Optimal. Leaf size=602 \[ \frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.32, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 296,
331, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {19}{96 a^2 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac {1}{8 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 296
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (19 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (95 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1045 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{2048 a^5 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 b \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a^5 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 b \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2048 a^{11/2} d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 b \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2048 a^{11/2} d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{4096 a^{11/2} \sqrt {b} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{4096 a^{11/2} \sqrt {b} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 213, normalized size = 0.35 \begin {gather*} \frac {x \left (a+b x^2\right ) \left (-4 a^{3/4} \left (2048 a^4+16967 a^3 b x^2+33345 a^2 b^2 x^4+26125 a b^3 x^6+7315 b^4 x^8\right )+21945 \sqrt {2} b^{3/4} x^{3/2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-21945 \sqrt {2} b^{3/4} x^{3/2} \left (a+b x^2\right )^4 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{23/4} (d x)^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1182\) vs.
\(2(391)=782\).
time = 0.08, size = 1183, normalized size = 1.97
method | result | size |
risch | \(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 a^{5} x \sqrt {d x}\, d^{2} \left (b \,x^{2}+a \right )}+\frac {\left (-\frac {2925 b \,d^{7} \sqrt {d x}}{1024 a^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}-\frac {7019 b^{2} d^{5} \left (d x \right )^{\frac {5}{2}}}{1024 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}-\frac {17933 b^{3} d^{3} \left (d x \right )^{\frac {9}{2}}}{3072 a^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}-\frac {5267 b^{4} d \left (d x \right )^{\frac {13}{2}}}{3072 a^{5} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}-\frac {7315 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{8192 a^{6} d}-\frac {7315 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{4096 a^{6} d}-\frac {7315 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{4096 a^{6} d}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{d^{2} \left (b \,x^{2}+a \right )}\) | \(368\) |
default | \(\text {Expression too large to display}\) | \(1183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 501, normalized size = 0.83 \begin {gather*} -\frac {87780 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{17} b d^{7} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {3}{4}} - \sqrt {a^{12} d^{6} \sqrt {-\frac {b^{3}}{a^{23} d^{10}}} + b^{2} d x} a^{17} d^{7} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 21945 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \log \left (7315 \, a^{6} d^{3} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} + 7315 \, \sqrt {d x} b\right ) - 21945 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \log \left (-7315 \, a^{6} d^{3} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} + 7315 \, \sqrt {d x} b\right ) + 4 \, {\left (7315 \, b^{4} x^{8} + 26125 \, a b^{3} x^{6} + 33345 \, a^{2} b^{2} x^{4} + 16967 \, a^{3} b x^{2} + 2048 \, a^{4}\right )} \sqrt {d x}}{12288 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.47, size = 397, normalized size = 0.66 \begin {gather*} -\frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{4096 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{4096 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{8192 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{8192 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {2}{3 \, \sqrt {d x} a^{5} d^{2} x \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5267 \, \sqrt {d x} b^{4} d^{6} x^{6} + 17933 \, \sqrt {d x} a b^{3} d^{6} x^{4} + 21057 \, \sqrt {d x} a^{2} b^{2} d^{6} x^{2} + 8775 \, \sqrt {d x} a^{3} b d^{6}}{3072 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5} d \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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