\(\int \frac {x^5}{(c+d x)^{5/2} (a-b x^2)^3} \, dx\) [733]

Optimal result

Integrand size = 23, antiderivative size = 412 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=-\frac {2 c^5}{3 \left (b c^2-a d^2\right )^3 (c+d x)^{3/2}}-\frac {2 c^4 \left (b c^2+5 a d^2\right )}{\left (b c^2-a d^2\right )^4 \sqrt {c+d x}}-\frac {a \sqrt {c+d x} \left (b^2 c^4 (16 c-53 d x)-a^2 d^4 (4 c-3 d x)+38 a b c^2 d^2 (2 c-d x)\right )}{16 b \left (b c^2-a d^2\right )^4 \left (a-b x^2\right )}+\frac {a^2 \sqrt {c+d x} \left (b c^2 (c-3 d x)+a d^2 (3 c-d x)\right )}{4 b \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )^2}+\frac {\left (32 b c^2+6 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 b^{7/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{9/2}}+\frac {\left (32 b c^2-6 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 b^{7/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{9/2}} \] Output:

-2/3*c^5/(-a*d^2+b*c^2)^3/(d*x+c)^(3/2)-2*c^4*(5*a*d^2+b*c^2)/(-a*d^2+b*c^ 
2)^4/(d*x+c)^(1/2)-1/16*a*(d*x+c)^(1/2)*(b^2*c^4*(-53*d*x+16*c)-a^2*d^4*(- 
3*d*x+4*c)+38*a*b*c^2*d^2*(-d*x+2*c))/b/(-a*d^2+b*c^2)^4/(-b*x^2+a)+1/4*a^ 
2*(d*x+c)^(1/2)*(b*c^2*(-3*d*x+c)+a*d^2*(-d*x+3*c))/b/(-a*d^2+b*c^2)^3/(-b 
*x^2+a)^2+1/32*(32*b*c^2+6*a^(1/2)*b^(1/2)*c*d-3*a*d^2)*arctanh(b^(1/4)*(d 
*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/b^(7/4)/(b^(1/2)*c-a^(1/2)*d)^(9/ 
2)+1/32*(32*b*c^2-6*a^(1/2)*b^(1/2)*c*d-3*a*d^2)*arctanh(b^(1/4)*(d*x+c)^( 
1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/b^(7/4)/(b^(1/2)*c+a^(1/2)*d)^(9/2)
 

Mathematica [A] (verified)

Time = 4.14 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.17 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {b} \left (-3 a^4 d^4 (-8 c+d x) (c+d x)^2+32 b^4 c^6 x^4 (4 c+3 d x)+a b^3 c^4 x^2 \left (-304 c^3-129 c^2 d x+718 c d^2 x^2+639 d^3 x^3\right )-a^3 b d^2 \left (-652 c^5-750 c^4 d x+60 c^3 d^2 x^2+123 c^2 d^3 x^3+6 c d^4 x^4+9 d^5 x^5\right )+a^2 b^2 c^2 \left (164 c^5+45 c^4 d x-1334 c^3 d^2 x^2-1425 c^2 d^3 x^3+114 d^5 x^5\right )\right )}{\left (b c^2-a d^2\right )^4 (c+d x)^{3/2} \left (a-b x^2\right )^2}+\frac {3 \left (32 b c^2-6 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^4 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {3 \left (32 b c^2+6 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^4 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{96 b^{3/2}} \] Input:

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

Output:

((-2*Sqrt[b]*(-3*a^4*d^4*(-8*c + d*x)*(c + d*x)^2 + 32*b^4*c^6*x^4*(4*c + 
3*d*x) + a*b^3*c^4*x^2*(-304*c^3 - 129*c^2*d*x + 718*c*d^2*x^2 + 639*d^3*x 
^3) - a^3*b*d^2*(-652*c^5 - 750*c^4*d*x + 60*c^3*d^2*x^2 + 123*c^2*d^3*x^3 
 + 6*c*d^4*x^4 + 9*d^5*x^5) + a^2*b^2*c^2*(164*c^5 + 45*c^4*d*x - 1334*c^3 
*d^2*x^2 - 1425*c^2*d^3*x^3 + 114*d^5*x^5)))/((b*c^2 - a*d^2)^4*(c + d*x)^ 
(3/2)*(a - b*x^2)^2) + (3*(32*b*c^2 - 6*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*Arc 
Tan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]* 
d)])/((Sqrt[b]*c + Sqrt[a]*d)^4*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]) + (3*(32 
*b*c^2 + 6*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sq 
rt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/((Sqrt[b]*c - Sqrt[a]*d) 
^4*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/(96*b^(3/2))
 

Rubi [A] (verified)

Time = 5.67 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.51, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1673

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 \left (\frac {8 a b c^5}{d^2}-\frac {8 a b \left (3 b c^2-5 a d^2\right ) (c+d x) c^4}{d^2 \left (b c^2-a d^2\right )}+\frac {4 a \left (6 b^3 c^6-18 a b^2 d^2 c^4+19 a^2 b d^4 c^2-a^3 d^6\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )^2}-\frac {a \left (8 b^3 c^6-24 a b^2 d^2 c^4+39 a^2 b d^4 c^2-3 a^3 d^6\right ) (c+d x)^3}{d^2 \left (b c^2-a d^2\right )^2}\right )}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \int \frac {\frac {8 a b c^5}{d^2}-\frac {8 a b \left (3 b c^2-5 a d^2\right ) (c+d x) c^4}{d^2 \left (b c^2-a d^2\right )}+\frac {4 a \left (6 b^3 c^6-18 a b^2 d^2 c^4+19 a^2 b d^4 c^2-a^3 d^6\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )^2}-\frac {a \left (8 b^3 c^6-24 a b^2 d^2 c^4+39 a^2 b d^4 c^2-3 a^3 d^6\right ) (c+d x)^3}{d^2 \left (b c^2-a d^2\right )^2}}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2198

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {d^4 \int -\frac {2 \left (\frac {32 a^2 b^2 c^5}{d^4}-\frac {32 a^2 b^2 \left (b c^2-5 a d^2\right ) (c+d x) c^4}{d^4 \left (b c^2-a d^2\right )}-\frac {a^3 b \left (5 b^2 c^4-190 a b d^2 c^2+9 a^2 d^4\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )^2}-\frac {a^3 b \left (53 b^2 c^4+38 a b d^2 c^2-3 a^2 d^4\right ) (c+d x)^3}{d^2 \left (b c^2-a d^2\right )^2}\right )}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (c \left (-7 a^2 d^4+114 a b c^2 d^2+69 b^2 c^4\right )-(c+d x) \left (-3 a^2 d^4+38 a b c^2 d^2+53 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \frac {\frac {32 a^2 b^2 c^5}{d^4}-\frac {32 a^2 b^2 \left (b c^2-5 a d^2\right ) (c+d x) c^4}{d^4 \left (b c^2-a d^2\right )}-\frac {a^3 b \left (5 b^2 c^4-190 a b d^2 c^2+9 a^2 d^4\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )^2}-\frac {a^3 b \left (53 b^2 c^4+38 a b d^2 c^2-3 a^2 d^4\right ) (c+d x)^3}{d^2 \left (b c^2-a d^2\right )^2}}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (c \left (-7 a^2 d^4+114 a b c^2 d^2+69 b^2 c^4\right )-(c+d x) \left (-3 a^2 d^4+38 a b c^2 d^2+53 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2195

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \left (\frac {32 a^2 b^2 c^5}{d^2 \left (a d^2-b c^2\right ) (c+d x)^2}-\frac {32 a^2 b^2 \left (b c^2+5 a d^2\right ) c^4}{d^2 \left (a d^2-b c^2\right )^2 (c+d x)}+\frac {a^2 b \left (\left (32 b^3 c^6+213 a b^2 d^2 c^4+38 a^2 b d^4 c^2-3 a^3 d^6\right ) (c+d x)-c \left (32 b^3 c^6+347 a b^2 d^2 c^4+190 a^2 b d^4 c^2-9 a^3 d^6\right )\right )}{d^2 \left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (c \left (-7 a^2 d^4+114 a b c^2 d^2+69 b^2 c^4\right )-(c+d x) \left (-3 a^2 d^4+38 a b c^2 d^2+53 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \left (-\frac {a^2 \sqrt [4]{b} \left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (6 \sqrt {a} \sqrt {b} c d-3 a d^2+32 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 d^2 \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}-\frac {a^2 \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (-6 \sqrt {a} \sqrt {b} c d-3 a d^2+32 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 d^2 \left (\sqrt {a} d+\sqrt {b} c\right )^{5/2}}+\frac {32 a^2 b^2 c^5}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}+\frac {32 a^2 b^2 c^4 \left (5 a d^2+b c^2\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )^2}\right )}{4 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (c \left (-7 a^2 d^4+114 a b c^2 d^2+69 b^2 c^4\right )-(c+d x) \left (-3 a^2 d^4+38 a b c^2 d^2+53 b^2 c^4\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{8 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

Input:

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

Output:

(-2*(-1/8*(a^2*d^6*Sqrt[c + d*x]*(4*c*(b*c^2 + a*d^2) - (3*b*c^2 + a*d^2)* 
(c + d*x)))/(b*(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^2*d^2*Sqrt[c + d*x]*(c*(69*b^2*c 
^4 + 114*a*b*c^2*d^2 - 7*a^2*d^4) - (53*b^2*c^4 + 38*a*b*c^2*d^2 - 3*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^5)/(3*d^2*(b*c^2 - a*d^2)* 
(c + d*x)^(3/2)) + (32*a^2*b^2*c^4*(b*c^2 + 5*a*d^2))/(d^2*(b*c^2 - a*d^2) 
^2*Sqrt[c + d*x]) - (a^2*b^(1/4)*(Sqrt[b]*c + Sqrt[a]*d)^2*(32*b*c^2 + 6*S 
qrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b] 
*c - Sqrt[a]*d]])/(2*d^2*(Sqrt[b]*c - Sqrt[a]*d)^(5/2)) - (a^2*b^(1/4)*(Sq 
rt[b]*c - Sqrt[a]*d)^2*(32*b*c^2 - 6*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTan 
h[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*d^2*(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^6
 

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 = 1.19 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.42

method	result	size
pseudoelliptic	\(\frac {\frac {3 \left (d x +c \right )^{\frac {3}{2}} \left (\left (\frac {1}{2} a^{3} d^{6}-\frac {19}{3} a^{2} b \,c^{2} d^{4}-\frac {71}{2} a \,b^{2} c^{4} d^{2}-\frac {16}{3} b^{3} c^{6}\right ) \sqrt {a b \,d^{2}}+b a c \,d^{2} \left (a^{2} d^{4}-\frac {76}{3} b \,c^{2} d^{2} a -\frac {67}{3} b^{2} c^{4}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{16}+\frac {3 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (d x +c \right )^{\frac {3}{2}} \left (-b \,x^{2}+a \right )^{2} \left (\left (-\frac {1}{2} a^{3} d^{6}+\frac {19}{3} a^{2} b \,c^{2} d^{4}+\frac {71}{2} a \,b^{2} c^{4} d^{2}+\frac {16}{3} b^{3} c^{6}\right ) \sqrt {a b \,d^{2}}+b a c \,d^{2} \left (a^{2} d^{4}-\frac {76}{3} b \,c^{2} d^{2} a -\frac {67}{3} b^{2} c^{4}\right )\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {8 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (d x +c \right )^{2} d^{4} \left (-\frac {d x}{8}+c \right ) a^{4}+\frac {163 d^{2} \left (-\frac {9}{652} x^{5} d^{5}-\frac {3}{326} x^{4} c \,d^{4}-\frac {123}{652} c^{2} d^{3} x^{3}-\frac {15}{163} c^{3} d^{2} x^{2}+\frac {375}{326} c^{4} d x +c^{5}\right ) a^{3} b}{6}+\frac {41 \left (\frac {57}{82} x^{5} d^{5}-\frac {1425}{164} c^{2} d^{3} x^{3}-\frac {667}{82} c^{3} d^{2} x^{2}+\frac {45}{164} c^{4} d x +c^{5}\right ) a^{2} c^{2} b^{2}}{6}-\frac {38 x^{2} \left (-\frac {639}{304} d^{3} x^{3}-\frac {359}{152} c \,d^{2} x^{2}+\frac {129}{304} c^{2} d x +c^{3}\right ) a \,c^{4} b^{3}}{3}+\frac {16 x^{4} b^{4} c^{6} \left (\frac {3 d x}{4}+c \right )}{3}\right )}{3}\right )}{16}}{\left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} b \left (d x +c \right )^{\frac {3}{2}}}\)	\(586\)
derivativedivides	\(\frac {2 c^{5}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 c^{4} \left (5 a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}-\frac {2 \left (\frac {\left (-\frac {3}{32} a^{3} d^{6}+\frac {19}{16} a^{2} b \,c^{2} d^{4}+\frac {53}{32} a \,b^{2} c^{4} d^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}+\frac {a c \,d^{2} \left (13 a^{2} d^{4}-190 b \,c^{2} d^{2} a -175 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {a \,d^{2} \left (a^{3} d^{6}+63 a^{2} b \,c^{2} d^{4}-225 a \,b^{2} c^{4} d^{2}-191 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b}+\frac {a c \,d^{2} \left (9 a^{3} d^{6}+121 a^{2} b \,c^{2} d^{4}-61 a \,b^{2} c^{4} d^{2}-69 b^{3} c^{6}\right ) \sqrt {d x +c}}{32 b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\left (-6 d^{6} c \,a^{3} b +152 a^{2} c^{3} d^{4} b^{2}+134 a \,c^{5} d^{2} b^{3}-3 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+38 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}+213 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (6 d^{6} c \,a^{3} b -152 a^{2} c^{3} d^{4} b^{2}-134 a \,c^{5} d^{2} b^{3}-3 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+38 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}+213 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4}}\)	\(644\)
default	\(\frac {2 c^{5}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 c^{4} \left (5 a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}-\frac {2 \left (\frac {\left (-\frac {3}{32} a^{3} d^{6}+\frac {19}{16} a^{2} b \,c^{2} d^{4}+\frac {53}{32} a \,b^{2} c^{4} d^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}+\frac {a c \,d^{2} \left (13 a^{2} d^{4}-190 b \,c^{2} d^{2} a -175 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {a \,d^{2} \left (a^{3} d^{6}+63 a^{2} b \,c^{2} d^{4}-225 a \,b^{2} c^{4} d^{2}-191 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b}+\frac {a c \,d^{2} \left (9 a^{3} d^{6}+121 a^{2} b \,c^{2} d^{4}-61 a \,b^{2} c^{4} d^{2}-69 b^{3} c^{6}\right ) \sqrt {d x +c}}{32 b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\left (-6 d^{6} c \,a^{3} b +152 a^{2} c^{3} d^{4} b^{2}+134 a \,c^{5} d^{2} b^{3}-3 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+38 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}+213 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (6 d^{6} c \,a^{3} b -152 a^{2} c^{3} d^{4} b^{2}-134 a \,c^{5} d^{2} b^{3}-3 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+38 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}+213 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4}}\)	\(644\)

Input:

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

Output:

3/16*((d*x+c)^(3/2)*((1/2*a^3*d^6-19/3*a^2*b*c^2*d^4-71/2*a*b^2*c^4*d^2-16 
/3*b^3*c^6)*(a*b*d^2)^(1/2)+b*a*c*d^2*(a^2*d^4-76/3*b*c^2*d^2*a-67/3*b^2*c 
^4))*(-b*x^2+a)^2*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*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*x+c) 
^(3/2)*(-b*x^2+a)^2*((-1/2*a^3*d^6+19/3*a^2*b*c^2*d^4+71/2*a*b^2*c^4*d^2+1 
6/3*b^3*c^6)*(a*b*d^2)^(1/2)+b*a*c*d^2*(a^2*d^4-76/3*b*c^2*d^2*a-67/3*b^2* 
c^4))*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))-8/3*(a*b*d^ 
2)^(1/2)*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*((d*x+c)^2*d^4*(-1/8*d*x+c)*a^4+1 
63/6*d^2*(-9/652*x^5*d^5-3/326*x^4*c*d^4-123/652*c^2*d^3*x^3-15/163*c^3*d^ 
2*x^2+375/326*c^4*d*x+c^5)*a^3*b+41/6*(57/82*x^5*d^5-1425/164*c^2*d^3*x^3- 
667/82*c^3*d^2*x^2+45/164*c^4*d*x+c^5)*a^2*c^2*b^2-38/3*x^2*(-639/304*d^3* 
x^3-359/152*c*d^2*x^2+129/304*c^2*d*x+c^3)*a*c^4*b^3+16/3*x^4*b^4*c^6*(3/4 
*d*x+c))))/(a*b*d^2)^(1/2)/(d*x+c)^(3/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/( 
(-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*d^2-b*c^2)^4/(-b*x^2+a)^2/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12386 vs. \(2 (352) = 704\).

Time = 54.84 (sec) , antiderivative size = 12386, normalized size of antiderivative = 30.06 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-integrate(x^5/((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. 2824 vs. \(2 (352) = 704\).

Time = 0.65 (sec) , antiderivative size = 2824, normalized size of antiderivative = 6.85 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/32*((b^5*c^8*d - 4*a*b^4*c^6*d^3 + 6*a^2*b^3*c^4*d^5 - 4*a^3*b^2*c^2*d^7 
 + a^4*b*d^9)^2*(32*sqrt(a*b)*b^3*c^6 + 213*sqrt(a*b)*a*b^2*c^4*d^2 + 38*s 
qrt(a*b)*a^2*b*c^2*d^4 - 3*sqrt(a*b)*a^3*d^6)*abs(b) - (32*b^9*c^15 + 219* 
a*b^8*c^13*d^2 - 1006*a^2*b^7*c^11*d^4 + 1185*a^3*b^6*c^9*d^6 - 180*a^4*b^ 
5*c^7*d^8 - 467*a^5*b^4*c^5*d^10 + 226*a^6*b^3*c^3*d^12 - 9*a^7*b^2*c*d^14 
)*abs(b^5*c^8*d - 4*a*b^4*c^6*d^3 + 6*a^2*b^3*c^4*d^5 - 4*a^3*b^2*c^2*d^7 
+ a^4*b*d^9)*abs(b) + 2*(67*sqrt(a*b)*b^13*c^22*d^2 - 460*sqrt(a*b)*a*b^12 
*c^20*d^4 + 1265*sqrt(a*b)*a^2*b^11*c^18*d^6 - 1600*sqrt(a*b)*a^3*b^10*c^1 
6*d^8 + 350*sqrt(a*b)*a^4*b^9*c^14*d^10 + 1736*sqrt(a*b)*a^5*b^8*c^12*d^12 
 - 2590*sqrt(a*b)*a^6*b^7*c^10*d^14 + 1760*sqrt(a*b)*a^7*b^6*c^8*d^16 - 62 
5*sqrt(a*b)*a^8*b^5*c^6*d^18 + 100*sqrt(a*b)*a^9*b^4*c^4*d^20 - 3*sqrt(a*b 
)*a^10*b^3*c^2*d^22)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^6*c^9 - 4*a*b^5 
*c^7*d^2 + 6*a^2*b^4*c^5*d^4 - 4*a^3*b^3*c^3*d^6 + a^4*b^2*c*d^8 + sqrt((b 
^6*c^9 - 4*a*b^5*c^7*d^2 + 6*a^2*b^4*c^5*d^4 - 4*a^3*b^3*c^3*d^6 + a^4*b^2 
*c*d^8)^2 - (b^6*c^10 - 5*a*b^5*c^8*d^2 + 10*a^2*b^4*c^6*d^4 - 10*a^3*b^3* 
c^4*d^6 + 5*a^4*b^2*c^2*d^8 - a^5*b*d^10)*(b^6*c^8 - 4*a*b^5*c^6*d^2 + 6*a 
^2*b^4*c^4*d^4 - 4*a^3*b^3*c^2*d^6 + a^4*b^2*d^8)))/(b^6*c^8 - 4*a*b^5*c^6 
*d^2 + 6*a^2*b^4*c^4*d^4 - 4*a^3*b^3*c^2*d^6 + a^4*b^2*d^8)))/((b^12*c^17 
- 8*a*b^11*c^15*d^2 + 28*a^2*b^10*c^13*d^4 - 56*a^3*b^9*c^11*d^6 + 70*a^4* 
b^8*c^9*d^8 - 56*a^5*b^7*c^7*d^10 + 28*a^6*b^6*c^5*d^12 - 8*a^7*b^5*c^3...
 

Mupad [B] (verification not implemented)

Time = 16.21 (sec) , antiderivative size = 17026, normalized size of antiderivative = 41.33 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((c + d*x)^(1/2)*(147456*a^19*b^5*d^38 + 16777216*a*b^23*c^36*d^2 + 
316211200*a^2*b^22*c^34*d^4 - 3652567040*a^3*b^21*c^32*d^6 + 13673693184*a 
^4*b^20*c^30*d^8 - 17213030400*a^5*b^19*c^28*d^10 - 35371614208*a^6*b^18*c 
^26*d^12 + 186760232960*a^7*b^17*c^24*d^14 - 374120120320*a^8*b^16*c^22*d^ 
16 + 439514431488*a^9*b^15*c^20*d^18 - 308867891200*a^10*b^14*c^18*d^20 + 
96254394368*a^11*b^13*c^16*d^22 + 37380423680*a^12*b^12*c^14*d^24 - 564127 
33440*a^13*b^11*c^12*d^26 + 27954511872*a^14*b^10*c^10*d^28 - 6948454400*a 
^15*b^9*c^8*d^30 + 700186624*a^16*b^8*c^6*d^32 + 20316160*a^17*b^7*c^4*d^3 
4 - 4915200*a^18*b^6*c^2*d^36) - ((1024*b^10*c^13 - 9*a^6*d^13*(a*b^7)^(1/ 
2) + 40164*a*b^9*c^11*d^2 + 45*a^6*b^4*c*d^12 - 9600*b^6*c^12*d*(a*b^7)^(1 
/2) + 155349*a^2*b^8*c^9*d^4 + 112044*a^3*b^7*c^7*d^6 + 6534*a^4*b^6*c^5*d 
^8 - 1560*a^5*b^5*c^3*d^10 - 163089*a^2*b^4*c^8*d^5*(a*b^7)^(1/2) - 45684* 
a^3*b^3*c^6*d^7*(a*b^7)^(1/2) + 3026*a^4*b^2*c^4*d^9*(a*b^7)^(1/2) - 98400 
*a*b^5*c^10*d^3*(a*b^7)^(1/2) + 156*a^5*b*c^2*d^11*(a*b^7)^(1/2))/(4096*(b 
^16*c^18 - a^9*b^7*d^18 - 9*a*b^15*c^16*d^2 + 36*a^2*b^14*c^14*d^4 - 84*a^ 
3*b^13*c^12*d^6 + 126*a^4*b^12*c^10*d^8 - 126*a^5*b^11*c^8*d^10 + 84*a^6*b 
^10*c^6*d^12 - 36*a^7*b^9*c^4*d^14 + 9*a^8*b^8*c^2*d^16)))^(1/2)*((c + d*x 
)^(1/2)*((1024*b^10*c^13 - 9*a^6*d^13*(a*b^7)^(1/2) + 40164*a*b^9*c^11*d^2 
 + 45*a^6*b^4*c*d^12 - 9600*b^6*c^12*d*(a*b^7)^(1/2) + 155349*a^2*b^8*c^9* 
d^4 + 112044*a^3*b^7*c^7*d^6 + 6534*a^4*b^6*c^5*d^8 - 1560*a^5*b^5*c^3*...
 

Reduce [B] (verification not implemented)

Time = 23.21 (sec) , antiderivative size = 9268, normalized size of antiderivative = 22.50 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(18*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 + 18*sqrt(a)*sqr 
t(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*a**5*d**8*x - 192*sqrt(a)*sqrt(c + d*x)*sqr 
t(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)))*a**4*b*c**3*d**5 - 192*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**2*d**6*x - 36*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*d**7*x**2 - 36*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 - 2190*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**5*d 
**3 - 2190*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 
 + 384*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**3*d**5*x**2 
+ 384*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**2*d**6*x**3 + 
 18*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + ...