\(\int \frac {x^3}{(c+d x)^{5/2} (a-b x^2)^3} \, dx\) [735]

Optimal result

Integrand size = 23, antiderivative size = 399 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=-\frac {2 c^3 d^2}{3 \left (b c^2-a d^2\right )^3 (c+d x)^{3/2}}-\frac {6 c^2 d^2 \left (b c^2+a d^2\right )}{\left (b c^2-a d^2\right )^4 \sqrt {c+d x}}+\frac {a \sqrt {c+d x} \left (b c^2 (c-3 d x)+a d^2 (3 c-d x)\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (b^2 c^4 (8 c-29 d x)+6 a b c^2 d^2 (10 c-9 d x)+5 a^2 d^4 (4 c-d x)\right )}{16 \left (b c^2-a d^2\right )^4 \left (a-b x^2\right )}+\frac {5 d \left (6 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 \sqrt {a} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{9/2}}-\frac {5 d \left (6 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 \sqrt {a} b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{9/2}} \] Output:

-2/3*c^3*d^2/(-a*d^2+b*c^2)^3/(d*x+c)^(3/2)-6*c^2*d^2*(a*d^2+b*c^2)/(-a*d^ 
2+b*c^2)^4/(d*x+c)^(1/2)+1/4*a*(d*x+c)^(1/2)*(b*c^2*(-3*d*x+c)+a*d^2*(-d*x 
+3*c))/(-a*d^2+b*c^2)^3/(-b*x^2+a)^2-1/16*(d*x+c)^(1/2)*(b^2*c^4*(-29*d*x+ 
8*c)+6*a*b*c^2*d^2*(-9*d*x+10*c)+5*a^2*d^4*(-d*x+4*c))/(-a*d^2+b*c^2)^4/(- 
b*x^2+a)+5/32*d*(6*b^(1/2)*c+a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^( 
1/2)*c-a^(1/2)*d)^(1/2))/a^(1/2)/b^(3/4)/(b^(1/2)*c-a^(1/2)*d)^(9/2)-5/32* 
d*(6*b^(1/2)*c-a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2) 
*d)^(1/2))/a^(1/2)/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(9/2)
 

Mathematica [A] (verified)

Time = 3.63 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\frac {b^3 c^4 x^2 \left (24 c^3-39 c^2 d x-470 c d^2 x^2-375 d^3 x^3\right )+a^3 d^4 \left (-352 c^3-453 c^2 d x-42 c d^2 x^2+27 d^3 x^3\right )+a b^2 c^2 \left (-12 c^5+27 c^4 d x+910 c^3 d^2 x^2+825 c^2 d^3 x^3-400 c d^4 x^4-450 d^5 x^5\right )+a^2 b d^2 \left (-476 c^5-414 c^4 d x+788 c^3 d^2 x^2+867 c^2 d^3 x^3+30 c d^4 x^4-15 d^5 x^5\right )}{48 \left (b c^2-a d^2\right )^4 (c+d x)^{3/2} \left (a-b x^2\right )^2}+\frac {5 d \left (-6 \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{32 \sqrt {a} \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )^4 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {5 d \left (6 \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{32 \sqrt {a} \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^4 \sqrt {-b c+\sqrt {a} \sqrt {b} d}} \] Input:

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

Output:

(b^3*c^4*x^2*(24*c^3 - 39*c^2*d*x - 470*c*d^2*x^2 - 375*d^3*x^3) + a^3*d^4 
*(-352*c^3 - 453*c^2*d*x - 42*c*d^2*x^2 + 27*d^3*x^3) + a*b^2*c^2*(-12*c^5 
 + 27*c^4*d*x + 910*c^3*d^2*x^2 + 825*c^2*d^3*x^3 - 400*c*d^4*x^4 - 450*d^ 
5*x^5) + a^2*b*d^2*(-476*c^5 - 414*c^4*d*x + 788*c^3*d^2*x^2 + 867*c^2*d^3 
*x^3 + 30*c*d^4*x^4 - 15*d^5*x^5))/(48*(b*c^2 - a*d^2)^4*(c + d*x)^(3/2)*( 
a - b*x^2)^2) + (5*d*(-6*Sqrt[b]*c + Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) - Sqrt 
[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/(32*Sqrt[a]*Sqrt[b 
]*(Sqrt[b]*c + Sqrt[a]*d)^4*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]) + (5*d*(6*Sq 
rt[b]*c + Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x 
])/(Sqrt[b]*c - Sqrt[a]*d)])/(32*Sqrt[a]*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)^4 
*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d])
 

Rubi [A] (verified)

Time = 4.98 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.49, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {561, 25, 27, 1673, 27, 2198, 27, 2195, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1673

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 \left (\frac {8 a b c^3}{d^2}-\frac {8 a b \left (b c^2-3 a d^2\right ) (c+d x) c^2}{d^2 \left (b c^2-a d^2\right )}+\frac {4 a^2 b \left (b c^2+5 a d^2\right ) (c+d x)^2 c}{\left (b c^2-a d^2\right )^2}-\frac {5 a^2 b \left (3 b c^2+a d^2\right ) (c+d x)^3}{\left (b c^2-a d^2\right )^2}\right )}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2198

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {d^4 \int -\frac {2 \left (\frac {32 a^2 b^2 c^3}{d^4}+\frac {32 a^2 b^2 \left (b c^2+3 a d^2\right ) (c+d x) c^2}{d^4 \left (b c^2-a d^2\right )}-\frac {a^2 b^2 \left (5 b^2 c^4-126 a b d^2 c^2-55 a^2 d^4\right ) (c+d x)^2 c}{d^4 \left (b c^2-a d^2\right )^2}-\frac {a^2 b^2 \left (29 b^2 c^4+54 a b d^2 c^2+5 a^2 d^4\right ) (c+d x)^3}{d^4 \left (b c^2-a d^2\right )^2}\right )}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a b \sqrt {c+d x} \left (c \left (25 a^2 d^4+114 a b c^2 d^2+37 b^2 c^4\right )-(c+d x) \left (5 a^2 d^4+54 a b c^2 d^2+29 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \frac {\frac {32 a^2 b^2 c^3}{d^4}+\frac {32 a^2 b^2 \left (b c^2+3 a d^2\right ) (c+d x) c^2}{d^4 \left (b c^2-a d^2\right )}-\frac {a^2 b^2 \left (5 b^2 c^4-126 a b d^2 c^2-55 a^2 d^4\right ) (c+d x)^2 c}{d^4 \left (b c^2-a d^2\right )^2}-\frac {a^2 b^2 \left (29 b^2 c^4+54 a b d^2 c^2+5 a^2 d^4\right ) (c+d x)^3}{d^4 \left (b c^2-a d^2\right )^2}}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a b \sqrt {c+d x} \left (c \left (25 a^2 d^4+114 a b c^2 d^2+37 b^2 c^4\right )-(c+d x) \left (5 a^2 d^4+54 a b c^2 d^2+29 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 2195

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \left (\frac {32 a^2 b^2 c^3}{d^2 \left (a d^2-b c^2\right ) (c+d x)^2}-\frac {96 a^2 b^2 \left (b c^2+a d^2\right ) c^2}{d^2 \left (a d^2-b c^2\right )^2 (c+d x)}+\frac {5 a^2 b^2 \left (\left (25 b^2 c^4+30 a b d^2 c^2+a^2 d^4\right ) (c+d x)-c \left (31 b^2 c^4+70 a b d^2 c^2+11 a^2 d^4\right )\right )}{d^2 \left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a b \sqrt {c+d x} \left (c \left (25 a^2 d^4+114 a b c^2 d^2+37 b^2 c^4\right )-(c+d x) \left (5 a^2 d^4+54 a b c^2 d^2+29 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {a b \sqrt {c+d x} \left (c \left (25 a^2 d^4+114 a b c^2 d^2+37 b^2 c^4\right )-(c+d x) \left (5 a^2 d^4+54 a b c^2 d^2+29 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {d^4 \left (-\frac {5 a^{3/2} b^{5/4} \left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (\sqrt {a} d+6 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 d^3 \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {5 a^{3/2} b^{5/4} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (6 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 d^3 \left (\sqrt {a} d+\sqrt {b} c\right )^{5/2}}+\frac {96 a^2 b^2 c^2 \left (a d^2+b c^2\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )^2}+\frac {32 a^2 b^2 c^3}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{4 a b \left (b c^2-a d^2\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

Input:

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

Output:

(-2*(-1/8*(a*d^4*Sqrt[c + d*x]*(4*c*(b*c^2 + a*d^2) - (3*b*c^2 + a*d^2)*(c 
 + d*x)))/((b*c^2 - a*d^2)^3*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b 
*(c + d*x)^2)/d^2)^2) - (d^4*(-1/4*(a*b*Sqrt[c + d*x]*(c*(37*b^2*c^4 + 114 
*a*b*c^2*d^2 + 25*a^2*d^4) - (29*b^2*c^4 + 54*a*b*c^2*d^2 + 5*a^2*d^4)*(c 
+ d*x)))/((b*c^2 - a*d^2)^3*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b* 
(c + d*x)^2)/d^2)) - (d^4*((32*a^2*b^2*c^3)/(3*d^2*(b*c^2 - a*d^2)*(c + d* 
x)^(3/2)) + (96*a^2*b^2*c^2*(b*c^2 + a*d^2))/(d^2*(b*c^2 - a*d^2)^2*Sqrt[c 
 + d*x]) - (5*a^(3/2)*b^(5/4)*(Sqrt[b]*c + Sqrt[a]*d)^2*(6*Sqrt[b]*c + Sqr 
t[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*d 
^3*(Sqrt[b]*c - Sqrt[a]*d)^(5/2)) + (5*a^(3/2)*b^(5/4)*(Sqrt[b]*c - Sqrt[a 
]*d)^2*(6*Sqrt[b]*c - Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt 
[b]*c + Sqrt[a]*d]])/(2*d^3*(Sqrt[b]*c + Sqrt[a]*d)^(5/2))))/(4*a*b*(b*c^2 
 - a*d^2))))/(8*a*b*(b*c^2 - a*d^2))))/d^4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1673

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q 
, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a 
*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), 
 x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[x^m*(a + b*x^2 + c*x^4)^(p + 
 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + 
 e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 
) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] 
&& ILtQ[m/2, 0]
 

rule 2009

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

rule 2195

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; 
FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
 

rule 2198

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol 
ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial 
Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) 
^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 
 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[x^m*(a + b*x^2 + 
 c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* 
p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - 
m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x 
^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
 

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.38

method	result	size
pseudoelliptic	\(-\frac {25 \left (d^{2} \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\frac {1}{10} a^{2} d^{4}+3 b \,c^{2} d^{2} a +\frac {5}{2} b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}+b c \left (a^{2} d^{4}+4 b \,c^{2} d^{2} a +\frac {3}{5} b^{2} c^{4}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d^{2} \left (d x +c \right )^{\frac {3}{2}} \left (\left (-\frac {1}{10} a^{2} d^{4}-3 b \,c^{2} d^{2} a -\frac {5}{2} b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}+b c \left (a^{2} d^{4}+4 b \,c^{2} d^{2} a +\frac {3}{5} b^{2} c^{4}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {352 \sqrt {a b \,d^{2}}\, \left (-\frac {27 x^{3} \left (-\frac {5 b \,x^{2}}{9}+a \right ) a^{2} d^{7}}{352}+\frac {21 x^{2} a^{2} \left (-\frac {5 b \,x^{2}}{7}+a \right ) c \,d^{6}}{176}+\frac {453 x a \,c^{2} \left (\frac {150}{151} b^{2} x^{4}-\frac {289}{151} a b \,x^{2}+a^{2}\right ) d^{5}}{352}+a \,c^{3} \left (a^{2}+\frac {25}{22} b^{2} x^{4}-\frac {197}{88} a b \,x^{2}\right ) d^{4}+\frac {207 x b \,c^{4} \left (\frac {125}{138} b^{2} x^{4}-\frac {275}{138} a b \,x^{2}+a^{2}\right ) d^{3}}{176}+\frac {119 b \,c^{5} \left (\frac {235}{238} b^{2} x^{4}-\frac {65}{34} a b \,x^{2}+a^{2}\right ) d^{2}}{88}-\frac {27 x \left (-\frac {13 b \,x^{2}}{9}+a \right ) b^{2} c^{6} d}{352}+\frac {3 b^{2} c^{7} \left (-2 b \,x^{2}+a \right )}{88}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{75}\right )\right )}{16 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right )^{4}}\)	\(551\)
derivativedivides	\(-2 d^{2} \left (\frac {\frac {\left (\frac {5}{32} a^{2} b \,d^{4}+\frac {27}{16} a \,b^{2} c^{2} d^{2}+\frac {29}{32} b^{3} c^{4}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {\left (35 a^{2} d^{4}+222 b \,c^{2} d^{2} a +95 b^{2} c^{4}\right ) b c \left (d x +c \right )^{\frac {5}{2}}}{32}+\left (-\frac {9}{32} a^{3} d^{6}-\frac {7}{32} a^{2} b \,c^{2} d^{4}+\frac {265}{32} a \,b^{2} c^{4} d^{2}+\frac {103}{32} b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {c \left (41 a^{3} d^{6}+89 a^{2} b \,c^{2} d^{4}-93 a \,b^{2} c^{4} d^{2}-37 b^{3} c^{6}\right ) \sqrt {d x +c}}{32}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {5 b \left (-\frac {\left (-10 a^{2} c \,d^{4} b -40 a \,b^{2} c^{3} d^{2}-6 b^{3} c^{5}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+30 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+25 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (10 a^{2} c \,d^{4} b +40 a \,b^{2} c^{3} d^{2}+6 b^{3} c^{5}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+30 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+25 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{\left (a \,d^{2}-b \,c^{2}\right )^{4}}-\frac {c^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {3 c^{2} \left (a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}\right )\)	\(573\)
default	\(2 d^{2} \left (-\frac {\frac {\left (\frac {5}{32} a^{2} b \,d^{4}+\frac {27}{16} a \,b^{2} c^{2} d^{2}+\frac {29}{32} b^{3} c^{4}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {\left (35 a^{2} d^{4}+222 b \,c^{2} d^{2} a +95 b^{2} c^{4}\right ) b c \left (d x +c \right )^{\frac {5}{2}}}{32}+\left (-\frac {9}{32} a^{3} d^{6}-\frac {7}{32} a^{2} b \,c^{2} d^{4}+\frac {265}{32} a \,b^{2} c^{4} d^{2}+\frac {103}{32} b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {c \left (41 a^{3} d^{6}+89 a^{2} b \,c^{2} d^{4}-93 a \,b^{2} c^{4} d^{2}-37 b^{3} c^{6}\right ) \sqrt {d x +c}}{32}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {5 b \left (-\frac {\left (-10 a^{2} c \,d^{4} b -40 a \,b^{2} c^{3} d^{2}-6 b^{3} c^{5}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+30 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+25 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (10 a^{2} c \,d^{4} b +40 a \,b^{2} c^{3} d^{2}+6 b^{3} c^{5}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+30 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+25 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{\left (a \,d^{2}-b \,c^{2}\right )^{4}}+\frac {c^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {3 c^{2} \left (a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}\right )\)	\(574\)

Input:

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

Output:

-25/16/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)/(d*x+c)^(3/2)*(d^2* 
(d*x+c)^(3/2)*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*((1/10*a^2*d^4+3*b*c^2*d^2*a 
+5/2*b^2*c^4)*(a*b*d^2)^(1/2)+b*c*(a^2*d^4+4*b*c^2*d^2*a+3/5*b^2*c^4))*(-b 
*x^2+a)^2*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+((-b*c+ 
(a*b*d^2)^(1/2))*b)^(1/2)*(d^2*(d*x+c)^(3/2)*((-1/10*a^2*d^4-3*b*c^2*d^2*a 
-5/2*b^2*c^4)*(a*b*d^2)^(1/2)+b*c*(a^2*d^4+4*b*c^2*d^2*a+3/5*b^2*c^4))*(-b 
*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+352/75* 
(a*b*d^2)^(1/2)*(-27/352*x^3*(-5/9*b*x^2+a)*a^2*d^7+21/176*x^2*a^2*(-5/7*b 
*x^2+a)*c*d^6+453/352*x*a*c^2*(150/151*b^2*x^4-289/151*a*b*x^2+a^2)*d^5+a* 
c^3*(a^2+25/22*b^2*x^4-197/88*a*b*x^2)*d^4+207/176*x*b*c^4*(125/138*b^2*x^ 
4-275/138*a*b*x^2+a^2)*d^3+119/88*b*c^5*(235/238*b^2*x^4-65/34*a*b*x^2+a^2 
)*d^2-27/352*x*(-13/9*b*x^2+a)*b^2*c^6*d+3/88*b^2*c^7*(-2*b*x^2+a))*((b*c+ 
(a*b*d^2)^(1/2))*b)^(1/2)))/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(-b*x^2+a)^2/ 
(a*d^2-b*c^2)^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11704 vs. \(2 (335) = 670\).

Time = 14.41 (sec) , antiderivative size = 11704, normalized size of antiderivative = 29.33 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2731 vs. \(2 (335) = 670\).

Time = 0.61 (sec) , antiderivative size = 2731, normalized size of antiderivative = 6.84 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

5/32*((b^4*c^8*d - 4*a*b^3*c^6*d^3 + 6*a^2*b^2*c^4*d^5 - 4*a^3*b*c^2*d^7 + 
 a^4*d^9)^2*(25*sqrt(a*b)*a*b^2*c^4*d^2 + 30*sqrt(a*b)*a^2*b*c^2*d^4 + sqr 
t(a*b)*a^3*d^6)*abs(b) - (31*a*b^7*c^13*d^2 - 54*a^2*b^6*c^11*d^4 - 83*a^3 
*b^5*c^9*d^6 + 252*a^4*b^4*c^7*d^8 - 183*a^5*b^3*c^5*d^10 + 26*a^6*b^2*c^3 
*d^12 + 11*a^7*b*c*d^14)*abs(b^4*c^8*d - 4*a*b^3*c^6*d^3 + 6*a^2*b^2*c^4*d 
^5 - 4*a^3*b*c^2*d^7 + a^4*d^9)*abs(b) + 2*(3*sqrt(a*b)*b^11*c^22*d^2 - 4* 
sqrt(a*b)*a*b^10*c^20*d^4 - 71*sqrt(a*b)*a^2*b^9*c^18*d^6 + 352*sqrt(a*b)* 
a^3*b^8*c^16*d^8 - 770*sqrt(a*b)*a^4*b^7*c^14*d^10 + 952*sqrt(a*b)*a^5*b^6 
*c^12*d^12 - 686*sqrt(a*b)*a^6*b^5*c^10*d^14 + 256*sqrt(a*b)*a^7*b^4*c^8*d 
^16 - 17*sqrt(a*b)*a^8*b^3*c^6*d^18 - 20*sqrt(a*b)*a^9*b^2*c^4*d^20 + 5*sq 
rt(a*b)*a^10*b*c^2*d^22)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^5*c^9 - 4*a 
*b^4*c^7*d^2 + 6*a^2*b^3*c^5*d^4 - 4*a^3*b^2*c^3*d^6 + a^4*b*c*d^8 + sqrt( 
(b^5*c^9 - 4*a*b^4*c^7*d^2 + 6*a^2*b^3*c^5*d^4 - 4*a^3*b^2*c^3*d^6 + a^4*b 
*c*d^8)^2 - (b^5*c^10 - 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 - 10*a^3*b^2* 
c^4*d^6 + 5*a^4*b*c^2*d^8 - a^5*d^10)*(b^5*c^8 - 4*a*b^4*c^6*d^2 + 6*a^2*b 
^3*c^4*d^4 - 4*a^3*b^2*c^2*d^6 + a^4*b*d^8)))/(b^5*c^8 - 4*a*b^4*c^6*d^2 + 
 6*a^2*b^3*c^4*d^4 - 4*a^3*b^2*c^2*d^6 + a^4*b*d^8)))/((a*b^10*c^17 - 8*a^ 
2*b^9*c^15*d^2 + 28*a^3*b^8*c^13*d^4 - 56*a^4*b^7*c^11*d^6 + 70*a^5*b^6*c^ 
9*d^8 - 56*a^6*b^5*c^7*d^10 + 28*a^7*b^4*c^5*d^12 - 8*a^8*b^3*c^3*d^14 + a 
^9*b^2*c*d^16 - sqrt(a*b)*a*b^9*c^16*d + 8*sqrt(a*b)*a^2*b^8*c^14*d^3 -...
 

Mupad [B] (verification not implemented)

Time = 15.38 (sec) , antiderivative size = 16439, normalized size of antiderivative = 41.20 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((c + d*x)^(1/2)*(409600*a^17*b^2*d^38 + 14745600*b^19*c^34*d^4 + 27 
5660800*a*b^18*c^32*d^6 - 3139174400*a^2*b^17*c^30*d^8 + 11517952000*a^3*b 
^16*c^28*d^10 - 13762560000*a^4*b^15*c^26*d^12 - 31275417600*a^5*b^14*c^24 
*d^14 + 154645299200*a^6*b^13*c^22*d^16 - 297621913600*a^7*b^12*c^20*d^18 
+ 332152832000*a^8*b^11*c^18*d^20 - 213159936000*a^9*b^10*c^16*d^22 + 4678 
6150400*a^10*b^9*c^14*d^24 + 42824499200*a^11*b^8*c^12*d^26 - 43948441600* 
a^12*b^7*c^10*d^28 + 17719296000*a^13*b^6*c^8*d^30 - 3047424000*a^14*b^5*c 
^6*d^32 - 42598400*a^15*b^4*c^4*d^34 + 60620800*a^16*b^3*c^2*d^36) - (-(25 
*(a^5*d^13*(a^3*b^3)^(1/2) - 36*a*b^7*c^11*d^2 - 21*a^6*b^2*c*d^12 - 1405* 
a^2*b^6*c^9*d^4 - 5580*a^3*b^5*c^7*d^6 - 4662*a^4*b^4*c^5*d^8 - 840*a^5*b^ 
3*c^3*d^10 + 336*b^5*c^10*d^3*(a^3*b^3)^(1/2) + 3465*a*b^4*c^8*d^5*(a^3*b^ 
3)^(1/2) + 180*a^4*b*c^2*d^11*(a^3*b^3)^(1/2) + 6132*a^2*b^3*c^6*d^7*(a^3* 
b^3)^(1/2) + 2430*a^3*b^2*c^4*d^9*(a^3*b^3)^(1/2)))/(4096*(a^2*b^12*c^18 - 
 a^11*b^3*d^18 - 9*a^3*b^11*c^16*d^2 + 36*a^4*b^10*c^14*d^4 - 84*a^5*b^9*c 
^12*d^6 + 126*a^6*b^8*c^10*d^8 - 126*a^7*b^7*c^8*d^10 + 84*a^8*b^6*c^6*d^1 
2 - 36*a^9*b^5*c^4*d^14 + 9*a^10*b^4*c^2*d^16)))^(1/2)*((c + d*x)^(1/2)*(- 
(25*(a^5*d^13*(a^3*b^3)^(1/2) - 36*a*b^7*c^11*d^2 - 21*a^6*b^2*c*d^12 - 14 
05*a^2*b^6*c^9*d^4 - 5580*a^3*b^5*c^7*d^6 - 4662*a^4*b^4*c^5*d^8 - 840*a^5 
*b^3*c^3*d^10 + 336*b^5*c^10*d^3*(a^3*b^3)^(1/2) + 3465*a*b^4*c^8*d^5*(a^3 
*b^3)^(1/2) + 180*a^4*b*c^2*d^11*(a^3*b^3)^(1/2) + 6132*a^2*b^3*c^6*d^7...
 

Reduce [B] (verification not implemented)

Time = 23.30 (sec) , antiderivative size = 8246, normalized size of antiderivative = 20.67 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 30*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + 
d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**5*c*d**7 - 30*sqrt(a)* 
sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b 
)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**5*d**8*x - 1200*sqrt(a)*sqrt(c + d*x) 
*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b 
)*sqrt(a)*d - b*c)))*a**4*b*c**3*d**5 - 1200*sqrt(a)*sqrt(c + d*x)*sqrt(sq 
rt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a 
)*d - b*c)))*a**4*b*c**2*d**6*x + 60*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b* 
c)))*a**4*b*c*d**7*x**2 + 60*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d 
- b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a** 
4*b*d**8*x**3 - 1950*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*a 
tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b**2*c 
**5*d**3 - 1950*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan(( 
sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b**2*c**4*d 
**4*x + 2400*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqr 
t(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b**2*c**3*d**5 
*x**2 + 2400*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqr 
t(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b**2*c**2*d**6 
*x**3 - 30*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sq...