\(\int \frac {\sqrt {c+d x}}{x (a-b x^2)^3} \, dx\) [694]

Optimal result

Integrand size = 23, antiderivative size = 300 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^3} \, dx=\frac {\sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (7 a d^2-b c (8 c-d x)\right )}{16 a^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}+\frac {\left (32 b c^2-54 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^3 \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\left (32 b c^2+54 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^3 \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

1/4*(d*x+c)^(1/2)/a/(-b*x^2+a)^2-1/16*(d*x+c)^(1/2)*(7*a*d^2-b*c*(-d*x+8*c 
))/a^2/(-a*d^2+b*c^2)/(-b*x^2+a)-2*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))/ 
a^3+1/32*(32*b*c^2-54*a^(1/2)*b^(1/2)*c*d+21*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^(1/4)/(b^(1/2)*c-a^(1/2)*d)^(3/ 
2)+1/32*(32*b*c^2+54*a^(1/2)*b^(1/2)*c*d+21*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^(1/4)/(b^(1/2)*c+a^(1/2)*d)^(3/2 
)
 

Mathematica [A] (verified)

Time = 3.36 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^3} \, dx=\frac {\frac {2 a \sqrt {c+d x} \left (11 a^2 d^2+b^2 c x^2 (8 c-d x)+a b \left (-12 c^2+c d x-7 d^2 x^2\right )\right )}{\left (-b c^2+a d^2\right ) \left (a-b x^2\right )^2}+\frac {\left (32 b c^2+54 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {\left (32 b c^2-54 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-64 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{32 a^3} \] Input:

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

Output:

((2*a*Sqrt[c + d*x]*(11*a^2*d^2 + b^2*c*x^2*(8*c - d*x) + a*b*(-12*c^2 + c 
*d*x - 7*d^2*x^2)))/((-(b*c^2) + a*d^2)*(a - b*x^2)^2) + ((32*b*c^2 + 54*S 
qrt[a]*Sqrt[b]*c*d + 21*a*d^2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sq 
rt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/((Sqrt[b]*c + Sqrt[a]*d)*Sqrt[-(b*c 
) - Sqrt[a]*Sqrt[b]*d]) + ((32*b*c^2 - 54*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)* 
ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[ 
a]*d)])/((Sqrt[b]*c - Sqrt[a]*d)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) - 64*Sq 
rt[c]*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(32*a^3)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(753\) vs. \(2(300)=600\).

Time = 2.26 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.51, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {561, 25, 27, 1652, 25, 25, 27, 1492, 27, 1405, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1652

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1492

\(\displaystyle -2 \left (\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}+\frac {\frac {d^4 \int -\frac {14 a b \left (b c^2-a d^2\right )}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \left (\frac {-\frac {7}{8} d^2 \int \frac {1}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}+\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}\right )\)

\(\Big \downarrow \) 1405

\(\displaystyle -2 \left (\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}+\frac {-\frac {7}{8} d^2 \left (\frac {d^4 \int \frac {2 b \left (b c^2+b (c+d x) c-3 a d^2\right )}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d^2+b c^2-b c (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \left (\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}+\frac {-\frac {7}{8} d^2 \left (\frac {\int \frac {b c^2+b (c+d x) c-3 a d^2}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d^2+b c^2-b c (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}\right )\)

\(\Big \downarrow \) 1480

\(\displaystyle -2 \left (\frac {-\frac {7}{8} d^2 \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {b} c d-3 a d^2+2 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\sqrt {b} \left (-\sqrt {a} \sqrt {b} c d-3 a d^2+2 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d^2+b c^2-b c (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}+\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -2 \left (\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}+\frac {-\frac {7}{8} d^2 \left (\frac {\frac {d \left (\sqrt {a} \sqrt {b} c d-3 a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-\sqrt {a} \sqrt {b} c d-3 a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d^2+b c^2-b c (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}\right )\)

\(\Big \downarrow \) 1567

\(\displaystyle -2 \left (\frac {c \int \left (-\frac {b x d^3}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )^2}-\frac {b x d}{a^2 \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a^2 x d}\right )d\sqrt {c+d x}}{a}+\frac {-\frac {7}{8} d^2 \left (\frac {\frac {d \left (\sqrt {a} \sqrt {b} c d-3 a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-\sqrt {a} \sqrt {b} c d-3 a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d^2+b c^2-b c (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )-\frac {d^2 \sqrt {c+d x}}{8 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a d^2}\right )\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-2*((c*((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 - S 
qrt[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)*ArcTa 
nh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*a^2*Sqrt[Sqrt[ 
b]*c + Sqrt[a]*d])))/a + (-1/8*(d^2*Sqrt[c + d*x])/(a - (b*c^2)/d^2 + (2*b 
*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)^2 - (7*d^2*(-1/4*(Sqrt[c + d*x]*( 
b*c^2 + a*d^2 - b*c*(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)) + (-1/2*(d*(2*b*c^2 - Sqrt[a]*Sq 
rt[b]*c*d - 3*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt 
[a]*d]])/(Sqrt[a]*b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) + (d*(2*b*c^2 + Sqr 
t[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*Sqrt[a]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d]))/(4*a*(b*c 
^2 - a*d^2))))/8)/(a*d^2))
 

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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

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 1652

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

rule 2009

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

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.29

method	result	size
pseudoelliptic	\(\frac {\frac {21 \left (\left (-\frac {11}{7} a \,d^{2} c +\frac {32}{21} b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a \,d^{2} \left (a \,d^{2}-\frac {22 b \,c^{2}}{21}\right )\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, b \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}+2 \left (\frac {21 b \left (\left (\frac {11}{7} a \,d^{2} c -\frac {32}{21} b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a \,d^{2} \left (a \,d^{2}-\frac {22 b \,c^{2}}{21}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64}+\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (\left (c^{\frac {5}{2}} b -a \,d^{2} \sqrt {c}\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\frac {11 \left (\frac {8 x^{2} c \left (-\frac {d x}{8}+c \right ) b^{2}}{11}-\frac {12 \left (\frac {7}{12} d^{2} x^{2}-\frac {1}{12} c d x +c^{2}\right ) a b}{11}+a^{2} d^{2}\right ) a \sqrt {d x +c}}{32}\right )\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} a^{3} \left (a \,d^{2}-b \,c^{2}\right )}\)	\(386\)
derivativedivides	\(-2 d^{6} \left (-\frac {\frac {-\frac {a \,b^{2} c \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}-\frac {\left (7 a \,d^{2}-11 b \,c^{2}\right ) a b \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {a b c \,d^{2} \left (15 a \,d^{2}-19 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 a \,d^{2}-32 b \,c^{2}}+\frac {\left (11 a \,d^{2}-9 b \,c^{2}\right ) a \,d^{2} \sqrt {d x +c}}{32}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (\frac {\left (21 a^{2} d^{4}-22 b \,c^{2} d^{2} a -33 \sqrt {a b \,d^{2}}\, a c \,d^{2}+32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-21 a^{2} d^{4}+22 b \,c^{2} d^{2} a -33 \sqrt {a b \,d^{2}}\, a c \,d^{2}+32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a \,d^{2}-32 b \,c^{2}}}{a^{3} d^{6}}+\frac {\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6}}\right )\)	\(433\)
default	\(2 d^{6} \left (\frac {\frac {-\frac {a \,b^{2} c \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}-\frac {\left (7 a \,d^{2}-11 b \,c^{2}\right ) a b \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {a b c \,d^{2} \left (15 a \,d^{2}-19 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 a \,d^{2}-32 b \,c^{2}}+\frac {\left (11 a \,d^{2}-9 b \,c^{2}\right ) a \,d^{2} \sqrt {d x +c}}{32}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (\frac {\left (21 a^{2} d^{4}-22 b \,c^{2} d^{2} a -33 \sqrt {a b \,d^{2}}\, a c \,d^{2}+32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-21 a^{2} d^{4}+22 b \,c^{2} d^{2} a -33 \sqrt {a b \,d^{2}}\, a c \,d^{2}+32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a \,d^{2}-32 b \,c^{2}}}{a^{3} d^{6}}-\frac {\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6}}\right )\)	\(433\)

Input:

int((d*x+c)^(1/2)/x/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

2/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)*(21/64*((-11/7*a*d^2*c+ 
32/21*b*c^3)*(a*b*d^2)^(1/2)+a*d^2*(a*d^2-22/21*b*c^2))*((b*c+(a*b*d^2)^(1 
/2))*b)^(1/2)*b*(-b*x^2+a)^2*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2) 
)*b)^(1/2))+(21/64*b*((11/7*a*d^2*c-32/21*b*c^3)*(a*b*d^2)^(1/2)+a*d^2*(a* 
d^2-22/21*b*c^2))*(-b*x^2+a)^2*arctanh(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)*(a*b*d^2)^(1/2)*((c^(5/2)*b- 
a*d^2*c^(1/2))*(-b*x^2+a)^2*arctanh((d*x+c)^(1/2)/c^(1/2))+11/32*(8/11*x^2 
*c*(-1/8*d*x+c)*b^2-12/11*(7/12*d^2*x^2-1/12*c*d*x+c^2)*a*b+a^2*d^2)*a*(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)/(-b*x^2+a)^2/a^3/(a*d^2-b*c^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4657 vs. \(2 (238) = 476\).

Time = 9.42 (sec) , antiderivative size = 9323, normalized size of antiderivative = 31.08 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/x/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((d*x+c)**(1/2)/x/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((d*x+c)^(1/2)/x/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate(sqrt(d*x + c)/((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. 1104 vs. \(2 (238) = 476\).

Time = 0.35 (sec) , antiderivative size = 1104, normalized size of antiderivative = 3.68 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/x/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/32*((a^3*b*c^2*d - a^4*d^3)^2*(32*b^2*c^3 - 33*a*b*c*d^2)*abs(b) - (32* 
sqrt(a*b)*a^2*b^3*c^6 - 87*sqrt(a*b)*a^3*b^2*c^4*d^2 + 76*sqrt(a*b)*a^4*b* 
c^2*d^4 - 21*sqrt(a*b)*a^5*d^6)*abs(a^3*b*c^2*d - a^4*d^3)*abs(b) - (22*a^ 
6*b^4*c^7*d^2 - 65*a^7*b^3*c^5*d^4 + 64*a^8*b^2*c^3*d^6 - 21*a^9*b*c*d^8)* 
abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a^3*b^2*c^3 - a^4*b*c*d^2 + sqrt((a^3* 
b^2*c^3 - a^4*b*c*d^2)^2 - (a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*(a^3* 
b^2*c^2 - a^4*b*d^2)))/(a^3*b^2*c^2 - a^4*b*d^2)))/((a^6*b^3*c^4*d - 2*a^7 
*b^2*c^2*d^3 + a^8*b*d^5 - sqrt(a*b)*a^5*b^3*c^5 + 2*sqrt(a*b)*a^6*b^2*c^3 
*d^2 - sqrt(a*b)*a^7*b*c*d^4)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a^3*b*c^2*d 
 - a^4*d^3)) - 1/32*((a^3*b*c^2*d - a^4*d^3)^2*(32*b^2*c^3 - 33*a*b*c*d^2) 
*abs(b) + (32*sqrt(a*b)*a^2*b^3*c^6 - 87*sqrt(a*b)*a^3*b^2*c^4*d^2 + 76*sq 
rt(a*b)*a^4*b*c^2*d^4 - 21*sqrt(a*b)*a^5*d^6)*abs(a^3*b*c^2*d - a^4*d^3)*a 
bs(b) - (22*a^6*b^4*c^7*d^2 - 65*a^7*b^3*c^5*d^4 + 64*a^8*b^2*c^3*d^6 - 21 
*a^9*b*c*d^8)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a^3*b^2*c^3 - a^4*b*c*d^ 
2 - sqrt((a^3*b^2*c^3 - a^4*b*c*d^2)^2 - (a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + 
a^5*d^4)*(a^3*b^2*c^2 - a^4*b*d^2)))/(a^3*b^2*c^2 - a^4*b*d^2)))/((a^6*b^3 
*c^4*d - 2*a^7*b^2*c^2*d^3 + a^8*b*d^5 + sqrt(a*b)*a^5*b^3*c^5 - 2*sqrt(a* 
b)*a^6*b^2*c^3*d^2 + sqrt(a*b)*a^7*b*c*d^4)*sqrt(-b^2*c + sqrt(a*b)*b*d)*a 
bs(a^3*b*c^2*d - a^4*d^3)) + 2*c*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt( 
-c)) + 1/16*((d*x + c)^(7/2)*b^2*c*d^2 - 11*(d*x + c)^(5/2)*b^2*c^2*d^2...
 

Mupad [B] (verification not implemented)

Time = 11.67 (sec) , antiderivative size = 13831, normalized size of antiderivative = 46.10 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^(1/2)/(x*(a - b*x^2)^3),x)
 

Output:

- (((19*b^2*c^3*d^2 - 15*a*b*c*d^4)*(c + d*x)^(3/2))/(16*a^2*(a*d^2 - b*c^ 
2)) - ((11*a*d^4 - 9*b*c^2*d^2)*(c + d*x)^(1/2))/(16*a^2) + (b*(7*a*d^4 - 
11*b*c^2*d^2)*(c + d*x)^(5/2))/(16*a^2*(a*d^2 - b*c^2)) + (b^2*c*d^2*(c + 
d*x)^(7/2))/(16*a^2*(a*d^2 - b*c^2)))/(b^2*(c + d*x)^4 + a^2*d^4 + b^2*c^4 
 + (6*b^2*c^2 - 2*a*b*d^2)*(c + d*x)^2 - (4*b^2*c^3 - 4*a*b*c*d^2)*(c + d* 
x) - 4*b^2*c*(c + d*x)^3 - 2*a*b*c^2*d^2) - atan(((((((14024704*a^14*b^4*c 
*d^14 - 12582912*a^11*b^7*c^7*d^8 + 38928384*a^12*b^6*c^5*d^10 - 40370176* 
a^13*b^5*c^3*d^12)/(32768*(a^12*d^4 + a^10*b^2*c^4 - 2*a^11*b*c^2*d^2)) - 
((c + d*x)^(1/2)*(-(1024*a^6*b^4*c^7 - 441*a^3*d^7*(a^13*b)^(1/2) + 384*b^ 
3*c^6*d*(a^13*b)^(1/2) - 3036*a^7*b^3*c^5*d^2 + 2961*a^8*b^2*c^3*d^4 - 945 
*a^9*b*c*d^6 - 1168*a*b^2*c^4*d^3*(a^13*b)^(1/2) + 1221*a^2*b*c^2*d^5*(a^1 
3*b)^(1/2))/(4096*(a^15*b*d^6 - a^12*b^4*c^6 + 3*a^13*b^3*c^4*d^2 - 3*a^14 
*b^2*c^2*d^4)))^(1/2)*(16777216*a^15*b^4*d^14 - 25165824*a^12*b^7*c^6*d^8 
+ 67108864*a^13*b^6*c^4*d^10 - 58720256*a^14*b^5*c^2*d^12))/(32768*(a^10*d 
^4 + a^8*b^2*c^4 - 2*a^9*b*c^2*d^2)))*(-(1024*a^6*b^4*c^7 - 441*a^3*d^7*(a 
^13*b)^(1/2) + 384*b^3*c^6*d*(a^13*b)^(1/2) - 3036*a^7*b^3*c^5*d^2 + 2961* 
a^8*b^2*c^3*d^4 - 945*a^9*b*c*d^6 - 1168*a*b^2*c^4*d^3*(a^13*b)^(1/2) + 12 
21*a^2*b*c^2*d^5*(a^13*b)^(1/2))/(4096*(a^15*b*d^6 - a^12*b^4*c^6 + 3*a^13 
*b^3*c^4*d^2 - 3*a^14*b^2*c^2*d^4)))^(1/2) + ((c + d*x)^(1/2)*(6838272*a^9 
*b^4*c*d^14 - 18874368*a^6*b^7*c^7*d^8 + 44982272*a^7*b^6*c^5*d^10 - 32...
 

Reduce [B] (verification not implemented)

Time = 6.03 (sec) , antiderivative size = 3035, normalized size of antiderivative = 10.12 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)/x/(-b*x^2+a)^3,x)
 

Output:

( - 24*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*b*c*d**3 + 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**2*c**3*d + 48*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**2*c*d**3 
*x**2 - 40*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(s 
qrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**3*c**3*d*x**2 - 24*sqrt(a)*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*b**3*c*d**3*x**4 + 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)))*b**4* 
c**3*d*x**4 + 42*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**4*d**4 - 110*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*b*c**2*d**2 - 84*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*b*d* 
*4*x**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**2*c**4 + 220*sqrt(b)*sqrt 
(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)))*a**2*b**2*c**2*d**2*x**2 + 42*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)...