\(\int \frac {(d x)^{17/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [653]

Optimal result

Integrand size = 30, antiderivative size = 448 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

-385/1024*d^7*(d*x)^(3/2)/b^4/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(15/2)/b/(b* 
x^2+a)^3/((b*x^2+a)^2)^(1/2)-5/32*d^3*(d*x)^(11/2)/b^2/(b*x^2+a)^2/((b*x^2 
+a)^2)^(1/2)-55/256*d^5*(d*x)^(7/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-1155 
/4096*d^(17/2)*(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^(1/4)/b^(19/4)/((b*x^2+a)^2)^(1/2)+1155/4096*d^(17/2)*(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^(1/4) 
/b^(19/4)/((b*x^2+a)^2)^(1/2)-1155/4096*d^(17/2)*(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^(1/4)/ 
b^(19/4)/((b*x^2+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.46 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^8 \sqrt {d x} \left (-4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (385 a^3+1375 a^2 b x^2+1755 a b^2 x^4+893 b^3 x^6\right )+1155 \sqrt {2} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-1155 \sqrt {2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{4096 \sqrt [4]{a} b^{19/4} \sqrt {x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \] Input:

Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(d^8*Sqrt[d*x]*(-4*a^(1/4)*b^(3/4)*x^(3/2)*(385*a^3 + 1375*a^2*b*x^2 + 175 
5*a*b^2*x^4 + 893*b^3*x^6) + 1155*Sqrt[2]*(a + b*x^2)^4*ArcTan[(-Sqrt[a] + 
 Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 1155*Sqrt[2]*(a + b*x^2)^ 
4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(4096 
*a^(1/4)*b^(19/4)*Sqrt[x]*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1384, 27, 252, 252, 252, 252, 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.

\(\displaystyle \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {(d x)^{17/2}}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{17/2}}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \int \frac {(d x)^{13/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \int \frac {(d x)^{9/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \int \frac {(d x)^{5/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \int \frac {\sqrt {d x}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {15 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

Input:

Int[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

((a + b*x^2)*(-1/8*(d*(d*x)^(15/2))/(b*(a + b*x^2)^4) + (15*d^2*(-1/6*(d*( 
d*x)^(11/2))/(b*(a + b*x^2)^3) + (11*d^2*(-1/4*(d*(d*x)^(7/2))/(b*(a + b*x 
^2)^2) + (7*d^2*(-1/2*(d*(d*x)^(3/2))/(b*(a + b*x^2)) + (3*d^3*((-(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])]/(S 
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a]*d + Sqrt[ 
b]*d*x - Sqrt[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])))/(2*b)))/(8 
*b)))/(12*b)))/(16*b)))/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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1045\) vs. \(2(291)=582\).

Time = 0.15 (sec) , antiderivative size = 1046, normalized size of antiderivative = 2.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1046\)

Input:

int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8192*(-1155*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*b^4* 
d^8*x^8-2310*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b 
)^(1/4))*b^4*d^8*x^8-2310*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1 
/4))/(a*d^2/b)^(1/4))*b^4*d^8*x^8+7144*(d*x)^(15/2)*(a*d^2/b)^(1/4)*b^4-46 
20*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a*b^3*d^8*x^6-92 
40*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a 
*b^3*d^8*x^6-9240*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a* 
d^2/b)^(1/4))*a*b^3*d^8*x^6+14040*(d*x)^(11/2)*(a*d^2/b)^(1/4)*a*b^3*d^2-6 
930*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a^2*b^2*d^8*x^4 
-13860*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4 
))*a^2*b^2*d^8*x^4-13860*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/ 
4))/(a*d^2/b)^(1/4))*a^2*b^2*d^8*x^4+11000*(d*x)^(7/2)*(a*d^2/b)^(1/4)*a^2 
*b^2*d^4-4620*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a^3*b 
*d^8*x^2-9240*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/ 
b)^(1/4))*a^3*b*d^8*x^2-9240*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b) 
^(1/4))/(a*d^2/b)^(1/4))*a^3*b*d^8*x^2+3080*(d*x)^(3/2)*(a*d^2/b)^(1/4)...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.06 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1155 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} + 1540798875 \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 1155 \, {\left (i \, b^{8} x^{8} + 4 i \, a b^{7} x^{6} + 6 i \, a^{2} b^{6} x^{4} + 4 i \, a^{3} b^{5} x^{2} + i \, a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} + 1540798875 i \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 1155 \, {\left (-i \, b^{8} x^{8} - 4 i \, a b^{7} x^{6} - 6 i \, a^{2} b^{6} x^{4} - 4 i \, a^{3} b^{5} x^{2} - i \, a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} - 1540798875 i \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 1155 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} - 1540798875 \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 4 \, {\left (893 \, b^{3} d^{8} x^{7} + 1755 \, a b^{2} d^{8} x^{5} + 1375 \, a^{2} b d^{8} x^{3} + 385 \, a^{3} d^{8} x\right )} \sqrt {d x}}{4096 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \] Input:

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas" 
)
 

Output:

1/4096*(1155*(b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4* 
b^4)*(-d^34/(a*b^19))^(1/4)*log(1540798875*sqrt(d*x)*d^25 + 1540798875*(-d 
^34/(a*b^19))^(3/4)*a*b^14) - 1155*(I*b^8*x^8 + 4*I*a*b^7*x^6 + 6*I*a^2*b^ 
6*x^4 + 4*I*a^3*b^5*x^2 + I*a^4*b^4)*(-d^34/(a*b^19))^(1/4)*log(1540798875 
*sqrt(d*x)*d^25 + 1540798875*I*(-d^34/(a*b^19))^(3/4)*a*b^14) - 1155*(-I*b 
^8*x^8 - 4*I*a*b^7*x^6 - 6*I*a^2*b^6*x^4 - 4*I*a^3*b^5*x^2 - I*a^4*b^4)*(- 
d^34/(a*b^19))^(1/4)*log(1540798875*sqrt(d*x)*d^25 - 1540798875*I*(-d^34/( 
a*b^19))^(3/4)*a*b^14) - 1155*(b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a 
^3*b^5*x^2 + a^4*b^4)*(-d^34/(a*b^19))^(1/4)*log(1540798875*sqrt(d*x)*d^25 
 - 1540798875*(-d^34/(a*b^19))^(3/4)*a*b^14) - 4*(893*b^3*d^8*x^7 + 1755*a 
*b^2*d^8*x^5 + 1375*a^2*b*d^8*x^3 + 385*a^3*d^8*x)*sqrt(d*x))/(b^8*x^8 + 4 
*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x)**(17/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {17}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima" 
)
 

Output:

d^(17/2)*integrate(sqrt(x)/(b^5*x^2 + a*b^4), x) - 893/8192*d^(17/2)*(2*sq 
rt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqr 
t(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)))/b^4 - 1/3 
072*(2679*b^3*d^(17/2)*x^(15/2) + 9441*a*b^2*d^(17/2)*x^(11/2) + 11645*a^2 
*b*d^(17/2)*x^(7/2) + 5267*a^3*d^(17/2)*x^(3/2))/(b^8*x^8 + 4*a*b^7*x^6 + 
6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4) + 1/192*((261*a*b^4*d^(17/2)*x^5 
+ 610*a^2*b^3*d^(17/2)*x^3 + 381*a^3*b^2*d^(17/2)*x)*x^(9/2) + 2*(191*a^2* 
b^3*d^(17/2)*x^5 + 462*a^3*b^2*d^(17/2)*x^3 + 303*a^4*b*d^(17/2)*x)*x^(5/2 
) + (153*a^3*b^2*d^(17/2)*x^5 + 378*a^4*b*d^(17/2)*x^3 + 257*a^5*d^(17/2)* 
x)*sqrt(x))/(a^3*b^7*x^6 + 3*a^4*b^6*x^4 + 3*a^5*b^5*x^2 + a^6*b^4 + (b^10 
*x^6 + 3*a*b^9*x^4 + 3*a^2*b^8*x^2 + a^3*b^7)*x^6 + 3*(a*b^9*x^6 + 3*a^2*b 
^8*x^4 + 3*a^3*b^7*x^2 + a^4*b^6)*x^4 + 3*(a^2*b^8*x^6 + 3*a^3*b^7*x^4 + 3 
*a^4*b^6*x^2 + a^5*b^5)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.85 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{8192} \, d^{8} {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {8 \, {\left (893 \, \sqrt {d x} b^{3} d^{8} x^{7} + 1755 \, \sqrt {d x} a b^{2} d^{8} x^{5} + 1375 \, \sqrt {d x} a^{2} b d^{8} x^{3} + 385 \, \sqrt {d x} a^{3} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \] Input:

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

1/8192*d^8*(2310*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*b^7*d*sgn(b*x^2 + a)) + 23 
10*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*b^7*d*sgn(b*x^2 + a)) - 1155*sqrt(2)*(a 
*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b 
))/(a*b^7*d*sgn(b*x^2 + a)) + 1155*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqr 
t(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^7*d*sgn(b*x^2 + a)) - 
 8*(893*sqrt(d*x)*b^3*d^8*x^7 + 1755*sqrt(d*x)*a*b^2*d^8*x^5 + 1375*sqrt(d 
*x)*a^2*b*d^8*x^3 + 385*sqrt(d*x)*a^3*d^8*x)/((b*d^2*x^2 + a*d^2)^4*b^4*sg 
n(b*x^2 + a)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{17/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d*x)^(17/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
 

Output:

int((d*x)^(17/2)/(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 = 825, normalized size of antiderivative = 1.84 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(sqrt(d)*d**8*( - 2310*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s 
qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4 - 9240*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 - 13860*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)*s 
qrt(2)))*a**2*b**2*x**4 - 9240*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 - 2310*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sq 
rt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**4*x**8 + 2310*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 + 9240*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 + 13860*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 + 9240*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 + 2310*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)*sq 
rt(2)))*b**4*x**8 + 1155*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 + 4620*b**(1/4)*a**(3/4)*sqr 
t(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*...