\(\int \frac {1}{(e x)^{3/2} (c+d x)^2 (a+b x^2)^3} \, dx\) [456]

Optimal result

Integrand size = 24, antiderivative size = 741 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=-\frac {2}{a^3 c^2 e \sqrt {e x}}-\frac {d^7 \sqrt {e x}}{c^2 \left (b c^2+a d^2\right )^3 e^2 (c+d x)}-\frac {b^2 \sqrt {e x} \left (2 a c d+\left (b c^2-a d^2\right ) x\right )}{4 a^2 \left (b c^2+a d^2\right )^2 e^2 \left (a+b x^2\right )^2}-\frac {b^2 \sqrt {e x} \left (13 b^2 c^4 x+a^2 d^3 (46 c-21 d x)+2 a b c^2 d (7 c+12 d x)\right )}{16 a^3 \left (b c^2+a d^2\right )^3 e^2 \left (a+b x^2\right )}-\frac {3 d^{13/2} \left (5 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{5/2} \left (b c^2+a d^2\right )^4 e^{3/2}}+\frac {3 b^{5/4} \left (15 b^3 c^6+55 a b^2 c^4 d^2+65 a^2 b c^2 d^4-39 a^3 d^6+2 \sqrt {a} \sqrt {b} c d \left (7 b^2 c^4+30 a b c^2 d^2+55 a^2 d^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{13/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {3 b^{5/4} \left (15 b^3 c^6+55 a b^2 c^4 d^2+65 a^2 b c^2 d^4-39 a^3 d^6+2 \sqrt {a} \sqrt {b} c d \left (7 b^2 c^4+30 a b c^2 d^2+55 a^2 d^4\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{13/4} \left (b c^2+a d^2\right )^4 e^{3/2}}+\frac {3 b^{5/4} \left (15 b^3 c^6-14 \sqrt {a} b^{5/2} c^5 d+55 a b^2 c^4 d^2-60 a^{3/2} b^{3/2} c^3 d^3+65 a^2 b c^2 d^4-110 a^{5/2} \sqrt {b} c d^5-39 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{13/4} \left (b c^2+a d^2\right )^4 e^{3/2}} \] Output:

-2/a^3/c^2/e/(e*x)^(1/2)-d^7*(e*x)^(1/2)/c^2/(a*d^2+b*c^2)^3/e^2/(d*x+c)-1 
/4*b^2*(e*x)^(1/2)*(2*a*c*d+(-a*d^2+b*c^2)*x)/a^2/(a*d^2+b*c^2)^2/e^2/(b*x 
^2+a)^2-1/16*b^2*(e*x)^(1/2)*(13*b^2*c^4*x+a^2*d^3*(-21*d*x+46*c)+2*a*b*c^ 
2*d*(12*d*x+7*c))/a^3/(a*d^2+b*c^2)^3/e^2/(b*x^2+a)-3*d^(13/2)*(a*d^2+5*b* 
c^2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(5/2)/(a*d^2+b*c^2)^4/e 
^(3/2)+3/64*b^(5/4)*(15*b^3*c^6+55*a*b^2*c^4*d^2+65*a^2*b*c^2*d^4-39*a^3*d 
^6+2*a^(1/2)*b^(1/2)*c*d*(55*a^2*d^4+30*a*b*c^2*d^2+7*b^2*c^4))*arctan(1-2 
^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(13/4)/(a*d^2+b*c^2) 
^4/e^(3/2)-3/64*b^(5/4)*(15*b^3*c^6+55*a*b^2*c^4*d^2+65*a^2*b*c^2*d^4-39*a 
^3*d^6+2*a^(1/2)*b^(1/2)*c*d*(55*a^2*d^4+30*a*b*c^2*d^2+7*b^2*c^4))*arctan 
(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(13/4)/(a*d^2+b* 
c^2)^4/e^(3/2)+3/64*b^(5/4)*(15*b^3*c^6-14*a^(1/2)*b^(5/2)*c^5*d+55*a*b^2* 
c^4*d^2-60*a^(3/2)*b^(3/2)*c^3*d^3+65*a^2*b*c^2*d^4-110*a^(5/2)*b^(1/2)*c* 
d^5-39*a^3*d^6)*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^(13/4)/(a*d^2+b*c^2)^4/e^(3/2)
 

Mathematica [A] (verified)

Time = 2.43 (sec) , antiderivative size = 643, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {x \left (-\frac {4 \left (b c^2+a d^2\right ) \left (45 b^5 c^6 x^4 (c+d x)+16 a^5 d^6 (2 c+3 d x)+32 a^4 b d^4 \left (3 c^3+3 c^2 d x+2 c d^2 x^2+3 d^3 x^3\right )+a b^4 c^4 x^2 \left (81 c^3+95 c^2 d x+134 c d^2 x^2+120 d^3 x^3\right )+a^3 b^2 d^2 \left (96 c^5+150 c^4 d x+221 c^3 d^2 x^2+167 c^2 d^3 x^3+32 c d^4 x^4+48 d^5 x^5\right )+a^2 b^3 c^2 \left (32 c^5+54 c^4 d x+238 c^3 d^2 x^2+262 c^2 d^3 x^3+121 c d^4 x^4+75 d^5 x^5\right )\right )}{a^3 c^2 (c+d x) \left (a+b x^2\right )^2}-\frac {192 d^{13/2} \left (5 b c^2+a d^2\right ) \sqrt {x} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{5/2}}+\frac {3 \sqrt {2} b^{5/4} \left (15 b^3 c^6+14 \sqrt {a} b^{5/2} c^5 d+55 a b^2 c^4 d^2+60 a^{3/2} b^{3/2} c^3 d^3+65 a^2 b c^2 d^4+110 a^{5/2} \sqrt {b} c d^5-39 a^3 d^6\right ) \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{13/4}}-\frac {3 \sqrt {2} b^{5/4} \left (-15 b^3 c^6+14 \sqrt {a} b^{5/2} c^5 d-55 a b^2 c^4 d^2+60 a^{3/2} b^{3/2} c^3 d^3-65 a^2 b c^2 d^4+110 a^{5/2} \sqrt {b} c d^5+39 a^3 d^6\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{13/4}}\right )}{64 \left (b c^2+a d^2\right )^4 (e x)^{3/2}} \] Input:

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

Output:

(x*((-4*(b*c^2 + a*d^2)*(45*b^5*c^6*x^4*(c + d*x) + 16*a^5*d^6*(2*c + 3*d* 
x) + 32*a^4*b*d^4*(3*c^3 + 3*c^2*d*x + 2*c*d^2*x^2 + 3*d^3*x^3) + a*b^4*c^ 
4*x^2*(81*c^3 + 95*c^2*d*x + 134*c*d^2*x^2 + 120*d^3*x^3) + a^3*b^2*d^2*(9 
6*c^5 + 150*c^4*d*x + 221*c^3*d^2*x^2 + 167*c^2*d^3*x^3 + 32*c*d^4*x^4 + 4 
8*d^5*x^5) + a^2*b^3*c^2*(32*c^5 + 54*c^4*d*x + 238*c^3*d^2*x^2 + 262*c^2* 
d^3*x^3 + 121*c*d^4*x^4 + 75*d^5*x^5)))/(a^3*c^2*(c + d*x)*(a + b*x^2)^2) 
- (192*d^(13/2)*(5*b*c^2 + a*d^2)*Sqrt[x]*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c] 
])/c^(5/2) + (3*Sqrt[2]*b^(5/4)*(15*b^3*c^6 + 14*Sqrt[a]*b^(5/2)*c^5*d + 5 
5*a*b^2*c^4*d^2 + 60*a^(3/2)*b^(3/2)*c^3*d^3 + 65*a^2*b*c^2*d^4 + 110*a^(5 
/2)*Sqrt[b]*c*d^5 - 39*a^3*d^6)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt 
[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(13/4) - (3*Sqrt[2]*b^(5/4)*(-15*b^3*c^6 
+ 14*Sqrt[a]*b^(5/2)*c^5*d - 55*a*b^2*c^4*d^2 + 60*a^(3/2)*b^(3/2)*c^3*d^3 
 - 65*a^2*b*c^2*d^4 + 110*a^(5/2)*Sqrt[b]*c*d^5 + 39*a^3*d^6)*Sqrt[x]*ArcT 
anh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(13/4)))/( 
64*(b*c^2 + a*d^2)^4*(e*x)^(3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1790\) vs. \(2(741)=1482\).

Time = 4.55 (sec) , antiderivative size = 1790, normalized size of antiderivative = 2.42, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{13/2}}{c^{5/2} \left (b c^2+a d^2\right )^3 e^{3/2}}-\frac {12 b \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{13/2}}{\sqrt {c} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {3 d^6}{c^2 \left (b c^2+a d^2\right )^3 e \sqrt {e x}}-\frac {12 b d^6}{\left (b c^2+a d^2\right )^4 e \sqrt {e x}}+\frac {d^6}{c \left (b c^2+a d^2\right )^3 e \sqrt {e x} (c+d x)}+\frac {b^{5/4} \left (5 b c^2+6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {b^{5/4} \left (5 b c^2+6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {b^{5/4} \left (5 b c^2-6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \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} a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}+\frac {b^{5/4} \left (5 b c^2-6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \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} a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {2 b \left (5 b c^2-a d^2\right ) d^4}{a \left (b c^2+a d^2\right )^4 e \sqrt {e x}}+\frac {b^{5/4} \left (15 b c^2+12 \sqrt {a} \sqrt {b} d c-5 a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3 e^{3/2}}-\frac {b^{5/4} \left (15 b c^2+12 \sqrt {a} \sqrt {b} d c-5 a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3 e^{3/2}}-\frac {b^{5/4} \left (15 b c^2-12 \sqrt {a} \sqrt {b} d c-5 a d^2\right ) \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}{8 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3 e^{3/2}}+\frac {b^{5/4} \left (15 b c^2-12 \sqrt {a} \sqrt {b} d c-5 a d^2\right ) \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}{8 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^3 e^{3/2}}-\frac {5 b \left (3 b c^2-a d^2\right ) d^2}{2 a^2 \left (b c^2+a d^2\right )^3 e \sqrt {e x}}+\frac {b \left (3 b c^2-4 b d x c-a d^2\right ) d^2}{2 a \left (b c^2+a d^2\right )^3 e \sqrt {e x} \left (b x^2+a\right )}+\frac {3 b^{5/4} \left (15 b c^2+14 \sqrt {a} \sqrt {b} d c-15 a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{13/4} \left (b c^2+a d^2\right )^2 e^{3/2}}-\frac {3 b^{5/4} \left (15 b c^2+14 \sqrt {a} \sqrt {b} d c-15 a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{13/4} \left (b c^2+a d^2\right )^2 e^{3/2}}-\frac {3 b^{5/4} \left (15 b c^2-14 \sqrt {a} \sqrt {b} d c-15 a d^2\right ) \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^{13/4} \left (b c^2+a d^2\right )^2 e^{3/2}}+\frac {3 b^{5/4} \left (15 b c^2-14 \sqrt {a} \sqrt {b} d c-15 a d^2\right ) \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^{13/4} \left (b c^2+a d^2\right )^2 e^{3/2}}-\frac {45 b \left (b c^2-a d^2\right )}{16 a^3 \left (b c^2+a d^2\right )^2 e \sqrt {e x}}+\frac {b \left (9 \left (b c^2-a d^2\right )-14 b c d x\right )}{16 a^2 \left (b c^2+a d^2\right )^2 e \sqrt {e x} \left (b x^2+a\right )}+\frac {b \left (b c^2-2 b d x c-a d^2\right )}{4 a \left (b c^2+a d^2\right )^2 e \sqrt {e x} \left (b x^2+a\right )^2}\)

Input:

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

Output:

(-12*b*d^6)/((b*c^2 + a*d^2)^4*e*Sqrt[e*x]) - (2*b*d^4*(5*b*c^2 - a*d^2))/ 
(a*(b*c^2 + a*d^2)^4*e*Sqrt[e*x]) - (3*d^6)/(c^2*(b*c^2 + a*d^2)^3*e*Sqrt[ 
e*x]) - (5*b*d^2*(3*b*c^2 - a*d^2))/(2*a^2*(b*c^2 + a*d^2)^3*e*Sqrt[e*x]) 
- (45*b*(b*c^2 - a*d^2))/(16*a^3*(b*c^2 + a*d^2)^2*e*Sqrt[e*x]) + d^6/(c*( 
b*c^2 + a*d^2)^3*e*Sqrt[e*x]*(c + d*x)) + (b*(b*c^2 - a*d^2 - 2*b*c*d*x))/ 
(4*a*(b*c^2 + a*d^2)^2*e*Sqrt[e*x]*(a + b*x^2)^2) + (b*(9*(b*c^2 - a*d^2) 
- 14*b*c*d*x))/(16*a^2*(b*c^2 + a*d^2)^2*e*Sqrt[e*x]*(a + b*x^2)) + (b*d^2 
*(3*b*c^2 - a*d^2 - 4*b*c*d*x))/(2*a*(b*c^2 + a*d^2)^3*e*Sqrt[e*x]*(a + b* 
x^2)) - (12*b*d^(13/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(Sqr 
t[c]*(b*c^2 + a*d^2)^4*e^(3/2)) - (3*d^(13/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/( 
Sqrt[c]*Sqrt[e])])/(c^(5/2)*(b*c^2 + a*d^2)^3*e^(3/2)) + (b^(5/4)*d^4*(5*b 
*c^2 + 6*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x 
])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^4*e^(3/2)) + (b^(5 
/4)*d^2*(15*b*c^2 + 12*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*ArcTan[1 - (Sqrt[2]* 
b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(9/4)*(b*c^2 + a*d^2)^ 
3*e^(3/2)) + (3*b^(5/4)*(15*b*c^2 + 14*Sqrt[a]*Sqrt[b]*c*d - 15*a*d^2)*Arc 
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sqrt[2]*a^(13/ 
4)*(b*c^2 + a*d^2)^2*e^(3/2)) - (b^(5/4)*d^4*(5*b*c^2 + 6*Sqrt[a]*Sqrt[b]* 
c*d - a*d^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(S 
qrt[2]*a^(5/4)*(b*c^2 + a*d^2)^4*e^(3/2)) - (b^(5/4)*d^2*(15*b*c^2 + 12...
 

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 = 1.53 (sec) , antiderivative size = 696, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(2 e^{7} \left (-\frac {1}{c^{2} e^{8} a^{3} \sqrt {e x}}-\frac {b^{2} \left (\frac {\left (-\frac {21}{32} a^{3} b \,d^{6}+\frac {3}{32} a^{2} b^{2} c^{2} d^{4}+\frac {37}{32} a \,b^{3} c^{4} d^{2}+\frac {13}{32} b^{4} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}+\left (\frac {23}{16} a^{3} b c \,d^{5} e +\frac {15}{8} c^{3} d^{3} b^{2} e \,a^{2}+\frac {7}{16} c^{5} d \,b^{3} e a \right ) \left (e x \right )^{\frac {5}{2}}-\frac {a \,e^{2} \left (25 a^{3} d^{6}+a^{2} b \,c^{2} d^{4}-41 a \,b^{2} c^{4} d^{2}-17 b^{3} c^{6}\right ) \left (e x \right )^{\frac {3}{2}}}{32}+\frac {a^{2} e^{3} c d \left (27 a^{2} d^{4}+38 b \,c^{2} d^{2} a +11 b^{2} c^{4}\right ) \sqrt {e x}}{16}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {3 \left (110 a^{3} c \,d^{5} e +60 a^{2} b \,c^{3} d^{3} e +14 a \,b^{2} c^{5} 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 )}{256 a \,e^{2}}+\frac {3 \left (-39 a^{3} d^{6}+65 a^{2} b \,c^{2} d^{4}+55 a \,b^{2} c^{4} d^{2}+15 b^{3} c^{6}\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 )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} e^{8} a^{3}}-\frac {d^{7} \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {3 \left (a \,d^{2}+5 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{c^{2} e^{8} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\right )\)	\(696\)
default	\(2 e^{7} \left (-\frac {1}{c^{2} e^{8} a^{3} \sqrt {e x}}-\frac {b^{2} \left (\frac {\left (-\frac {21}{32} a^{3} b \,d^{6}+\frac {3}{32} a^{2} b^{2} c^{2} d^{4}+\frac {37}{32} a \,b^{3} c^{4} d^{2}+\frac {13}{32} b^{4} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}+\left (\frac {23}{16} a^{3} b c \,d^{5} e +\frac {15}{8} c^{3} d^{3} b^{2} e \,a^{2}+\frac {7}{16} c^{5} d \,b^{3} e a \right ) \left (e x \right )^{\frac {5}{2}}-\frac {a \,e^{2} \left (25 a^{3} d^{6}+a^{2} b \,c^{2} d^{4}-41 a \,b^{2} c^{4} d^{2}-17 b^{3} c^{6}\right ) \left (e x \right )^{\frac {3}{2}}}{32}+\frac {a^{2} e^{3} c d \left (27 a^{2} d^{4}+38 b \,c^{2} d^{2} a +11 b^{2} c^{4}\right ) \sqrt {e x}}{16}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {3 \left (110 a^{3} c \,d^{5} e +60 a^{2} b \,c^{3} d^{3} e +14 a \,b^{2} c^{5} 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 )}{256 a \,e^{2}}+\frac {3 \left (-39 a^{3} d^{6}+65 a^{2} b \,c^{2} d^{4}+55 a \,b^{2} c^{4} d^{2}+15 b^{3} c^{6}\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 )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} e^{8} a^{3}}-\frac {d^{7} \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {3 \left (a \,d^{2}+5 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{c^{2} e^{8} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\right )\)	\(696\)
risch	\(-\frac {2}{a^{3} c^{2} e \sqrt {e x}}-\frac {-\frac {2 b^{2} c^{2} \left (\frac {\left (\frac {21}{32} a^{3} b \,d^{6}-\frac {3}{32} a^{2} b^{2} c^{2} d^{4}-\frac {37}{32} a \,b^{3} c^{4} d^{2}-\frac {13}{32} b^{4} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}+\left (-\frac {23}{16} a^{3} b c \,d^{5} e -\frac {15}{8} c^{3} d^{3} b^{2} e \,a^{2}-\frac {7}{16} c^{5} d \,b^{3} e a \right ) \left (e x \right )^{\frac {5}{2}}+\frac {a \,e^{2} \left (25 a^{3} d^{6}+a^{2} b \,c^{2} d^{4}-41 a \,b^{2} c^{4} d^{2}-17 b^{3} c^{6}\right ) \left (e x \right )^{\frac {3}{2}}}{32}-\frac {a^{2} e^{3} c d \left (27 a^{2} d^{4}+38 b \,c^{2} d^{2} a +11 b^{2} c^{4}\right ) \sqrt {e x}}{16}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {3 \left (-110 a^{3} c \,d^{5} e -60 a^{2} b \,c^{3} d^{3} e -14 a \,b^{2} c^{5} 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 )}{256 a \,e^{2}}+\frac {3 \left (39 a^{3} d^{6}-65 a^{2} b \,c^{2} d^{4}-55 a \,b^{2} c^{4} d^{2}-15 b^{3} c^{6}\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 )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {2 a^{3} d^{7} \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {3 \left (a \,d^{2}+5 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}}{a^{3} c^{2} e}\)	\(697\)
pseudoelliptic	\(\frac {-\frac {165 d \left (d x +c \right ) \left (a^{2} d^{4}+\frac {6}{11} b \,c^{2} d^{2} a +\frac {7}{55} b^{2} c^{4}\right ) \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 ) b^{2} \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} c^{3} \sqrt {d e c}\, \sqrt {e x}\, \sqrt {\frac {a \,e^{2}}{b}}}{64}+\frac {117 e \left (\left (-\frac {128 a^{3} d^{7} \sqrt {e x}\, \left (d x +c \right ) \left (b \,x^{2}+a \right )^{2} \left (a \,d^{2}+5 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{39}-\frac {256 \left (\frac {45 c^{6} x^{4} \left (d x +c \right ) b^{5}}{32}+\frac {81 \left (d x +c \right ) x^{2} \left (\frac {40}{27} d^{2} x^{2}+\frac {14}{81} c d x +c^{2}\right ) a \,c^{4} b^{4}}{32}+\left (d x +c \right ) \left (\frac {75}{32} d^{4} x^{4}+\frac {23}{16} c \,d^{3} x^{3}+\frac {27}{4} d^{2} c^{2} x^{2}+\frac {11}{16} c^{3} d x +c^{4}\right ) a^{2} c^{2} b^{3}+3 d^{2} a^{3} \left (\frac {1}{2} x^{5} d^{5}+\frac {1}{3} x^{4} c \,d^{4}+\frac {167}{96} c^{2} d^{3} x^{3}+\frac {221}{96} c^{3} d^{2} x^{2}+\frac {25}{16} c^{4} d x +c^{5}\right ) b^{2}+3 d^{4} a^{4} \left (d^{3} x^{3}+\frac {2}{3} c \,d^{2} x^{2}+c^{2} d x +c^{3}\right ) b +a^{5} d^{6} \left (\frac {3 d x}{2}+c \right )\right ) \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}{117}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+\left (d x +c \right ) \left (a^{3} d^{6}-\frac {5}{3} a^{2} b \,c^{2} d^{4}-\frac {55}{39} a \,b^{2} c^{4} d^{2}-\frac {5}{13} b^{3} c^{6}\right ) \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 ) b \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} c^{2} \sqrt {d e c}\, \sqrt {e x}\right )}{128}}{e^{2} \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right )^{4} \left (b \,x^{2}+a \right )^{2} \left (d x +c \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a^{3} c^{2} \sqrt {e x}}\)	\(750\)

Input:

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

Output:

2*e^7*(-1/c^2/e^8/a^3/(e*x)^(1/2)-b^2/(a*d^2+b*c^2)^4/e^8/a^3*(((-21/32*a^ 
3*b*d^6+3/32*a^2*b^2*c^2*d^4+37/32*a*b^3*c^4*d^2+13/32*b^4*c^6)*(e*x)^(7/2 
)+(23/16*a^3*b*c*d^5*e+15/8*c^3*d^3*b^2*e*a^2+7/16*c^5*d*b^3*e*a)*(e*x)^(5 
/2)-1/32*a*e^2*(25*a^3*d^6+a^2*b*c^2*d^4-41*a*b^2*c^4*d^2-17*b^3*c^6)*(e*x 
)^(3/2)+1/16*a^2*e^3*c*d*(27*a^2*d^4+38*a*b*c^2*d^2+11*b^2*c^4)*(e*x)^(1/2 
))/(b*e^2*x^2+a*e^2)^2+3/256*(110*a^3*c*d^5*e+60*a^2*b*c^3*d^3*e+14*a*b^2* 
c^5*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))+3/256*(-39*a^3*d^6+65*a^2*b*c^2*d^4+55* 
a*b^2*c^4*d^2+15*b^3*c^6)/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^7/c^2/e^8/(a*d^2+b*c^2) 
^4*((1/2*a*d^2+1/2*b*c^2)*(e*x)^(1/2)/(d*e*x+c*e)+3/2*(a*d^2+5*b*c^2)/(d*e 
*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2))))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x)^(3/2)/(d*x+c)^2/(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. 1666 vs. \(2 (632) = 1264\).

Time = 0.28 (sec) , antiderivative size = 1666, normalized size of antiderivative = 2.25 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/64*(6*(14*(a*b^3*e^2)^(1/4)*a*b^4*c^5*d*e + 60*(a*b^3*e^2)^(1/4)*a^2*b^ 
3*c^3*d^3*e + 110*(a*b^3*e^2)^(1/4)*a^3*b^2*c*d^5*e + 15*(a*b^3*e^2)^(3/4) 
*b^3*c^6 + 55*(a*b^3*e^2)^(3/4)*a*b^2*c^4*d^2 + 65*(a*b^3*e^2)^(3/4)*a^2*b 
*c^2*d^4 - 39*(a*b^3*e^2)^(3/4)*a^3*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^4*b^5*c^8*e^2 + 4*sq 
rt(2)*a^5*b^4*c^6*d^2*e^2 + 6*sqrt(2)*a^6*b^3*c^4*d^4*e^2 + 4*sqrt(2)*a^7* 
b^2*c^2*d^6*e^2 + sqrt(2)*a^8*b*d^8*e^2) + 6*(14*(a*b^3*e^2)^(1/4)*a*b^4*c 
^5*d*e + 60*(a*b^3*e^2)^(1/4)*a^2*b^3*c^3*d^3*e + 110*(a*b^3*e^2)^(1/4)*a^ 
3*b^2*c*d^5*e + 15*(a*b^3*e^2)^(3/4)*b^3*c^6 + 55*(a*b^3*e^2)^(3/4)*a*b^2* 
c^4*d^2 + 65*(a*b^3*e^2)^(3/4)*a^2*b*c^2*d^4 - 39*(a*b^3*e^2)^(3/4)*a^3*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^4*b^5*c^8*e^2 + 4*sqrt(2)*a^5*b^4*c^6*d^2*e^2 + 6*sqrt(2) 
*a^6*b^3*c^4*d^4*e^2 + 4*sqrt(2)*a^7*b^2*c^2*d^6*e^2 + sqrt(2)*a^8*b*d^8*e 
^2) + 3*(14*(a*b^3*e^2)^(1/4)*a*b^4*c^5*d*e + 60*(a*b^3*e^2)^(1/4)*a^2*b^3 
*c^3*d^3*e + 110*(a*b^3*e^2)^(1/4)*a^3*b^2*c*d^5*e - 15*(a*b^3*e^2)^(3/4)* 
b^3*c^6 - 55*(a*b^3*e^2)^(3/4)*a*b^2*c^4*d^2 - 65*(a*b^3*e^2)^(3/4)*a^2*b* 
c^2*d^4 + 39*(a*b^3*e^2)^(3/4)*a^3*d^6)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)* 
sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^4*b^5*c^8*e^2 + 4*sqrt(2)*a^5*b^4*c^ 
6*d^2*e^2 + 6*sqrt(2)*a^6*b^3*c^4*d^4*e^2 + 4*sqrt(2)*a^7*b^2*c^2*d^6*e^2 
+ sqrt(2)*a^8*b*d^8*e^2) - 3*(14*(a*b^3*e^2)^(1/4)*a*b^4*c^5*d*e + 60*(...
 

Mupad [B] (verification not implemented)

Time = 15.91 (sec) , antiderivative size = 9831, normalized size of antiderivative = 13.27 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(23576504333053722624*a^31*b^17*c^32*d^39*e^12 - 64497254400000* 
a^17*b^31*c^60*d^11*e^12 - 1651703021568000*a^18*b^30*c^58*d^13*e^12 - 199 
87499754455040*a^19*b^29*c^56*d^15*e^12 - 151876169103310848*a^20*b^28*c^5 
4*d^17*e^12 - 811553277594304512*a^21*b^27*c^52*d^19*e^12 - 32349436787011 
09248*a^22*b^26*c^50*d^21*e^12 - 9952574646814507008*a^23*b^25*c^48*d^23*e 
^12 - 24102693409476575232*a^24*b^24*c^46*d^25*e^12 - 46376217926185254912 
*a^25*b^23*c^44*d^27*e^12 - 70852267750673350656*a^26*b^22*c^42*d^29*e^12 
- 84728732561329618944*a^27*b^21*c^40*d^31*e^12 - 76001271188211892224*a^2 
8*b^20*c^38*d^33*e^12 - 44560140321618395136*a^29*b^19*c^36*d^35*e^12 - 50 
63093038620868608*a^30*b^18*c^34*d^37*e^12 - root(1073741824*a^28*b*c^7*d^ 
30*e^9*g^6 + 8053063680*a^27*b^2*c^9*d^28*e^9*g^6 + 8053063680*a^15*b^14*c 
^33*d^4*e^9*g^6 + 37580963840*a^26*b^3*c^11*d^26*e^9*g^6 + 767725404160*a^ 
22*b^7*c^19*d^18*e^9*g^6 + 767725404160*a^20*b^9*c^23*d^14*e^9*g^6 + 37580 
963840*a^16*b^13*c^31*d^6*e^9*g^6 + 122138132480*a^25*b^4*c^13*d^24*e^9*g^ 
6 + 122138132480*a^17*b^12*c^29*d^8*e^9*g^6 + 293131517952*a^24*b^5*c^15*d 
^22*e^9*g^6 + 293131517952*a^18*b^11*c^27*d^10*e^9*g^6 + 1073741824*a^14*b 
^15*c^35*d^2*e^9*g^6 + 537407782912*a^23*b^6*c^17*d^20*e^9*g^6 + 537407782 
912*a^19*b^10*c^25*d^12*e^9*g^6 + 863691079680*a^21*b^8*c^21*d^16*e^9*g^6 
+ 67108864*a^29*c^5*d^32*e^9*g^6 + 67108864*a^13*b^16*c^37*e^9*g^6 + 73013 
1333120*a^14*b^9*c^18*d^15*e^6*g^4 + 674159394816*a^13*b^10*c^20*d^13*e...
 

Reduce [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 8573, normalized size of antiderivative = 11.57 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*( - 234*sqrt(x)*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** 
6 - 234*sqrt(x)*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**7*x + 390 
*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqr 
t(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b**2*c**6*d**4 + 390*sqrt( 
x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*s 
qrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b**2*c**5*d**5*x - 468*sqrt(x)*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**6*x**2 - 468*sqrt(x)*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**7*x**3 + 330*sqrt(x)*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**8*d**2 + 330*sqrt(x)*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**7*d**3*x + 780*sqrt(x)*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**6*d**4*x**2 + 780*sqrt(x)*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**5*d**5*x**3 - 234*sqrt(x)*b**(1/4)*a**...