)/a^2*(e*x)^(7/2)+3/32*b*d*e*(5*a^3*d^6-21*a^2*b*c 
^2*d^4-25*a*b^2*c^4*d^2+b^3*c^6)/a*(e*x)^(5/2)+3/32*b*c*e^2*(25*a^3*d^6+15 
*a^2*b*c^2*d^4-13*a*b^2*c^4*d^2-3*b^3*c^6)/a*(e*x)^(3/2)+(19/32*e^3*a^3*d^ 
7-67/32*b*c^2*d^5*e^3*a^2-95/32*a*b^2*c^4*d^3*e^3-9/32*b^3*c^6*d*e^3)*(e*x 
)^(1/2))/(b*e^2*x^2+a*e^2)^2+1/32/a^2*(1/8*(77*a^4*d^7*e-605*a^3*b*c^2*d^5 
*e+95*a^2*b^2*c^4*d^3*e+9*a*b^3*c^6*d*e)*(a*e^2/b)^(1/4)/a/e^2*2^(1/2)*(ln 
((e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^ 
(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/ 
4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+1/8*(35 
1*a^3*b*c*d^6-455*a^2*b^2*c^3*d^4-43*a*b^3*c^5*d^2-5*b^4*c^7)/b/(a*e^2/b)^ 
(1/4)*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2) 
)/(e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1 
/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^ 
(1/2)-1))))+d^5/e^7/(a*d^2+b*c^2)^5*((1/8*d*(a^2*d^4-22*a*b*c^2*d^2-23*b^2 
*c^4)/c*(e*x)^(3/2)-1/8*e*(a^2*d^4+26*a*b*c^2*d^2+25*b^2*c^4)*(e*x)^(1/2)) 
/(d*e*x+c*e)^2+1/8*(a^2*d^4+50*a*b*c^2*d^2-143*b^2*c^4)/c/(d*e*c)^(1/2)*ar 
ctan(d*(e*x)^(1/2)/(d*e*c)^(1/2))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2027 vs. \(2 (741) = 1482\).

Time = 0.32 (sec) , antiderivative size = 2027, normalized size of antiderivative = 2.29 \[ \int \frac {\sqrt {e x}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/64*(2*(9*(a*b^3*e^2)^(1/4)*a*b^4*c^6*d*e + 95*(a*b^3*e^2)^(1/4)*a^2*b^3 
*c^4*d^3*e - 605*(a*b^3*e^2)^(1/4)*a^3*b^2*c^2*d^5*e + 77*(a*b^3*e^2)^(1/4 
)*a^4*b*d^7*e - 5*(a*b^3*e^2)^(3/4)*b^3*c^7 - 43*(a*b^3*e^2)^(3/4)*a*b^2*c 
^5*d^2 - 455*(a*b^3*e^2)^(3/4)*a^2*b*c^3*d^4 + 351*(a*b^3*e^2)^(3/4)*a^3*c 
*d^6)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b) 
^(1/4))/(sqrt(2)*a^3*b^6*c^10 + 5*sqrt(2)*a^4*b^5*c^8*d^2 + 10*sqrt(2)*a^5 
*b^4*c^6*d^4 + 10*sqrt(2)*a^6*b^3*c^4*d^6 + 5*sqrt(2)*a^7*b^2*c^2*d^8 + sq 
rt(2)*a^8*b*d^10) + 2*(9*(a*b^3*e^2)^(1/4)*a*b^4*c^6*d*e + 95*(a*b^3*e^2)^ 
(1/4)*a^2*b^3*c^4*d^3*e - 605*(a*b^3*e^2)^(1/4)*a^3*b^2*c^2*d^5*e + 77*(a* 
b^3*e^2)^(1/4)*a^4*b*d^7*e - 5*(a*b^3*e^2)^(3/4)*b^3*c^7 - 43*(a*b^3*e^2)^ 
(3/4)*a*b^2*c^5*d^2 - 455*(a*b^3*e^2)^(3/4)*a^2*b*c^3*d^4 + 351*(a*b^3*e^2 
)^(3/4)*a^3*c*d^6)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e 
*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a^3*b^6*c^10 + 5*sqrt(2)*a^4*b^5*c^8*d^2 + 
10*sqrt(2)*a^5*b^4*c^6*d^4 + 10*sqrt(2)*a^6*b^3*c^4*d^6 + 5*sqrt(2)*a^7*b^ 
2*c^2*d^8 + sqrt(2)*a^8*b*d^10) + (9*(a*b^3*e^2)^(1/4)*a*b^4*c^6*d*e + 95* 
(a*b^3*e^2)^(1/4)*a^2*b^3*c^4*d^3*e - 605*(a*b^3*e^2)^(1/4)*a^3*b^2*c^2*d^ 
5*e + 77*(a*b^3*e^2)^(1/4)*a^4*b*d^7*e + 5*(a*b^3*e^2)^(3/4)*b^3*c^7 + 43* 
(a*b^3*e^2)^(3/4)*a*b^2*c^5*d^2 + 455*(a*b^3*e^2)^(3/4)*a^2*b*c^3*d^4 - 35 
1*(a*b^3*e^2)^(3/4)*a^3*c*d^6)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) 
 + sqrt(a*e^2/b))/(sqrt(2)*a^3*b^6*c^10 + 5*sqrt(2)*a^4*b^5*c^8*d^2 + 1...
 

Mupad [B] (verification not implemented)

Time = 15.56 (sec) , antiderivative size = 10804, normalized size of antiderivative = 12.19 \[ \int \frac {\sqrt {e x}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(- root(21474836480*a^28*b*c^5*d^38*g^6 + 204010946560*a^27*b^2* 
c^7*d^36*g^6 + 135259257569280*a^21*b^8*c^19*d^24*g^6 + 135259257569280*a^ 
17*b^12*c^27*d^16*g^6 + 204010946560*a^11*b^18*c^39*d^4*g^6 + 520227913728 
0*a^25*b^4*c^11*d^32*g^6 + 5202279137280*a^13*b^16*c^35*d^8*g^6 + 12240656 
79360*a^26*b^3*c^9*d^34*g^6 + 16647293239296*a^24*b^5*c^13*d^30*g^6 + 4161 
8233098240*a^23*b^6*c^15*d^28*g^6 + 83236466196480*a^22*b^7*c^17*d^26*g^6 
+ 180345676759040*a^20*b^9*c^21*d^22*g^6 + 198380244434944*a^19*b^10*c^23* 
d^20*g^6 + 180345676759040*a^18*b^11*c^25*d^18*g^6 + 83236466196480*a^16*b 
^13*c^29*d^14*g^6 + 41618233098240*a^15*b^14*c^31*d^12*g^6 + 1664729323929 
6*a^14*b^15*c^33*d^10*g^6 + 1224065679360*a^12*b^17*c^37*d^6*g^6 + 2147483 
6480*a^10*b^19*c^41*d^2*g^6 + 1073741824*a^29*c^3*d^40*g^6 + 1073741824*a^ 
9*b^20*c^43*g^6 - 1178339377152*a^19*b^4*c^8*d^29*e*g^4 + 83016810496*a^21 
*b^2*c^4*d^33*e*g^4 + 4601319260160*a^10*b^13*c^26*d^11*e*g^4 - 1375731712 
*a^6*b^17*c^34*d^3*e*g^4 - 7637726920704*a^18*b^5*c^10*d^27*e*g^4 + 233035 
530240*a^20*b^3*c^6*d^31*e*g^4 + 1845493760*a^22*b*c^2*d^35*e*g^4 + 253316 
24837120*a^15*b^8*c^16*d^21*e*g^4 + 150227386368*a^9*b^14*c^28*d^9*e*g^4 - 
 115527909376*a^8*b^15*c^30*d^7*e*g^4 + 21727744622592*a^11*b^12*c^24*d^13 
*e*g^4 - 16567500800*a^7*b^16*c^32*d^5*e*g^4 - 9151132467200*a^16*b^7*c^14 
*d^23*e*g^4 - 47185920*a^5*b^18*c^36*d*e*g^4 + 66338387132416*a^14*b^9*c^1 
8*d^19*e*g^4 + 77260289736704*a^13*b^10*c^20*d^17*e*g^4 + 5303663853568...
 

Reduce [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 11950, normalized size of antiderivative = 13.49 \[ \int \frac {\sqrt {e x}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*(702*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 
2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5*b*c**5*d**6 + 1404*b* 
*(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b 
))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5*b*c**4*d**7*x + 702*b**(1/4)*a**(3/4) 
*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a* 
*(1/4)*sqrt(2)))*a**5*b*c**3*d**8*x**2 - 910*b**(1/4)*a**(3/4)*sqrt(2)*ata 
n((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 
2)))*a**4*b**2*c**7*d**4 - 1820*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a 
**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b** 
2*c**6*d**5*x + 494*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt 
(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b**2*c**5*d**6* 
x**2 + 2808*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2* 
sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b**2*c**4*d**7*x**3 + 1 
404*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)* 
sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b**2*c**3*d**8*x**4 - 86*b**(1/ 
4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/( 
b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b**3*c**9*d**2 - 172*b**(1/4)*a**(3/4)*sq 
rt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1 
/4)*sqrt(2)))*a**3*b**3*c**8*d**3*x - 1906*b**(1/4)*a**(3/4)*sqrt(2)*atan( 
(b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt...