\(\int \frac {(e x)^{5/2}}{(c+d x)^3 (a+b x^2)^3} \, dx\) [457]

Optimal result

Integrand size = 24, antiderivative size = 860 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {c^2 d^3 e^2 \sqrt {e x}}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {3 c d^3 \left (5 b c^2-3 a d^2\right ) e^2 \sqrt {e x}}{4 \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {e^2 \sqrt {e x} \left (7 a^3 d^5+3 b^3 c^5 x-a^2 b c d^3 (62 c-39 d x)+27 a b^2 c^3 d (c-2 d x)\right )}{16 a \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}+\frac {e^2 \sqrt {e x} \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{4 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}-\frac {3 \sqrt {c} d^{5/2} \left (3 b c^2-5 a d^2\right ) \left (7 b c^2-a d^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 \left (b c^2+a d^2\right )^5}-\frac {3 \left (b^{7/2} c^7-5 \sqrt {a} b^3 c^6 d+47 a b^{5/2} c^5 d^2+125 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4-119 a^{5/2} b c^2 d^5+45 a^3 \sqrt {b} c d^6+7 a^{7/2} d^7\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^5}+\frac {3 \left (b^{7/2} c^7-5 \sqrt {a} b^3 c^6 d+47 a b^{5/2} c^5 d^2+125 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4-119 a^{5/2} b c^2 d^5+45 a^3 \sqrt {b} c d^6+7 a^{7/2} d^7\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^5}-\frac {3 \left (\sqrt {a} d \left (5 b^3 c^6-125 a b^2 c^4 d^2+119 a^2 b c^2 d^4-7 a^3 d^6\right )+\sqrt {b} c \left (b^3 c^6+47 a b^2 c^4 d^2-165 a^2 b c^2 d^4+45 a^3 d^6\right )\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^5} \] Output:

-1/2*c^2*d^3*e^2*(e*x)^(1/2)/(a*d^2+b*c^2)^3/(d*x+c)^2-3/4*c*d^3*(-3*a*d^2 
+5*b*c^2)*e^2*(e*x)^(1/2)/(a*d^2+b*c^2)^4/(d*x+c)+1/16*e^2*(e*x)^(1/2)*(7* 
a^3*d^5+3*b^3*c^5*x-a^2*b*c*d^3*(-39*d*x+62*c)+27*a*b^2*c^3*d*(-2*d*x+c))/ 
a/(a*d^2+b*c^2)^4/(b*x^2+a)+1/4*e^2*(e*x)^(1/2)*(a^2*d^3-b^2*c^3*x-3*a*b*c 
*d*(-d*x+c))/(a*d^2+b*c^2)^3/(b*x^2+a)^2-3/4*c^(1/2)*d^(5/2)*(-5*a*d^2+3*b 
*c^2)*(-a*d^2+7*b*c^2)*e^(5/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2)) 
/(a*d^2+b*c^2)^5-3/64*(b^(7/2)*c^7-5*a^(1/2)*b^3*c^6*d+47*a*b^(5/2)*c^5*d^ 
2+125*a^(3/2)*b^2*c^4*d^3-165*a^2*b^(3/2)*c^3*d^4-119*a^(5/2)*b*c^2*d^5+45 
*a^3*b^(1/2)*c*d^6+7*a^(7/2)*d^7)*e^(5/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^( 
1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/b^(1/4)/(a*d^2+b*c^2)^5+3/64*(b^(7/2 
)*c^7-5*a^(1/2)*b^3*c^6*d+47*a*b^(5/2)*c^5*d^2+125*a^(3/2)*b^2*c^4*d^3-165 
*a^2*b^(3/2)*c^3*d^4-119*a^(5/2)*b*c^2*d^5+45*a^3*b^(1/2)*c*d^6+7*a^(7/2)* 
d^7)*e^(5/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2) 
/a^(5/4)/b^(1/4)/(a*d^2+b*c^2)^5-3/64*(a^(1/2)*d*(-7*a^3*d^6+119*a^2*b*c^2 
*d^4-125*a*b^2*c^4*d^2+5*b^3*c^6)+b^(1/2)*c*(45*a^3*d^6-165*a^2*b*c^2*d^4+ 
47*a*b^2*c^4*d^2+b^3*c^6))*e^(5/2)*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^(5/4)/b^(1/4)/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 3.67 (sec) , antiderivative size = 649, normalized size of antiderivative = 0.75 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {(e x)^{5/2} \left (\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} \left (3 b^4 c^5 x^3 (c+d x)^2+a^4 d^5 \left (39 c^2+58 c d x+11 d^2 x^2\right )+a^3 b d^3 \left (-138 c^4-149 c^3 d x+95 c^2 d^2 x^2+137 c d^3 x^3+7 d^4 x^4\right )-a b^3 c^3 x \left (c^4-25 c^3 d x+c^2 d^2 x^2+149 c d^3 x^3+114 d^4 x^4\right )+a^2 b^2 c d \left (15 c^5-16 c^4 d x-275 c^3 d^2 x^2-251 c^2 d^3 x^3+44 c d^4 x^4+75 d^5 x^5\right )\right )}{a (c+d x)^2 \left (a+b x^2\right )^2}-48 \sqrt {c} d^{5/2} \left (21 b^2 c^4-38 a b c^2 d^2+5 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )-\frac {3 \sqrt {2} \left (b^{7/2} c^7-5 \sqrt {a} b^3 c^6 d+47 a b^{5/2} c^5 d^2+125 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4-119 a^{5/2} b c^2 d^5+45 a^3 \sqrt {b} c d^6+7 a^{7/2} d^7\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} \sqrt [4]{b}}-\frac {3 \sqrt {2} \left (b^{7/2} c^7+5 \sqrt {a} b^3 c^6 d+47 a b^{5/2} c^5 d^2-125 a^{3/2} b^2 c^4 d^3-165 a^2 b^{3/2} c^3 d^4+119 a^{5/2} b c^2 d^5+45 a^3 \sqrt {b} c d^6-7 a^{7/2} d^7\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} \sqrt [4]{b}}\right )}{64 \left (b c^2+a d^2\right )^5 x^{5/2}} \] Input:

Integrate[(e*x)^(5/2)/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

((e*x)^(5/2)*((4*(b*c^2 + a*d^2)*Sqrt[x]*(3*b^4*c^5*x^3*(c + d*x)^2 + a^4* 
d^5*(39*c^2 + 58*c*d*x + 11*d^2*x^2) + a^3*b*d^3*(-138*c^4 - 149*c^3*d*x + 
 95*c^2*d^2*x^2 + 137*c*d^3*x^3 + 7*d^4*x^4) - a*b^3*c^3*x*(c^4 - 25*c^3*d 
*x + c^2*d^2*x^2 + 149*c*d^3*x^3 + 114*d^4*x^4) + a^2*b^2*c*d*(15*c^5 - 16 
*c^4*d*x - 275*c^3*d^2*x^2 - 251*c^2*d^3*x^3 + 44*c*d^4*x^4 + 75*d^5*x^5)) 
)/(a*(c + d*x)^2*(a + b*x^2)^2) - 48*Sqrt[c]*d^(5/2)*(21*b^2*c^4 - 38*a*b* 
c^2*d^2 + 5*a^2*d^4)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] - (3*Sqrt[2]*(b^(7/ 
2)*c^7 - 5*Sqrt[a]*b^3*c^6*d + 47*a*b^(5/2)*c^5*d^2 + 125*a^(3/2)*b^2*c^4* 
d^3 - 165*a^2*b^(3/2)*c^3*d^4 - 119*a^(5/2)*b*c^2*d^5 + 45*a^3*Sqrt[b]*c*d 
^6 + 7*a^(7/2)*d^7)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)* 
Sqrt[x])])/(a^(5/4)*b^(1/4)) - (3*Sqrt[2]*(b^(7/2)*c^7 + 5*Sqrt[a]*b^3*c^6 
*d + 47*a*b^(5/2)*c^5*d^2 - 125*a^(3/2)*b^2*c^4*d^3 - 165*a^2*b^(3/2)*c^3* 
d^4 + 119*a^(5/2)*b*c^2*d^5 + 45*a^3*Sqrt[b]*c*d^6 - 7*a^(7/2)*d^7)*ArcTan 
h[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(5/4)*b^(1/ 
4))))/(64*(b*c^2 + a*d^2)^5*x^(5/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2275\) vs. \(2(860)=1720\).

Time = 5.51 (sec) , antiderivative size = 2275, normalized size of antiderivative = 2.65, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 a \left (7 b c^2-a d^2\right ) e^2 \sqrt {e x} d^5}{\left (b c^2+a d^2\right )^5}-\frac {6 b c (e x)^{5/2} d^5}{\left (b c^2+a d^2\right )^4 (c+d x)}-\frac {(e x)^{5/2} d^5}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {10 b c e (e x)^{3/2} d^4}{\left (b c^2+a d^2\right )^4}+\frac {2 b c \left (5 b c^2-3 a d^2\right ) e (e x)^{3/2} d^4}{\left (b c^2+a d^2\right )^5}-\frac {2 b c \left (7 b c^2-a d^2\right ) e (e x)^{3/2} d^4}{\left (b c^2+a d^2\right )^5}+\frac {3 a^{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 ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^5}-\frac {3 a^{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 ) e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^5}-\frac {3 a^{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 ) e^{5/2} \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} \sqrt [4]{b} \left (b c^2+a d^2\right )^5}+\frac {3 a^{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 ) e^{5/2} \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} \sqrt [4]{b} \left (b c^2+a d^2\right )^5}-\frac {5 e (e x)^{3/2} d^4}{4 \left (b c^2+a d^2\right )^3 (c+d x)}+\frac {15 e^2 \sqrt {e x} d^3}{4 \left (b c^2+a d^2\right )^3}-\frac {30 b c^2 e^2 \sqrt {e x} d^3}{\left (b c^2+a d^2\right )^4}-\frac {5 \left (5 b c^2-a d^2\right ) e^2 \sqrt {e x} d^3}{\left (b c^2+a d^2\right )^4}+\frac {6 b c^2 \left (7 b c^2-a d^2\right ) e^2 \sqrt {e x} d^3}{\left (b c^2+a d^2\right )^5}-\frac {15 \sqrt {c} e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{5/2}}{4 \left (b c^2+a d^2\right )^3}+\frac {30 b c^{5/2} e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{5/2}}{\left (b c^2+a d^2\right )^4}-\frac {6 b c^{5/2} \left (7 b c^2-a d^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{5/2}}{\left (b c^2+a d^2\right )^5}-\frac {\left (9 b^{3/2} c^3+25 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}+\frac {\left (9 b^{3/2} c^3+25 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}+\frac {\left (9 b^{3/2} c^3-25 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) e^{5/2} \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} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}-\frac {\left (9 b^{3/2} c^3-25 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) e^{5/2} \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} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}-\frac {b e (e x)^{3/2} \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 {\left (3 b^{3/2} c^3-15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (3 b^{3/2} c^3-15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (3 b^{3/2} c^3+15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) e^{5/2} \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^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\left (3 b^{3/2} c^3+15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) e^{5/2} \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^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {e^2 \sqrt {e x} \left (5 a d \left (3 b c^2-a d^2\right )+3 b c \left (b c^2-3 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 (e x)^{3/2} \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)^(5/2)/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

(6*b*c^2*d^3*(7*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(b*c^2 + a*d^2)^5 + (6*a*d^5 
*(7*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(b*c^2 + a*d^2)^5 - (30*b*c^2*d^3*e^2*Sq 
rt[e*x])/(b*c^2 + a*d^2)^4 - (5*d^3*(5*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(b*c^ 
2 + a*d^2)^4 + (15*d^3*e^2*Sqrt[e*x])/(4*(b*c^2 + a*d^2)^3) + (2*b*c*d^4*( 
5*b*c^2 - 3*a*d^2)*e*(e*x)^(3/2))/(b*c^2 + a*d^2)^5 - (2*b*c*d^4*(7*b*c^2 
- a*d^2)*e*(e*x)^(3/2))/(b*c^2 + a*d^2)^5 + (10*b*c*d^4*e*(e*x)^(3/2))/(b* 
c^2 + a*d^2)^4 - (d^5*(e*x)^(5/2))/(2*(b*c^2 + a*d^2)^3*(c + d*x)^2) - (5* 
d^4*e*(e*x)^(3/2))/(4*(b*c^2 + a*d^2)^3*(c + d*x)) - (6*b*c*d^5*(e*x)^(5/2 
))/((b*c^2 + a*d^2)^4*(c + d*x)) - (b*e*(e*x)^(3/2)*(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) + (e^2*Sqrt[e 
*x]*(5*a*d*(3*b*c^2 - a*d^2) + 3*b*c*(b*c^2 - 3*a*d^2)*x))/(16*a*(b*c^2 + 
a*d^2)^3*(a + b*x^2)) - (b*d^2*e*(e*x)^(3/2)*(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^(5/2)*d^(5/2)*( 
7*b*c^2 - a*d^2)*e^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b 
*c^2 + a*d^2)^5 + (30*b*c^(5/2)*d^(5/2)*e^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e*x]) 
/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^2)^4 - (15*Sqrt[c]*d^(5/2)*e^(5/2)*ArcTa 
n[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(4*(b*c^2 + a*d^2)^3) - (d^2*(9* 
b^(3/2)*c^3 + 25*Sqrt[a]*b*c^2*d - 9*a*Sqrt[b]*c*d^2 - 5*a^(3/2)*d^3)*e^(5 
/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(2*Sqrt[2]* 
a^(1/4)*b^(1/4)*(b*c^2 + a*d^2)^4) + (3*a^(3/4)*d^4*(5*b^(3/2)*c^3 + 7*...
 

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 = 3.73 (sec) , antiderivative size = 800, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(2 e^{8} \left (-\frac {c \,d^{3} \left (\frac {\left (-\frac {9}{8} a^{2} d^{5}+\frac {3}{4} d^{3} a \,c^{2} b +\frac {15}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}-\frac {c e \left (7 a^{2} d^{4}-10 b \,c^{2} d^{2} a -17 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {3 \left (5 a^{2} d^{4}-38 b \,c^{2} d^{2} a +21 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {\frac {\frac {3 b^{2} c \left (13 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-17 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a}+\left (\frac {7}{32} b \,d^{7} e \,a^{3}-\frac {55}{32} a^{2} c^{2} d^{5} e \,b^{2}-\frac {35}{32} a \,c^{4} d^{3} e \,b^{3}+\frac {27}{32} c^{6} d e \,b^{4}\right ) \left (e x \right )^{\frac {5}{2}}+\left (\frac {51}{32} a^{3} c \,d^{6} e^{2} b +\frac {5}{32} a^{2} c^{3} d^{4} e^{2} b^{2}-\frac {47}{32} a \,c^{5} d^{2} e^{2} b^{3}-\frac {1}{32} c^{7} e^{2} b^{4}\right ) \left (e x \right )^{\frac {3}{2}}+\frac {a \,e^{3} d \left (11 a^{3} d^{6}-59 a^{2} b \,c^{2} d^{4}-55 a \,b^{2} c^{4} d^{2}+15 b^{3} c^{6}\right ) \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {3 \left (7 a^{4} d^{7} e -119 a^{3} b \,c^{2} d^{5} e +125 a^{2} b^{2} c^{4} d^{3} e -5 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 )}{256 a \,e^{2}}+\frac {3 \left (45 d^{6} c \,a^{3} b -165 a^{2} c^{3} d^{4} b^{2}+47 a \,c^{5} d^{2} b^{3}+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 )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{a}}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\right )\)	\(800\)
default	\(2 e^{8} \left (-\frac {c \,d^{3} \left (\frac {\left (-\frac {9}{8} a^{2} d^{5}+\frac {3}{4} d^{3} a \,c^{2} b +\frac {15}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}-\frac {c e \left (7 a^{2} d^{4}-10 b \,c^{2} d^{2} a -17 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {3 \left (5 a^{2} d^{4}-38 b \,c^{2} d^{2} a +21 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {\frac {\frac {3 b^{2} c \left (13 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-17 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a}+\left (\frac {7}{32} b \,d^{7} e \,a^{3}-\frac {55}{32} a^{2} c^{2} d^{5} e \,b^{2}-\frac {35}{32} a \,c^{4} d^{3} e \,b^{3}+\frac {27}{32} c^{6} d e \,b^{4}\right ) \left (e x \right )^{\frac {5}{2}}+\left (\frac {51}{32} a^{3} c \,d^{6} e^{2} b +\frac {5}{32} a^{2} c^{3} d^{4} e^{2} b^{2}-\frac {47}{32} a \,c^{5} d^{2} e^{2} b^{3}-\frac {1}{32} c^{7} e^{2} b^{4}\right ) \left (e x \right )^{\frac {3}{2}}+\frac {a \,e^{3} d \left (11 a^{3} d^{6}-59 a^{2} b \,c^{2} d^{4}-55 a \,b^{2} c^{4} d^{2}+15 b^{3} c^{6}\right ) \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {3 \left (7 a^{4} d^{7} e -119 a^{3} b \,c^{2} d^{5} e +125 a^{2} b^{2} c^{4} d^{3} e -5 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 )}{256 a \,e^{2}}+\frac {3 \left (45 d^{6} c \,a^{3} b -165 a^{2} c^{3} d^{4} b^{2}+47 a \,c^{5} d^{2} b^{3}+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 )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{a}}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\right )\)	\(800\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(922\)

Input:

int((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^8*(-c*d^3/e^5/(a*d^2+b*c^2)^5*(((-9/8*a^2*d^5+3/4*d^3*a*c^2*b+15/8*b^2 
*c^4*d)*(e*x)^(3/2)-1/8*c*e*(7*a^2*d^4-10*a*b*c^2*d^2-17*b^2*c^4)*(e*x)^(1 
/2))/(d*e*x+c*e)^2+3/8*(5*a^2*d^4-38*a*b*c^2*d^2+21*b^2*c^4)/(d*e*c)^(1/2) 
*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))+1/e^5/(a*d^2+b*c^2)^5*((3/32*b^2*c*( 
13*a^3*d^6-5*a^2*b*c^2*d^4-17*a*b^2*c^4*d^2+b^3*c^6)/a*(e*x)^(7/2)+(7/32*b 
*d^7*e*a^3-55/32*a^2*c^2*d^5*e*b^2-35/32*a*c^4*d^3*e*b^3+27/32*c^6*d*e*b^4 
)*(e*x)^(5/2)+(51/32*a^3*c*d^6*e^2*b+5/32*a^2*c^3*d^4*e^2*b^2-47/32*a*c^5* 
d^2*e^2*b^3-1/32*c^7*e^2*b^4)*(e*x)^(3/2)+1/32*a*e^3*d*(11*a^3*d^6-59*a^2* 
b*c^2*d^4-55*a*b^2*c^4*d^2+15*b^3*c^6)*(e*x)^(1/2))/(b*e^2*x^2+a*e^2)^2+3/ 
32/a*(1/8*(7*a^4*d^7*e-119*a^3*b*c^2*d^5*e+125*a^2*b^2*c^4*d^3*e-5*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*(45*a^3*b*c*d^6-165*a^2*b^2*c^3*d^4+ 
47*a*b^3*c^5*d^2+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)))))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(5/2)/(d*x+c)**3/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(5/2)/(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. 1952 vs. \(2 (730) = 1460\).

Time = 0.33 (sec) , antiderivative size = 1952, normalized size of antiderivative = 2.27 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/64*e^2*(6*(5*(a*b^3*e^2)^(1/4)*a*b^4*c^6*d*e - 125*(a*b^3*e^2)^(1/4)*a^ 
2*b^3*c^4*d^3*e + 119*(a*b^3*e^2)^(1/4)*a^3*b^2*c^2*d^5*e - 7*(a*b^3*e^2)^ 
(1/4)*a^4*b*d^7*e - (a*b^3*e^2)^(3/4)*b^3*c^7 - 47*(a*b^3*e^2)^(3/4)*a*b^2 
*c^5*d^2 + 165*(a*b^3*e^2)^(3/4)*a^2*b*c^3*d^4 - 45*(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^2*b^7*c^10*e + 5*sqrt(2)*a^3*b^6*c^8*d^2*e + 10*sqrt(2 
)*a^4*b^5*c^6*d^4*e + 10*sqrt(2)*a^5*b^4*c^4*d^6*e + 5*sqrt(2)*a^6*b^3*c^2 
*d^8*e + sqrt(2)*a^7*b^2*d^10*e) + 6*(5*(a*b^3*e^2)^(1/4)*a*b^4*c^6*d*e - 
125*(a*b^3*e^2)^(1/4)*a^2*b^3*c^4*d^3*e + 119*(a*b^3*e^2)^(1/4)*a^3*b^2*c^ 
2*d^5*e - 7*(a*b^3*e^2)^(1/4)*a^4*b*d^7*e - (a*b^3*e^2)^(3/4)*b^3*c^7 - 47 
*(a*b^3*e^2)^(3/4)*a*b^2*c^5*d^2 + 165*(a*b^3*e^2)^(3/4)*a^2*b*c^3*d^4 - 4 
5*(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^2*b^7*c^10*e + 5*sqrt(2)*a^3* 
b^6*c^8*d^2*e + 10*sqrt(2)*a^4*b^5*c^6*d^4*e + 10*sqrt(2)*a^5*b^4*c^4*d^6* 
e + 5*sqrt(2)*a^6*b^3*c^2*d^8*e + sqrt(2)*a^7*b^2*d^10*e) + 3*(5*(a*b^3*e^ 
2)^(1/4)*a*b^4*c^6*d*e - 125*(a*b^3*e^2)^(1/4)*a^2*b^3*c^4*d^3*e + 119*(a* 
b^3*e^2)^(1/4)*a^3*b^2*c^2*d^5*e - 7*(a*b^3*e^2)^(1/4)*a^4*b*d^7*e + (a*b^ 
3*e^2)^(3/4)*b^3*c^7 + 47*(a*b^3*e^2)^(3/4)*a*b^2*c^5*d^2 - 165*(a*b^3*e^2 
)^(3/4)*a^2*b*c^3*d^4 + 45*(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^2*b^7*c^10*e + 5*...
 

Mupad [B] (verification not implemented)

Time = 13.27 (sec) , antiderivative size = 10463, normalized size of antiderivative = 12.17 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((e*x)^(5/2)/((a + b*x^2)^3*(c + d*x)^3),x)
 

Output:

(((e*x)^(5/2)*(11*a^3*d^7*e^6 + 25*b^3*c^6*d*e^6 - 275*a*b^2*c^4*d^3*e^6 + 
 95*a^2*b*c^2*d^5*e^6))/(16*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b 
*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) - ((e*x)^(3/2)*(b^3*c^7*e^7 - 58*a^3*c*d^6* 
e^7 + 16*a*b^2*c^5*d^2*e^7 + 149*a^2*b*c^3*d^4*e^7))/(16*(a^4*d^8 + b^4*c^ 
8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) + (3*(e*x)^(1/ 
2)*(13*a^3*c^2*d^5*e^8 + 5*a*b^2*c^6*d*e^8 - 46*a^2*b*c^4*d^3*e^8))/(16*(a 
^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) 
 + ((e*x)^(9/2)*(7*a^3*b*d^7*e^4 + 6*b^4*c^6*d*e^4 + 44*a^2*b^2*c^2*d^5*e^ 
4 - 149*a*b^3*c^4*d^3*e^4))/(16*a*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4 
*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) + (3*d^2*(e*x)^(11/2)*(b^4*c^5*e^3 - 
38*a*b^3*c^3*d^2*e^3 + 25*a^2*b^2*c*d^4*e^3))/(16*a*(a^4*d^8 + b^4*c^8 + 4 
*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) + (b*(e*x)^(7/2)*(3 
*b^3*c^7*e^5 + 137*a^3*c*d^6*e^5 - a*b^2*c^5*d^2*e^5 - 251*a^2*b*c^3*d^4*e 
^5))/(16*a*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 + 6*a^2* 
b^2*c^4*d^4)))/(a^2*c^2*e^6 + e^2*x^2*(a^2*d^2*e^4 + 2*a*b*c^2*e^4) + e^4* 
x^4*(b^2*c^2*e^2 + 2*a*b*d^2*e^2) + b^2*d^2*e^6*x^6 + 2*a^2*c*d*e^6*x + 2* 
b^2*c*d*e^6*x^5 + 4*a*b*c*d*e^6*x^3) + symsum(log(root(204010946560*a^23*b 
^3*c^4*d^36*g^6 + 135259257569280*a^17*b^9*c^16*d^24*g^6 + 135259257569280 
*a^13*b^13*c^24*d^16*g^6 + 204010946560*a^7*b^19*c^36*d^4*g^6 + 5202279137 
280*a^21*b^5*c^8*d^32*g^6 + 5202279137280*a^9*b^17*c^32*d^8*g^6 + 21474...
 

Reduce [F]

\[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\int \frac {\left (e x \right )^{\frac {5}{2}}}{\left (d x +c \right )^{3} \left (b \,x^{2}+a \right )^{3}}d x \] Input:

int((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a)^3,x)
 

Output:

int((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a)^3,x)