\(\int \frac {(c+d x)^{3/2}}{x (a-b x^2)^3} \, dx\) [702]

Optimal result

Integrand size = 23, antiderivative size = 275 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^3} \, dx=\frac {(c+d x)^{3/2}}{4 a \left (a-b x^2\right )^2}+\frac {\sqrt {c+d x} (8 c+5 d x)}{16 a^2 \left (a-b x^2\right )}-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}+\frac {\left (32 b c^2-34 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^3 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\left (32 b c^2+34 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^3 b^{3/4} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

1/4*(d*x+c)^(3/2)/a/(-b*x^2+a)^2+1/16*(d*x+c)^(1/2)*(5*d*x+8*c)/a^2/(-b*x^ 
2+a)-2*c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3+1/32*(32*b*c^2-34*a^(1/2 
)*b^(1/2)*c*d+5*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d) 
^(1/2))/a^3/b^(3/4)/(b^(1/2)*c-a^(1/2)*d)^(1/2)+1/32*(32*b*c^2+34*a^(1/2)* 
b^(1/2)*c*d+5*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^( 
1/2))/a^3/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 2.79 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 a \sqrt {c+d x} \left (-3 a (4 c+3 d x)+b x^2 (8 c+5 d x)\right )}{\left (a-b x^2\right )^2}+\frac {\left (32 b c^2+34 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {\left (32 b c^2-34 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-64 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{32 a^3} \] Input:

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

Output:

((-2*a*Sqrt[c + d*x]*(-3*a*(4*c + 3*d*x) + b*x^2*(8*c + 5*d*x)))/(a - b*x^ 
2)^2 + ((32*b*c^2 + 34*Sqrt[a]*Sqrt[b]*c*d + 5*a*d^2)*ArcTan[(Sqrt[-(b*c) 
- Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[b]*Sqr 
t[-(b*c) - Sqrt[a]*Sqrt[b]*d]) + ((32*b*c^2 - 34*Sqrt[a]*Sqrt[b]*c*d + 5*a 
*d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - 
 Sqrt[a]*d)])/(Sqrt[b]*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) - 64*c^(3/2)*ArcT 
anh[Sqrt[c + d*x]/Sqrt[c]])/(32*a^3)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(795\) vs. \(2(275)=550\).

Time = 2.39 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.89, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {561, 25, 27, 1650, 1492, 27, 1492, 27, 1480, 221, 1567, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1650

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\int \frac {c \left (a-\frac {b c^2}{d^2}\right )+\left (\frac {b c^2}{d^2}+a\right ) (c+d x)}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{a}\right )\)

\(\Big \downarrow \) 1492

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {d^4 \int \frac {2 a b \left (b c^2-a d^2\right ) (8 c+5 (c+d x))}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {1}{8} \int \frac {8 c+5 (c+d x)}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 1492

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {1}{8} \left (\frac {d^4 \int \frac {2 b \left (c \left (13 b c^2-29 a d^2\right )+\left (13 b c^2-5 a d^2\right ) (c+d x)\right )}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (3 a d^2+13 b c^2\right )-(c+d x) \left (13 b c^2-5 a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {1}{8} \left (\frac {\int \frac {c \left (13 b c^2-29 a d^2\right )+\left (13 b c^2-5 a d^2\right ) (c+d x)}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (3 a d^2+13 b c^2\right )-(c+d x) \left (13 b c^2-5 a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 1480

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {1}{8} \left (\frac {\frac {1}{2} \left (-\frac {26 b^{3/2} c^3}{\sqrt {a} d}+34 \sqrt {a} \sqrt {b} c d-5 a d^2+13 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (39 \sqrt {a} \sqrt {b} c d+5 a d^2+26 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (3 a d^2+13 b c^2\right )-(c+d x) \left (13 b c^2-5 a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -2 \left (\frac {c^2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {1}{8} \left (\frac {\frac {d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (39 \sqrt {a} \sqrt {b} c d+5 a d^2+26 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}+\frac {d^2 \left (-\frac {26 b^{3/2} c^3}{\sqrt {a} d}+34 \sqrt {a} \sqrt {b} c d-5 a d^2+13 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (3 a d^2+13 b c^2\right )-(c+d x) \left (13 b c^2-5 a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 1567

\(\displaystyle -2 \left (\frac {c^2 \int \left (-\frac {b x d^3}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )^2}-\frac {b x d}{a^2 \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a^2 x d}\right )d\sqrt {c+d x}}{a}-\frac {\frac {1}{8} \left (\frac {\frac {d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (39 \sqrt {a} \sqrt {b} c d+5 a d^2+26 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}+\frac {d^2 \left (-\frac {26 b^{3/2} c^3}{\sqrt {a} d}+34 \sqrt {a} \sqrt {b} c d-5 a d^2+13 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (3 a d^2+13 b c^2\right )-(c+d x) \left (13 b c^2-5 a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )+\frac {(c+d x)^{3/2}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {c^2 \left (\frac {b (c-d x) \sqrt {c+d x} d^2}{4 a \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right ) d}{8 a^{3/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right ) d}{8 a^{3/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^2 \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 a^2 \sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a}-\frac {\frac {(c+d x)^{3/2}}{8 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}+\frac {1}{8} \left (\frac {\frac {\left (-\frac {26 b^{3/2} c^3}{\sqrt {a} d}+13 b c^2+34 \sqrt {a} \sqrt {b} d c-5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right ) d^2}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (26 b c^2+39 \sqrt {a} \sqrt {b} d c+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right ) d}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {b} c+\sqrt {a} d}}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (13 b c^2+3 a d^2\right )-\left (13 b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}\right )}{a}\right )\)

Input:

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

Output:

-2*((c^2*((b*d^2*(c - d*x)*Sqrt[c + d*x])/(4*a*(b*c^2 - a*d^2)*(b*c^2 - a* 
d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)) + ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/( 
a^2*Sqrt[c]) + (b^(1/4)*d*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - 
 Sqrt[a]*d]])/(8*a^(3/2)*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)) - (b^(1/4)*ArcTanh 
[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^2*Sqrt[Sqrt[b] 
*c - Sqrt[a]*d]) - (b^(1/4)*d*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b] 
*c + Sqrt[a]*d]])/(8*a^(3/2)*(Sqrt[b]*c + Sqrt[a]*d)^(3/2)) - (b^(1/4)*Arc 
Tanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*a^2*Sqrt[Sqr 
t[b]*c + Sqrt[a]*d])))/a - ((c + d*x)^(3/2)/(8*(a - (b*c^2)/d^2 + (2*b*c*( 
c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)^2) + (-1/4*(Sqrt[c + d*x]*(c*(13*b*c^ 
2 + 3*a*d^2) - (13*b*c^2 - 5*a*d^2)*(c + d*x)))/(a*(b*c^2 - a*d^2)*(a - (b 
*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) + ((d^2*(13*b*c^ 
2 - (26*b^(3/2)*c^3)/(Sqrt[a]*d) + 34*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*ArcTa 
nh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(3/4)*Sqrt[S 
qrt[b]*c - Sqrt[a]*d]) + (d*(Sqrt[b]*c - Sqrt[a]*d)*(26*b*c^2 + 39*Sqrt[a] 
*Sqrt[b]*c*d + 5*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + S 
qrt[a]*d]])/(2*Sqrt[a]*b^(3/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d]))/(4*a*(b*c^2 - 
 a*d^2)))/8)/a)
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

rule 1567

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] 
 && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
 

rule 1650

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + 
 (e_.)*(x_)^2), x_Symbol] :> Simp[-f^4/(c*d^2 - b*d*e + a*e^2)   Int[(f*x)^ 
(m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Simp[d^2*(f 
^4/(c*d^2 - b*d*e + a*e^2))   Int[(f*x)^(m - 4)*((a + b*x^2 + c*x^4)^(p + 1 
)/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 
0] && LtQ[p, -1] && GtQ[m, 2]
 

rule 2009

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {\frac {17 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (-\frac {5 a \,d^{2}}{34}-\frac {16 b \,c^{2}}{17}\right ) \sqrt {a b \,d^{2}}+a b c \,d^{2}\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{16}+\frac {17 \left (\left (\left (\frac {5 a \,d^{2}}{34}+\frac {16 b \,c^{2}}{17}\right ) \sqrt {a b \,d^{2}}+a b c \,d^{2}\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {12 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-\frac {8 c^{\frac {3}{2}} \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{3}+\left (-\frac {2 \left (\frac {5 d x}{8}+c \right ) x^{2} b}{3}+a \left (\frac {3 d x}{4}+c \right )\right ) a \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}}{17}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{16}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} a^{3}}\)	\(312\)
derivativedivides	\(-2 d^{6} \left (\frac {c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6}}-\frac {\frac {-\frac {5 a b \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{32}+\frac {7 a b c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32}+\frac {a \,d^{2} \left (9 a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {3 a \,d^{2} c \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {d x +c}}{32}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (-\frac {\left (-34 a b c \,d^{2}-5 \sqrt {a b \,d^{2}}\, a \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (34 a b c \,d^{2}-5 \sqrt {a b \,d^{2}}\, a \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{a^{3} d^{6}}\right )\)	\(343\)
default	\(2 d^{6} \left (-\frac {c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6}}+\frac {\frac {-\frac {5 a b \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{32}+\frac {7 a b c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32}+\frac {a \,d^{2} \left (9 a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {3 a \,d^{2} c \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {d x +c}}{32}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (-\frac {\left (-34 a b c \,d^{2}-5 \sqrt {a b \,d^{2}}\, a \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (34 a b c \,d^{2}-5 \sqrt {a b \,d^{2}}\, a \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{a^{3} d^{6}}\right )\)	\(343\)

Input:

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

Output:

17/16/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1/ 
2))*b)^(1/2)*(((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*((-5/34*a*d^2-16/17*b*c^2)*( 
a*b*d^2)^(1/2)+a*b*c*d^2)*(-b*x^2+a)^2*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b* 
d^2)^(1/2))*b)^(1/2))+(((5/34*a*d^2+16/17*b*c^2)*(a*b*d^2)^(1/2)+a*b*c*d^2 
)*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+12 
/17*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(-8/3*c^(3/2)*(-b*x^2+a)^2*arctanh((d* 
x+c)^(1/2)/c^(1/2))+(-2/3*(5/8*d*x+c)*x^2*b+a*(3/4*d*x+c))*a*(d*x+c)^(1/2) 
)*(a*b*d^2)^(1/2))*((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))/(-b*x^2+a)^2/a^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2604 vs. \(2 (215) = 430\).

Time = 2.44 (sec) , antiderivative size = 5217, normalized size of antiderivative = 18.97 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (215) = 430\).

Time = 0.32 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^3} \, dx=\frac {2 \, c^{2} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} + \frac {{\left (34 \, a^{2} b c^{2} d^{2} {\left | b \right |} - {\left (32 \, b c^{2} + 5 \, a d^{2}\right )} a^{2} d^{2} {\left | b \right |} + {\left (32 \, \sqrt {a b} b c^{3} - 29 \, \sqrt {a b} a c d^{2}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{3} b c + \sqrt {a^{6} b^{2} c^{2} - {\left (a^{3} b c^{2} - a^{4} d^{2}\right )} a^{3} b}}{a^{3} b}}}\right )}{32 \, {\left (a^{4} b d - \sqrt {a b} a^{3} b c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {{\left (34 \, \sqrt {a b} a^{2} b c^{2} d^{2} {\left | b \right |} - {\left (32 \, \sqrt {a b} b c^{2} + 5 \, \sqrt {a b} a d^{2}\right )} a^{2} d^{2} {\left | b \right |} - {\left (32 \, a b^{2} c^{3} - 29 \, a^{2} b c d^{2}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{3} b c - \sqrt {a^{6} b^{2} c^{2} - {\left (a^{3} b c^{2} - a^{4} d^{2}\right )} a^{3} b}}{a^{3} b}}}\right )}{32 \, {\left (a^{4} b^{2} c + \sqrt {a b} a^{4} b d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} - \frac {5 \, {\left (d x + c\right )}^{\frac {7}{2}} b d^{2} - 7 \, {\left (d x + c\right )}^{\frac {5}{2}} b c d^{2} - {\left (d x + c\right )}^{\frac {3}{2}} b c^{2} d^{2} + 3 \, \sqrt {d x + c} b c^{3} d^{2} - 9 \, {\left (d x + c\right )}^{\frac {3}{2}} a d^{4} - 3 \, \sqrt {d x + c} a c d^{4}}{16 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )}^{2} a^{2}} \] Input:

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

Output:

2*c^2*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) + 1/32*(34*a^2*b*c^2*d 
^2*abs(b) - (32*b*c^2 + 5*a*d^2)*a^2*d^2*abs(b) + (32*sqrt(a*b)*b*c^3 - 29 
*sqrt(a*b)*a*c*d^2)*abs(a)*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqrt(-(a^3* 
b*c + sqrt(a^6*b^2*c^2 - (a^3*b*c^2 - a^4*d^2)*a^3*b))/(a^3*b)))/((a^4*b*d 
 - sqrt(a*b)*a^3*b*c)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a)*abs(d)) + 1/32*( 
34*sqrt(a*b)*a^2*b*c^2*d^2*abs(b) - (32*sqrt(a*b)*b*c^2 + 5*sqrt(a*b)*a*d^ 
2)*a^2*d^2*abs(b) - (32*a*b^2*c^3 - 29*a^2*b*c*d^2)*abs(a)*abs(b)*abs(d))* 
arctan(sqrt(d*x + c)/sqrt(-(a^3*b*c - sqrt(a^6*b^2*c^2 - (a^3*b*c^2 - a^4* 
d^2)*a^3*b))/(a^3*b)))/((a^4*b^2*c + sqrt(a*b)*a^4*b*d)*sqrt(-b^2*c + sqrt 
(a*b)*b*d)*abs(a)*abs(d)) - 1/16*(5*(d*x + c)^(7/2)*b*d^2 - 7*(d*x + c)^(5 
/2)*b*c*d^2 - (d*x + c)^(3/2)*b*c^2*d^2 + 3*sqrt(d*x + c)*b*c^3*d^2 - 9*(d 
*x + c)^(3/2)*a*d^4 - 3*sqrt(d*x + c)*a*c*d^4)/(((d*x + c)^2*b - 2*(d*x + 
c)*b*c + b*c^2 - a*d^2)^2*a^2)
 

Mupad [B] (verification not implemented)

Time = 10.90 (sec) , antiderivative size = 7976, normalized size of antiderivative = 29.00 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((4000*a^8*b^2*c*d^14 - 3145728*a^5*b^5*c^7*d^8 + 1241088*a^6*b^4*c 
^5*d^10 + 2472320*a^7*b^3*c^3*d^12)/(32768*a^10) + (((12582912*a^11*b^5*c^ 
4*d^8 - 12976128*a^12*b^4*c^2*d^10)/(32768*a^10) - ((16777216*a^13*b^4*d^1 
0 - 25165824*a^12*b^5*c^2*d^8)*(c + d*x)^(1/2)*(-(25*a^2*d^5*(a^13*b^3)^(1 
/2) - 1024*a^6*b^4*c^5 + 315*a^8*b^2*c*d^4 + 700*a^7*b^3*c^3*d^2 - 1152*b^ 
2*c^4*d*(a^13*b^3)^(1/2) + 1136*a*b*c^2*d^3*(a^13*b^3)^(1/2))/(4096*(a^12* 
b^4*c^2 - a^13*b^3*d^2)))^(1/2))/(32768*a^8))*(-(25*a^2*d^5*(a^13*b^3)^(1/ 
2) - 1024*a^6*b^4*c^5 + 315*a^8*b^2*c*d^4 + 700*a^7*b^3*c^3*d^2 - 1152*b^2 
*c^4*d*(a^13*b^3)^(1/2) + 1136*a*b*c^2*d^3*(a^13*b^3)^(1/2))/(4096*(a^12*b 
^4*c^2 - a^13*b^3*d^2)))^(1/2) + ((c + d*x)^(1/2)*(2529280*a^8*b^3*c*d^12 
- 18874368*a^6*b^5*c^5*d^8 + 2711552*a^7*b^4*c^3*d^10))/(32768*a^8))*(-(25 
*a^2*d^5*(a^13*b^3)^(1/2) - 1024*a^6*b^4*c^5 + 315*a^8*b^2*c*d^4 + 700*a^7 
*b^3*c^3*d^2 - 1152*b^2*c^4*d*(a^13*b^3)^(1/2) + 1136*a*b*c^2*d^3*(a^13*b^ 
3)^(1/2))/(4096*(a^12*b^4*c^2 - a^13*b^3*d^2)))^(1/2))*(-(25*a^2*d^5*(a^13 
*b^3)^(1/2) - 1024*a^6*b^4*c^5 + 315*a^8*b^2*c*d^4 + 700*a^7*b^3*c^3*d^2 - 
 1152*b^2*c^4*d*(a^13*b^3)^(1/2) + 1136*a*b*c^2*d^3*(a^13*b^3)^(1/2))/(409 
6*(a^12*b^4*c^2 - a^13*b^3*d^2)))^(1/2) + ((c + d*x)^(1/2)*(625*a^4*b*d^16 
 + 3145728*b^5*c^8*d^8 + 1310720*a*b^4*c^6*d^10 + 801296*a^2*b^3*c^4*d^12 
- 41800*a^3*b^2*c^2*d^14))/(32768*a^8))*(-(25*a^2*d^5*(a^13*b^3)^(1/2) - 1 
024*a^6*b^4*c^5 + 315*a^8*b^2*c*d^4 + 700*a^7*b^3*c^3*d^2 - 1152*b^2*c^...
 

Reduce [B] (verification not implemented)

Time = 5.61 (sec) , antiderivative size = 2278, normalized size of antiderivative = 8.28 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 10*sqrt(a)*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*d**3 + 4*sqrt(a)*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**2*b*c**2*d + 20*sqrt(a)*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**2*b*d**3*x**2 - 8*sqr 
t(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sq 
rt(b)*sqrt(a)*d - b*c)))*a*b**2*c**2*d*x**2 - 10*sqrt(a)*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*b**2*d**3*x**4 + 4*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c 
 + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**3*c**2*d*x**4 + 58* 
sqrt(b)*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*c*d**2 - 64*sqrt(b)*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 
**2*b*c**3 - 116*sqrt(b)*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**2*b*c*d**2*x**2 + 128*sqrt 
(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqr 
t(b)*sqrt(a)*d - b*c)))*a*b**2*c**3*x**2 + 58*sqrt(b)*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*b**2*c*d**2*x**4 - 64*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c 
 + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**3*c**3*x**4 - 5*...