\(\int \frac {1}{(d x)^{3/2} (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [663]

Optimal result

Integrand size = 30, antiderivative size = 496 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

663/1024/a^4/d/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)+1/8/a/d/(d*x)^(1/2)/(b*x^2+ 
a)^3/((b*x^2+a)^2)^(1/2)+17/96/a^2/d/(d*x)^(1/2)/(b*x^2+a)^2/((b*x^2+a)^2) 
^(1/2)+221/768/a^3/d/(d*x)^(1/2)/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-3315/1024*( 
b*x^2+a)/a^5/d/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)+3315/4096*b^(1/4)*(b*x^2+a) 
*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(21/4)/d^ 
(3/2)/((b*x^2+a)^2)^(1/2)-3315/4096*b^(1/4)*(b*x^2+a)*arctan(1+2^(1/2)*b^( 
1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(21/4)/d^(3/2)/((b*x^2+a)^2)^( 
1/2)+3315/4096*b^(1/4)*(b*x^2+a)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(d*x)^(1/ 
2)/d^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(21/4)/d^(3/2)/((b*x^2+a)^2)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {x \left (a+b x^2\right ) \left (-4 \sqrt [4]{a} \left (6144 a^4+31501 a^3 b x^2+52819 a^2 b^2 x^4+37791 a b^3 x^6+9945 b^4 x^8\right )+9945 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+9945 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{21/4} (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \] Input:

Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
 

Output:

(x*(a + b*x^2)*(-4*a^(1/4)*(6144*a^4 + 31501*a^3*b*x^2 + 52819*a^2*b^2*x^4 
 + 37791*a*b^3*x^6 + 9945*b^4*x^8) + 9945*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x 
^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 99 
45*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)* 
Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(12288*a^(21/4)*(d*x)^(3/2)*((a + b*x^2) 
^2)^(5/2))
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.92, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {1384, 27, 253, 253, 253, 253, 264, 266, 27, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1384

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^4}dx}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^3}dx}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {b \int \frac {\sqrt {d x}}{b x^2+a}dx}{a d^2}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d^3}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

Input:

Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
 

Output:

((a + b*x^2)*(1/(8*a*d*Sqrt[d*x]*(a + b*x^2)^4) + (17*(1/(6*a*d*Sqrt[d*x]* 
(a + b*x^2)^3) + (13*(1/(4*a*d*Sqrt[d*x]*(a + b*x^2)^2) + (9*(1/(2*a*d*Sqr 
t[d*x]*(a + b*x^2)) + (5*(-2/(a*d*Sqrt[d*x]) - (2*b*((-(ArcTan[1 - (Sqrt[2 
]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) 
 + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1 
/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sq 
rt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) 
 + Log[Sqrt[a]*d + Sqrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x] 
]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b])))/(a*d)))/(4*a)))/(8*a) 
))/(12*a)))/(16*a)))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1384

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac 
Part[p]))   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n 
- 1)] && NeQ[u, x^(2*n - 1)] &&  !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
 

rule 1476

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

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.55

method	result	size
risch	\(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}}{a^{5} \sqrt {d x}\, d \left (b \,x^{2}+a \right )}-\frac {b \left (\frac {\frac {6925 a^{3} d^{6} \left (d x \right )^{\frac {3}{2}}}{3072}+\frac {15955 a^{2} d^{4} b \left (d x \right )^{\frac {7}{2}}}{3072}+\frac {4405 a \,d^{2} b^{2} \left (d x \right )^{\frac {11}{2}}}{1024}+\frac {1267 b^{3} \left (d x \right )^{\frac {15}{2}}}{1024}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {3315 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8192 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{a^{5} d \left (b \,x^{2}+a \right )}\)	\(273\)
default	\(\text {Expression too large to display}\)	\(1081\)

Input:

int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/a^5/(d*x)^(1/2)/d*((b*x^2+a)^2)^(1/2)/(b*x^2+a)-b/a^5*(2*(6925/6144*a^3 
*d^6*(d*x)^(3/2)+15955/6144*a^2*d^4*b*(d*x)^(7/2)+4405/2048*a*d^2*b^2*(d*x 
)^(11/2)+1267/2048*b^3*(d*x)^(15/2))/(b*d^2*x^2+a*d^2)^4+3315/8192/b/(a*d^ 
2/b)^(1/4)*2^(1/2)*(ln((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^ 
(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan 
(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*( 
d*x)^(1/2)-1)))/d*((b*x^2+a)^2)^(1/2)/(b*x^2+a)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {9945 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \log \left (36429280875 \, a^{16} d^{5} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {3}{4}} + 36429280875 \, \sqrt {d x} b\right ) + 9945 \, {\left (-i \, a^{5} b^{4} d^{2} x^{9} - 4 i \, a^{6} b^{3} d^{2} x^{7} - 6 i \, a^{7} b^{2} d^{2} x^{5} - 4 i \, a^{8} b d^{2} x^{3} - i \, a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \log \left (36429280875 i \, a^{16} d^{5} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {3}{4}} + 36429280875 \, \sqrt {d x} b\right ) + 9945 \, {\left (i \, a^{5} b^{4} d^{2} x^{9} + 4 i \, a^{6} b^{3} d^{2} x^{7} + 6 i \, a^{7} b^{2} d^{2} x^{5} + 4 i \, a^{8} b d^{2} x^{3} + i \, a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \log \left (-36429280875 i \, a^{16} d^{5} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {3}{4}} + 36429280875 \, \sqrt {d x} b\right ) - 9945 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \log \left (-36429280875 \, a^{16} d^{5} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {3}{4}} + 36429280875 \, \sqrt {d x} b\right ) + 4 \, {\left (9945 \, b^{4} x^{8} + 37791 \, a b^{3} x^{6} + 52819 \, a^{2} b^{2} x^{4} + 31501 \, a^{3} b x^{2} + 6144 \, a^{4}\right )} \sqrt {d x}}{12288 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )}} \] Input:

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas 
")
 

Output:

-1/12288*(9945*(a^5*b^4*d^2*x^9 + 4*a^6*b^3*d^2*x^7 + 6*a^7*b^2*d^2*x^5 + 
4*a^8*b*d^2*x^3 + a^9*d^2*x)*(-b/(a^21*d^6))^(1/4)*log(36429280875*a^16*d^ 
5*(-b/(a^21*d^6))^(3/4) + 36429280875*sqrt(d*x)*b) + 9945*(-I*a^5*b^4*d^2* 
x^9 - 4*I*a^6*b^3*d^2*x^7 - 6*I*a^7*b^2*d^2*x^5 - 4*I*a^8*b*d^2*x^3 - I*a^ 
9*d^2*x)*(-b/(a^21*d^6))^(1/4)*log(36429280875*I*a^16*d^5*(-b/(a^21*d^6))^ 
(3/4) + 36429280875*sqrt(d*x)*b) + 9945*(I*a^5*b^4*d^2*x^9 + 4*I*a^6*b^3*d 
^2*x^7 + 6*I*a^7*b^2*d^2*x^5 + 4*I*a^8*b*d^2*x^3 + I*a^9*d^2*x)*(-b/(a^21* 
d^6))^(1/4)*log(-36429280875*I*a^16*d^5*(-b/(a^21*d^6))^(3/4) + 3642928087 
5*sqrt(d*x)*b) - 9945*(a^5*b^4*d^2*x^9 + 4*a^6*b^3*d^2*x^7 + 6*a^7*b^2*d^2 
*x^5 + 4*a^8*b*d^2*x^3 + a^9*d^2*x)*(-b/(a^21*d^6))^(1/4)*log(-36429280875 
*a^16*d^5*(-b/(a^21*d^6))^(3/4) + 36429280875*sqrt(d*x)*b) + 4*(9945*b^4*x 
^8 + 37791*a*b^3*x^6 + 52819*a^2*b^2*x^4 + 31501*a^3*b*x^2 + 6144*a^4)*sqr 
t(d*x))/(a^5*b^4*d^2*x^9 + 4*a^6*b^3*d^2*x^7 + 6*a^7*b^2*d^2*x^5 + 4*a^8*b 
*d^2*x^3 + a^9*d^2*x)
 

Sympy [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Integral(1/((d*x)**(3/2)*((a + b*x**2)**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}} \left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima 
")
 

Output:

-1/3072*(3801*b^4*x^(15/2) + 8079*a*b^3*x^(11/2) + 6515*a^2*b^2*x^(7/2) + 
1853*a^3*b*x^(3/2))/(a^5*b^4*d^(3/2)*x^8 + 4*a^6*b^3*d^(3/2)*x^6 + 6*a^7*b 
^2*d^(3/2)*x^4 + 4*a^8*b*d^(3/2)*x^2 + a^9*d^(3/2)) - 1/192*((321*b^5*sqrt 
(d)*x^5 + 490*a*b^4*sqrt(d)*x^3 + 201*a^2*b^3*sqrt(d)*x)*x^(9/2) + 2*(371* 
a*b^4*sqrt(d)*x^5 + 582*a^2*b^3*sqrt(d)*x^3 + 243*a^3*b^2*sqrt(d)*x)*x^(5/ 
2) + (453*a^2*b^3*sqrt(d)*x^5 + 738*a^3*b^2*sqrt(d)*x^3 + 317*a^4*b*sqrt(d 
)*x)*sqrt(x))/(a^7*b^3*d^2*x^6 + 3*a^8*b^2*d^2*x^4 + 3*a^9*b*d^2*x^2 + a^1 
0*d^2 + (a^4*b^6*d^2*x^6 + 3*a^5*b^5*d^2*x^4 + 3*a^6*b^4*d^2*x^2 + a^7*b^3 
*d^2)*x^6 + 3*(a^5*b^5*d^2*x^6 + 3*a^6*b^4*d^2*x^4 + 3*a^7*b^3*d^2*x^2 + a 
^8*b^2*d^2)*x^4 + 3*(a^6*b^4*d^2*x^6 + 3*a^7*b^3*d^2*x^4 + 3*a^8*b^2*d^2*x 
^2 + a^9*b*d^2)*x^2) - 1267/8192*b*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)* 
a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)* 
sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) 
 - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b 
)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a 
^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x 
 + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^5*d^(3/2)) + integrate(1/((a^4*b*d^(3/2) 
*x^2 + a^5*d^(3/2))*x^(3/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {\frac {49152}{\sqrt {d x} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (3801 \, \sqrt {d x} b^{4} d^{7} x^{7} + 13215 \, \sqrt {d x} a b^{3} d^{7} x^{5} + 15955 \, \sqrt {d x} a^{2} b^{2} d^{7} x^{3} + 6925 \, \sqrt {d x} a^{3} b d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}}{24576 \, d} \] Input:

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

-1/24576*(49152/(sqrt(d*x)*a^5*sgn(b*x^2 + a)) + 19890*sqrt(2)*(a*b^3*d^2) 
^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b 
)^(1/4))/(a^6*b^2*d^2*sgn(b*x^2 + a)) + 19890*sqrt(2)*(a*b^3*d^2)^(3/4)*ar 
ctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4)) 
/(a^6*b^2*d^2*sgn(b*x^2 + a)) - 9945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + s 
qrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b^2*d^2*sgn(b*x^2 + 
 a)) + 9945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sq 
rt(d*x) + sqrt(a*d^2/b))/(a^6*b^2*d^2*sgn(b*x^2 + a)) + 8*(3801*sqrt(d*x)* 
b^4*d^7*x^7 + 13215*sqrt(d*x)*a*b^3*d^7*x^5 + 15955*sqrt(d*x)*a^2*b^2*d^7* 
x^3 + 6925*sqrt(d*x)*a^3*b*d^7*x)/((b*d^2*x^2 + a*d^2)^4*a^5*sgn(b*x^2 + a 
)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:

int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)
 

Output:

int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 865, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(sqrt(d)*(19890*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)* 
sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4 + 79560*sqr 
t(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x) 
*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b*x**2 + 119340*sqrt(x)*b**(1/ 
4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/( 
b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*x**4 + 79560*sqrt(x)*b**(1/4)*a**(3/ 
4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)* 
a**(1/4)*sqrt(2)))*a*b**3*x**6 + 19890*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*a 
tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr 
t(2)))*b**4*x**8 - 19890*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)* 
a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4 - 
79560*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 
2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b*x**2 - 119340*sqrt( 
x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*s 
qrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*x**4 - 79560*sqrt(x)*b**(1/ 
4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/( 
b**(1/4)*a**(1/4)*sqrt(2)))*a*b**3*x**6 - 19890*sqrt(x)*b**(1/4)*a**(3/4)* 
sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a** 
(1/4)*sqrt(2)))*b**4*x**8 - 9945*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*log( - 
sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**4 - 39780*s...