\(\int \frac {(e x)^{3/2}}{(c+d x)^3 (a+b x^2)^3} \, dx\) [458]

Optimal result

Integrand size = 24, antiderivative size = 840 \[ \int \frac {(e x)^{3/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {c d^4 e \sqrt {e x}}{\left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {d^4 \left (13 b c^2-3 a d^2\right ) e \sqrt {e x}}{2 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {e \sqrt {e x} (c-d x)}{4 \left (b c^2+a d^2\right ) (c+d x)^2 \left (a+b x^2\right )^2}-\frac {b e \sqrt {e x} \left (2 a b c^2 d^2 (15 c-23 d x)-b^2 c^4 (c-9 d x)-3 a^2 d^4 (11 c-3 d x)\right )}{16 a \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}+\frac {3 d^{7/2} \left (33 b^2 c^4-30 a b c^2 d^2+a^2 d^4\right ) e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 \sqrt {c} \left (b c^2+a d^2\right )^5}-\frac {3 \sqrt [4]{b} \left (b^{7/2} c^7-3 \sqrt {a} b^3 c^6 d+15 a b^{5/2} c^5 d^2-85 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4+159 a^{5/2} b c^2 d^5+77 a^3 \sqrt {b} c d^6-15 a^{7/2} d^7\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^5}+\frac {3 \sqrt [4]{b} \left (b^{7/2} c^7-3 \sqrt {a} b^3 c^6 d+15 a b^{5/2} c^5 d^2-85 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4+159 a^{5/2} b c^2 d^5+77 a^3 \sqrt {b} c d^6-15 a^{7/2} d^7\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^5}+\frac {3 \sqrt [4]{b} \left (b^{7/2} c^7+3 \sqrt {a} b^3 c^6 d+15 a b^{5/2} c^5 d^2+85 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4-159 a^{5/2} b c^2 d^5+77 a^3 \sqrt {b} c d^6+15 a^{7/2} d^7\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^5} \] Output:

c*d^4*e*(e*x)^(1/2)/(a*d^2+b*c^2)^3/(d*x+c)^2+1/2*d^4*(-3*a*d^2+13*b*c^2)* 
e*(e*x)^(1/2)/(a*d^2+b*c^2)^4/(d*x+c)-1/4*e*(e*x)^(1/2)*(-d*x+c)/(a*d^2+b* 
c^2)/(d*x+c)^2/(b*x^2+a)^2-1/16*b*e*(e*x)^(1/2)*(2*a*b*c^2*d^2*(-23*d*x+15 
*c)-b^2*c^4*(-9*d*x+c)-3*a^2*d^4*(-3*d*x+11*c))/a/(a*d^2+b*c^2)^4/(b*x^2+a 
)+3/4*d^(7/2)*(a^2*d^4-30*a*b*c^2*d^2+33*b^2*c^4)*e^(3/2)*arctan(d^(1/2)*( 
e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(1/2)/(a*d^2+b*c^2)^5-3/64*b^(1/4)*(b^(7/2)* 
c^7-3*a^(1/2)*b^3*c^6*d+15*a*b^(5/2)*c^5*d^2-85*a^(3/2)*b^2*c^4*d^3-165*a^ 
2*b^(3/2)*c^3*d^4+159*a^(5/2)*b*c^2*d^5+77*a^3*b^(1/2)*c*d^6-15*a^(7/2)*d^ 
7)*e^(3/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a 
^(7/4)/(a*d^2+b*c^2)^5+3/64*b^(1/4)*(b^(7/2)*c^7-3*a^(1/2)*b^3*c^6*d+15*a* 
b^(5/2)*c^5*d^2-85*a^(3/2)*b^2*c^4*d^3-165*a^2*b^(3/2)*c^3*d^4+159*a^(5/2) 
*b*c^2*d^5+77*a^3*b^(1/2)*c*d^6-15*a^(7/2)*d^7)*e^(3/2)*arctan(1+2^(1/2)*b 
^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(7/4)/(a*d^2+b*c^2)^5+3/64*b 
^(1/4)*(b^(7/2)*c^7+3*a^(1/2)*b^3*c^6*d+15*a*b^(5/2)*c^5*d^2+85*a^(3/2)*b^ 
2*c^4*d^3-165*a^2*b^(3/2)*c^3*d^4-159*a^(5/2)*b*c^2*d^5+77*a^3*b^(1/2)*c*d 
^6+15*a^(7/2)*d^7)*e^(3/2)*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^(7/4)/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 4.65 (sec) , antiderivative size = 646, normalized size of antiderivative = 0.77 \[ \int \frac {(e x)^{3/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {(e x)^{3/2} \left (-\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} \left (-b^4 c^4 x^2 (c-9 d x) (c+d x)^2+4 a^4 d^6 (3 c+5 d x)+a^3 b d^4 \left (-141 c^3-173 c^2 d x+c d^2 x^2+57 d^3 x^3\right )+a b^3 c^2 \left (3 c^5+3 c^4 d x+47 c^3 d^2 x^2+23 c^2 d^3 x^3-182 c d^4 x^4-150 d^5 x^5\right )+a^2 b^2 d^2 \left (42 c^5+2 c^4 d x-335 c^3 d^2 x^2-311 c^2 d^3 x^3-7 c d^4 x^4+33 d^5 x^5\right )\right )}{a (c+d x)^2 \left (a+b x^2\right )^2}+\frac {48 d^{7/2} \left (33 b^2 c^4-30 a b c^2 d^2+a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {3 \sqrt {2} \sqrt [4]{b} \left (-b^{7/2} c^7+3 \sqrt {a} b^3 c^6 d-15 a b^{5/2} c^5 d^2+85 a^{3/2} b^2 c^4 d^3+165 a^2 b^{3/2} c^3 d^4-159 a^{5/2} b c^2 d^5-77 a^3 \sqrt {b} c d^6+15 a^{7/2} d^7\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4}}+\frac {3 \sqrt {2} \sqrt [4]{b} \left (b^{7/2} c^7+3 \sqrt {a} b^3 c^6 d+15 a b^{5/2} c^5 d^2+85 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4-159 a^{5/2} b c^2 d^5+77 a^3 \sqrt {b} c d^6+15 a^{7/2} d^7\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4}}\right )}{64 \left (b c^2+a d^2\right )^5 x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*((-4*(b*c^2 + a*d^2)*Sqrt[x]*(-(b^4*c^4*x^2*(c - 9*d*x)*(c + 
d*x)^2) + 4*a^4*d^6*(3*c + 5*d*x) + a^3*b*d^4*(-141*c^3 - 173*c^2*d*x + c* 
d^2*x^2 + 57*d^3*x^3) + a*b^3*c^2*(3*c^5 + 3*c^4*d*x + 47*c^3*d^2*x^2 + 23 
*c^2*d^3*x^3 - 182*c*d^4*x^4 - 150*d^5*x^5) + a^2*b^2*d^2*(42*c^5 + 2*c^4* 
d*x - 335*c^3*d^2*x^2 - 311*c^2*d^3*x^3 - 7*c*d^4*x^4 + 33*d^5*x^5)))/(a*( 
c + d*x)^2*(a + b*x^2)^2) + (48*d^(7/2)*(33*b^2*c^4 - 30*a*b*c^2*d^2 + a^2 
*d^4)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/Sqrt[c] + (3*Sqrt[2]*b^(1/4)*(-(b 
^(7/2)*c^7) + 3*Sqrt[a]*b^3*c^6*d - 15*a*b^(5/2)*c^5*d^2 + 85*a^(3/2)*b^2* 
c^4*d^3 + 165*a^2*b^(3/2)*c^3*d^4 - 159*a^(5/2)*b*c^2*d^5 - 77*a^3*Sqrt[b] 
*c*d^6 + 15*a^(7/2)*d^7)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^( 
1/4)*Sqrt[x])])/a^(7/4) + (3*Sqrt[2]*b^(1/4)*(b^(7/2)*c^7 + 3*Sqrt[a]*b^3* 
c^6*d + 15*a*b^(5/2)*c^5*d^2 + 85*a^(3/2)*b^2*c^4*d^3 - 165*a^2*b^(3/2)*c^ 
3*d^4 - 159*a^(5/2)*b*c^2*d^5 + 77*a^3*Sqrt[b]*c*d^6 + 15*a^(7/2)*d^7)*Arc 
Tanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(7/4)))/( 
64*(b*c^2 + a*d^2)^5*x^(3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2057\) vs. \(2(840)=1680\).

Time = 4.99 (sec) , antiderivative size = 2057, normalized size of antiderivative = 2.45, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 b c (e x)^{3/2} d^5}{\left (b c^2+a d^2\right )^4 (c+d x)}-\frac {(e x)^{3/2} d^5}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {3 \sqrt [4]{a} \sqrt [4]{b} \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 ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} \left (b c^2+a d^2\right )^5}-\frac {3 \sqrt [4]{a} \sqrt [4]{b} \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 ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} \left (b c^2+a d^2\right )^5}+\frac {3 \sqrt [4]{a} \sqrt [4]{b} \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 ) e^{3/2} \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} \left (b c^2+a d^2\right )^5}-\frac {3 \sqrt [4]{a} \sqrt [4]{b} \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 ) e^{3/2} \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} \left (b c^2+a d^2\right )^5}+\frac {18 b c e \sqrt {e x} d^4}{\left (b c^2+a d^2\right )^4}+\frac {6 b c \left (5 b c^2-3 a d^2\right ) e \sqrt {e x} d^4}{\left (b c^2+a d^2\right )^5}-\frac {6 b c \left (7 b c^2-a d^2\right ) e \sqrt {e x} d^4}{\left (b c^2+a d^2\right )^5}-\frac {3 e \sqrt {e x} d^4}{4 \left (b c^2+a d^2\right )^3 (c+d x)}+\frac {3 e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{7/2}}{4 \sqrt {c} \left (b c^2+a d^2\right )^3}-\frac {18 b c^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{7/2}}{\left (b c^2+a d^2\right )^4}+\frac {6 b c^{3/2} \left (7 b c^2-a d^2\right ) e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{7/2}}{\left (b c^2+a d^2\right )^5}-\frac {3 \sqrt [4]{b} \left (b^{3/2} c^3-5 \sqrt {a} b d c^2-a \sqrt {b} d^2 c+a^{3/2} d^3\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{2 \sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^4}+\frac {3 \sqrt [4]{b} \left (b^{3/2} c^3-5 \sqrt {a} b d c^2-a \sqrt {b} d^2 c+a^{3/2} d^3\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{2 \sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^4}-\frac {3 \sqrt [4]{b} \left (b^{3/2} c^3+5 \sqrt {a} b d c^2-a \sqrt {b} d^2 c-a^{3/2} d^3\right ) e^{3/2} \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^{3/4} \left (b c^2+a d^2\right )^4}+\frac {3 \sqrt [4]{b} \left (b^{3/2} c^3+5 \sqrt {a} b d c^2-a \sqrt {b} d^2 c-a^{3/2} d^3\right ) e^{3/2} \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^{3/4} \left (b c^2+a d^2\right )^4}-\frac {b e \sqrt {e x} \left (3 c \left (b c^2-a d^2\right )-d \left (5 b c^2-a d^2\right ) x\right ) d^2}{\left (b c^2+a d^2\right )^4 \left (b x^2+a\right )}-\frac {3 \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} d c+a d^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^3}+\frac {3 \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} d c+a d^2\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^3}-\frac {3 \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+4 \sqrt {a} \sqrt {b} d c+a d^2\right ) e^{3/2} \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^{7/4} \left (b c^2+a d^2\right )^3}+\frac {3 \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+4 \sqrt {a} \sqrt {b} d c+a d^2\right ) e^{3/2} \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^{7/4} \left (b c^2+a d^2\right )^3}+\frac {b e \sqrt {e x} \left (c \left (b c^2-3 a d^2\right )-3 d \left (3 b c^2-a d^2\right ) x\right )}{16 a \left (b c^2+a d^2\right )^3 \left (b x^2+a\right )}-\frac {b e \sqrt {e x} \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{4 \left (b c^2+a d^2\right )^3 \left (b x^2+a\right )^2}\)

Input:

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

Output:

(6*b*c*d^4*(5*b*c^2 - 3*a*d^2)*e*Sqrt[e*x])/(b*c^2 + a*d^2)^5 - (6*b*c*d^4 
*(7*b*c^2 - a*d^2)*e*Sqrt[e*x])/(b*c^2 + a*d^2)^5 + (18*b*c*d^4*e*Sqrt[e*x 
])/(b*c^2 + a*d^2)^4 - (d^5*(e*x)^(3/2))/(2*(b*c^2 + a*d^2)^3*(c + d*x)^2) 
 - (3*d^4*e*Sqrt[e*x])/(4*(b*c^2 + a*d^2)^3*(c + d*x)) - (6*b*c*d^5*(e*x)^ 
(3/2))/((b*c^2 + a*d^2)^4*(c + d*x)) - (b*e*Sqrt[e*x]*(c*(b*c^2 - 3*a*d^2) 
 - d*(3*b*c^2 - a*d^2)*x))/(4*(b*c^2 + a*d^2)^3*(a + b*x^2)^2) + (b*e*Sqrt 
[e*x]*(c*(b*c^2 - 3*a*d^2) - 3*d*(3*b*c^2 - a*d^2)*x))/(16*a*(b*c^2 + a*d^ 
2)^3*(a + b*x^2)) - (b*d^2*e*Sqrt[e*x]*(3*c*(b*c^2 - a*d^2) - d*(5*b*c^2 - 
 a*d^2)*x))/((b*c^2 + a*d^2)^4*(a + b*x^2)) + (6*b*c^(3/2)*d^(7/2)*(7*b*c^ 
2 - a*d^2)*e^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + 
 a*d^2)^5 - (18*b*c^(3/2)*d^(7/2)*e^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt 
[c]*Sqrt[e])])/(b*c^2 + a*d^2)^4 + (3*d^(7/2)*e^(3/2)*ArcTan[(Sqrt[d]*Sqrt 
[e*x])/(Sqrt[c]*Sqrt[e])])/(4*Sqrt[c]*(b*c^2 + a*d^2)^3) - (3*b^(1/4)*(Sqr 
t[b]*c + Sqrt[a]*d)*(b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*e^(3/2)*ArcTan 
[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sqrt[2]*a^(7/4)*( 
b*c^2 + a*d^2)^3) + (3*a^(1/4)*b^(1/4)*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)*e^(3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4) 
*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*(b*c^2 + a*d^2)^5) - (3*b^(1/4)*d 
^2*(b^(3/2)*c^3 - 5*Sqrt[a]*b*c^2*d - a*Sqrt[b]*c*d^2 + a^(3/2)*d^3)*e^(3/ 
2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(2*Sqrt[2...
 

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 = 1.65 (sec) , antiderivative size = 794, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(2 e^{8} \left (\frac {d^{4} \left (\frac {\left (-\frac {5}{8} a^{2} d^{5}+\frac {7}{4} d^{3} a \,c^{2} b +\frac {19}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {9}{4} a b \,d^{2} e \,c^{3}+\frac {21}{8} b^{2} e \,c^{5}-\frac {3}{8} c e \,a^{2} d^{4}\right ) \sqrt {e x}}{\left (d e x +c e \right )^{2}}+\frac {3 \left (a^{2} d^{4}-30 b \,c^{2} d^{2} a +33 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {b \left (\frac {-\frac {b d \left (13 a^{3} d^{6}-61 a^{2} b \,c^{2} d^{4}-65 a \,b^{2} c^{4} d^{2}+9 b^{3} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a}+\frac {b c e \left (45 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-49 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a}+\left (-\frac {17}{32} e^{2} a^{3} d^{7}+\frac {65}{32} b \,c^{2} d^{5} e^{2} a^{2}+\frac {85}{32} a \,b^{2} c^{4} d^{3} e^{2}+\frac {3}{32} b^{3} c^{6} d \,e^{2}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {57}{32} c \,e^{3} a^{3} d^{6}+\frac {15}{32} b \,c^{3} d^{4} e^{3} a^{2}-\frac {45}{32} a \,b^{2} c^{5} d^{2} e^{3}-\frac {3}{32} b^{3} c^{7} e^{3}\right ) \sqrt {e x}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {3 \left (77 a^{3} c \,d^{6} e -165 a^{2} b \,c^{3} d^{4} e +15 a \,b^{2} c^{5} d^{2} e +b^{3} c^{7} 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 (-15 a^{3} d^{7}+159 a^{2} b \,c^{2} d^{5}-85 a \,b^{2} c^{4} d^{3}-3 b^{3} c^{6} d \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}}}}{a}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\right )\)	\(794\)
default	\(2 e^{8} \left (\frac {d^{4} \left (\frac {\left (-\frac {5}{8} a^{2} d^{5}+\frac {7}{4} d^{3} a \,c^{2} b +\frac {19}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {9}{4} a b \,d^{2} e \,c^{3}+\frac {21}{8} b^{2} e \,c^{5}-\frac {3}{8} c e \,a^{2} d^{4}\right ) \sqrt {e x}}{\left (d e x +c e \right )^{2}}+\frac {3 \left (a^{2} d^{4}-30 b \,c^{2} d^{2} a +33 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {b \left (\frac {-\frac {b d \left (13 a^{3} d^{6}-61 a^{2} b \,c^{2} d^{4}-65 a \,b^{2} c^{4} d^{2}+9 b^{3} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a}+\frac {b c e \left (45 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-49 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a}+\left (-\frac {17}{32} e^{2} a^{3} d^{7}+\frac {65}{32} b \,c^{2} d^{5} e^{2} a^{2}+\frac {85}{32} a \,b^{2} c^{4} d^{3} e^{2}+\frac {3}{32} b^{3} c^{6} d \,e^{2}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {57}{32} c \,e^{3} a^{3} d^{6}+\frac {15}{32} b \,c^{3} d^{4} e^{3} a^{2}-\frac {45}{32} a \,b^{2} c^{5} d^{2} e^{3}-\frac {3}{32} b^{3} c^{7} e^{3}\right ) \sqrt {e x}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {3 \left (77 a^{3} c \,d^{6} e -165 a^{2} b \,c^{3} d^{4} e +15 a \,b^{2} c^{5} d^{2} e +b^{3} c^{7} 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 (-15 a^{3} d^{7}+159 a^{2} b \,c^{2} d^{5}-85 a \,b^{2} c^{4} d^{3}-3 b^{3} c^{6} d \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}}}}{a}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\right )\)	\(794\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(929\)

Input:

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

Output:

2*e^8*(d^4/e^6/(a*d^2+b*c^2)^5*(((-5/8*a^2*d^5+7/4*d^3*a*c^2*b+19/8*b^2*c^ 
4*d)*(e*x)^(3/2)+(9/4*a*b*d^2*e*c^3+21/8*b^2*e*c^5-3/8*c*e*a^2*d^4)*(e*x)^ 
(1/2))/(d*e*x+c*e)^2+3/8*(a^2*d^4-30*a*b*c^2*d^2+33*b^2*c^4)/(d*e*c)^(1/2) 
*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))+b/e^6/(a*d^2+b*c^2)^5*((-1/32*b*d*(1 
3*a^3*d^6-61*a^2*b*c^2*d^4-65*a*b^2*c^4*d^2+9*b^3*c^6)/a*(e*x)^(7/2)+1/32* 
b*c*e*(45*a^3*d^6-5*a^2*b*c^2*d^4-49*a*b^2*c^4*d^2+b^3*c^6)/a*(e*x)^(5/2)+ 
(-17/32*e^2*a^3*d^7+65/32*b*c^2*d^5*e^2*a^2+85/32*a*b^2*c^4*d^3*e^2+3/32*b 
^3*c^6*d*e^2)*(e*x)^(3/2)+(57/32*c*e^3*a^3*d^6+15/32*b*c^3*d^4*e^3*a^2-45/ 
32*a*b^2*c^5*d^2*e^3-3/32*b^3*c^7*e^3)*(e*x)^(1/2))/(b*e^2*x^2+a*e^2)^2+3/ 
32/a*(1/8*(77*a^3*c*d^6*e-165*a^2*b*c^3*d^4*e+15*a*b^2*c^5*d^2*e+b^3*c^7*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*(-15*a^3*d^7+159*a^2*b*c^2*d^5-85*a*b^2*c^ 
4*d^3-3*b^3*c^6*d)/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)))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)/(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. 1969 vs. \(2 (700) = 1400\).

Time = 0.32 (sec) , antiderivative size = 1969, normalized size of antiderivative = 2.34 \[ \int \frac {(e x)^{3/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*(6*((a*b^3*e^2)^(1/4)*b^5*c^7*e^2 + 15*(a*b^3*e^2)^(1/4)*a*b^4*c^5*d^ 
2*e^2 - 165*(a*b^3*e^2)^(1/4)*a^2*b^3*c^3*d^4*e^2 + 77*(a*b^3*e^2)^(1/4)*a 
^3*b^2*c*d^6*e^2 - 3*(a*b^3*e^2)^(3/4)*b^3*c^6*d*e - 85*(a*b^3*e^2)^(3/4)* 
a*b^2*c^4*d^3*e + 159*(a*b^3*e^2)^(3/4)*a^2*b*c^2*d^5*e - 15*(a*b^3*e^2)^( 
3/4)*a^3*d^7*e)*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^2*b^7*c^10 + 5*sqrt(2)*a^3*b^6*c^8*d^2 + 10*s 
qrt(2)*a^4*b^5*c^6*d^4 + 10*sqrt(2)*a^5*b^4*c^4*d^6 + 5*sqrt(2)*a^6*b^3*c^ 
2*d^8 + sqrt(2)*a^7*b^2*d^10) + 6*((a*b^3*e^2)^(1/4)*b^5*c^7*e^2 + 15*(a*b 
^3*e^2)^(1/4)*a*b^4*c^5*d^2*e^2 - 165*(a*b^3*e^2)^(1/4)*a^2*b^3*c^3*d^4*e^ 
2 + 77*(a*b^3*e^2)^(1/4)*a^3*b^2*c*d^6*e^2 - 3*(a*b^3*e^2)^(3/4)*b^3*c^6*d 
*e - 85*(a*b^3*e^2)^(3/4)*a*b^2*c^4*d^3*e + 159*(a*b^3*e^2)^(3/4)*a^2*b*c^ 
2*d^5*e - 15*(a*b^3*e^2)^(3/4)*a^3*d^7*e)*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^2*b^7*c^10 + 5*sqr 
t(2)*a^3*b^6*c^8*d^2 + 10*sqrt(2)*a^4*b^5*c^6*d^4 + 10*sqrt(2)*a^5*b^4*c^4 
*d^6 + 5*sqrt(2)*a^6*b^3*c^2*d^8 + sqrt(2)*a^7*b^2*d^10) + 3*((a*b^3*e^2)^ 
(1/4)*b^5*c^7*e^2 + 15*(a*b^3*e^2)^(1/4)*a*b^4*c^5*d^2*e^2 - 165*(a*b^3*e^ 
2)^(1/4)*a^2*b^3*c^3*d^4*e^2 + 77*(a*b^3*e^2)^(1/4)*a^3*b^2*c*d^6*e^2 + 3* 
(a*b^3*e^2)^(3/4)*b^3*c^6*d*e + 85*(a*b^3*e^2)^(3/4)*a*b^2*c^4*d^3*e - 159 
*(a*b^3*e^2)^(3/4)*a^2*b*c^2*d^5*e + 15*(a*b^3*e^2)^(3/4)*a^3*d^7*e)*log(e 
*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^2*b^...
 

Mupad [B] (verification not implemented)

Time = 13.84 (sec) , antiderivative size = 10604, normalized size of antiderivative = 12.62 \[ \int \frac {(e x)^{3/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((3*(3007125*a^9*b^5*d^26*e^18 + 2673*b^14*c^18*d^8*e^18 + 33670 
404*a^2*b^12*c^14*d^12*e^18 + 670265604*a^3*b^11*c^12*d^14*e^18 + 28122602 
22*a^4*b^10*c^10*d^16*e^18 - 7196416650*a^5*b^9*c^8*d^18*e^18 + 5460052212 
*a^6*b^8*c^6*d^20*e^18 - 1519252524*a^7*b^7*c^4*d^22*e^18 - 40387815*a^8*b 
^6*c^2*d^24*e^18 + 540189*a*b^13*c^16*d^10*e^18))/(262144*(a^20*d^32 + a^4 
*b^16*c^32 + 16*a^19*b*c^2*d^30 + 16*a^5*b^15*c^30*d^2 + 120*a^6*b^14*c^28 
*d^4 + 560*a^7*b^13*c^26*d^6 + 1820*a^8*b^12*c^24*d^8 + 4368*a^9*b^11*c^22 
*d^10 + 8008*a^10*b^10*c^20*d^12 + 11440*a^11*b^9*c^18*d^14 + 12870*a^12*b 
^8*c^16*d^16 + 11440*a^13*b^7*c^14*d^18 + 8008*a^14*b^6*c^12*d^20 + 4368*a 
^15*b^5*c^10*d^22 + 1820*a^16*b^4*c^8*d^24 + 560*a^17*b^3*c^6*d^26 + 120*a 
^18*b^2*c^4*d^28)) - root(21474836480*a^26*b*c^3*d^38*g^6 + 204010946560*a 
^25*b^2*c^5*d^36*g^6 + 135259257569280*a^19*b^8*c^17*d^24*g^6 + 1352592575 
69280*a^15*b^12*c^25*d^16*g^6 + 204010946560*a^9*b^18*c^37*d^4*g^6 + 52022 
79137280*a^23*b^4*c^9*d^32*g^6 + 5202279137280*a^11*b^16*c^33*d^8*g^6 + 12 
24065679360*a^24*b^3*c^7*d^34*g^6 + 16647293239296*a^22*b^5*c^11*d^30*g^6 
+ 41618233098240*a^21*b^6*c^13*d^28*g^6 + 83236466196480*a^20*b^7*c^15*d^2 
6*g^6 + 180345676759040*a^18*b^9*c^19*d^22*g^6 + 198380244434944*a^17*b^10 
*c^21*d^20*g^6 + 180345676759040*a^16*b^11*c^23*d^18*g^6 + 83236466196480* 
a^14*b^13*c^27*d^14*g^6 + 41618233098240*a^13*b^14*c^29*d^12*g^6 + 1664729 
3239296*a^12*b^15*c^31*d^10*g^6 + 1224065679360*a^10*b^17*c^35*d^6*g^6 ...
 

Reduce [F]

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

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

Output:

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