Optimal. Leaf size=404 \[ -\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {33649 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}+\frac {33649 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}} \]
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Rubi [A]
time = 0.33, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 296, 331,
335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {33649 b^{3/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}+\frac {33649 b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 217
Rule 296
Rule 331
Rule 335
Rule 631
Rule 642
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 )^3} \, dx &=b^6 \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {\left (23 b^5\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {\left (437 b^4\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^4} \, dx}{320 a^2}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {\left (437 b^3\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {\left (4807 b^2\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {(33649 b) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (33649 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^6 d^2}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (33649 b^2\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^6 d^3}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (33649 b^2\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 )}{8192 a^{13/2} d^4}-\frac {\left (33649 b^2\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 )}{8192 a^{13/2} d^4}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {\left (33649 b^{3/4}\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 )}{16384 \sqrt {2} a^{27/4} d^{5/2}}+\frac {\left (33649 b^{3/4}\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 )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {\left (33649 \sqrt {b}\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 )}{16384 a^{13/2} d^2}-\frac {\left (33649 \sqrt {b}\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 )}{16384 a^{13/2} d^2}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {33649 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {\left (33649 b^{3/4}\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 )}{8192 \sqrt {2} a^{27/4} d^{5/2}}+\frac {\left (33649 b^{3/4}\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 )}{8192 \sqrt {2} a^{27/4} d^{5/2}}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {33649 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}+\frac {33649 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.73, size = 195, normalized size = 0.48 \begin {gather*} \frac {x \left (-\frac {4 a^{3/4} \left (40960 a^5+437345 a^4 b x^2+1157176 a^3 b^2 x^4+1367810 a^2 b^3 x^6+769120 a b^4 x^8+168245 b^5 x^{10}\right )}{\left (a+b x^2\right )^5}+504735 \sqrt {2} b^{3/4} x^{3/2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-504735 \sqrt {2} b^{3/4} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{245760 a^{27/4} (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 256, normalized size = 0.63
method | result | size |
derivativedivides | \(2 d^{11} \left (-\frac {b \left (\frac {\frac {15503 a^{4} d^{8} \sqrt {d x}}{8192}+\frac {31149 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}+\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{12288}+\frac {3527 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{768}+\frac {25457 b^{4} \left (d x \right )^{\frac {17}{2}}}{24576}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{a^{6} d^{12}}-\frac {1}{3 a^{6} d^{12} \left (d x \right )^{\frac {3}{2}}}\right )\) | \(256\) |
default | \(2 d^{11} \left (-\frac {b \left (\frac {\frac {15503 a^{4} d^{8} \sqrt {d x}}{8192}+\frac {31149 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}+\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{12288}+\frac {3527 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{768}+\frac {25457 b^{4} \left (d x \right )^{\frac {17}{2}}}{24576}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{a^{6} d^{12}}-\frac {1}{3 a^{6} d^{12} \left (d x \right )^{\frac {3}{2}}}\right )\) | \(256\) |
risch | \(-\frac {2}{3 a^{6} x \sqrt {d x}\, d^{2}}+\frac {-\frac {15503 b \,d^{9} \sqrt {d x}}{4096 a^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}-\frac {31149 b^{2} d^{7} \left (d x \right )^{\frac {5}{2}}}{2560 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}-\frac {95821 b^{3} d^{5} \left (d x \right )^{\frac {9}{2}}}{6144 a^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}-\frac {3527 b^{4} d^{3} \left (d x \right )^{\frac {13}{2}}}{384 a^{5} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}-\frac {25457 b^{5} d \left (d x \right )^{\frac {17}{2}}}{12288 a^{6} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}-\frac {33649 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 a^{7} d}-\frac {33649 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{7} d}-\frac {33649 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{7} d}}{d^{2}}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 395, normalized size = 0.98 \begin {gather*} -\frac {\frac {8 \, {\left (168245 \, b^{5} d^{10} x^{10} + 769120 \, a b^{4} d^{10} x^{8} + 1367810 \, a^{2} b^{3} d^{10} x^{6} + 1157176 \, a^{3} b^{2} d^{10} x^{4} + 437345 \, a^{4} b d^{10} x^{2} + 40960 \, a^{5} d^{10}\right )}}{\left (d x\right )^{\frac {23}{2}} a^{6} b^{5} + 5 \, \left (d x\right )^{\frac {19}{2}} a^{7} b^{4} d^{2} + 10 \, \left (d x\right )^{\frac {15}{2}} a^{8} b^{3} d^{4} + 10 \, \left (d x\right )^{\frac {11}{2}} a^{9} b^{2} d^{6} + 5 \, \left (d x\right )^{\frac {7}{2}} a^{10} b d^{8} + \left (d x\right )^{\frac {3}{2}} a^{11} d^{10}} + \frac {504735 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d}\right )}}{a^{6}}}{491520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 568, normalized size = 1.41 \begin {gather*} -\frac {2018940 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{20} b d^{7} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {3}{4}} - \sqrt {a^{14} d^{6} \sqrt {-\frac {b^{3}}{a^{27} d^{10}}} + b^{2} d x} a^{20} d^{7} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 504735 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (33649 \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) - 504735 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (-33649 \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) + 4 \, {\left (168245 \, b^{5} x^{10} + 769120 \, a b^{4} x^{8} + 1367810 \, a^{2} b^{3} x^{6} + 1157176 \, a^{3} b^{2} x^{4} + 437345 \, a^{4} b x^{2} + 40960 \, a^{5}\right )} \sqrt {d x}}{245760 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} 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 (a + b x^{2}\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.20, size = 356, normalized size = 0.88 \begin {gather*} -\frac {33649 \, \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 )}{16384 \, a^{7} d^{3}} - \frac {33649 \, \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 )}{16384 \, a^{7} d^{3}} - \frac {33649 \, \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 )}{32768 \, a^{7} d^{3}} + \frac {33649 \, \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 )}{32768 \, a^{7} d^{3}} - \frac {2}{3 \, \sqrt {d x} a^{6} d^{2} x} - \frac {127285 \, \sqrt {d x} b^{5} d^{8} x^{8} + 564320 \, \sqrt {d x} a b^{4} d^{8} x^{6} + 958210 \, \sqrt {d x} a^{2} b^{3} d^{8} x^{4} + 747576 \, \sqrt {d x} a^{3} b^{2} d^{8} x^{2} + 232545 \, \sqrt {d x} a^{4} b d^{8}}{61440 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.46, size = 226, normalized size = 0.56 \begin {gather*} \frac {33649\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{27/4}\,d^{5/2}}-\frac {\frac {2\,d^9}{3\,a}+\frac {87469\,b\,d^9\,x^2}{12288\,a^2}+\frac {144647\,b^2\,d^9\,x^4}{7680\,a^3}+\frac {136781\,b^3\,d^9\,x^6}{6144\,a^4}+\frac {4807\,b^4\,d^9\,x^8}{384\,a^5}+\frac {33649\,b^5\,d^9\,x^{10}}{12288\,a^6}}{b^5\,{\left (d\,x\right )}^{23/2}+a^5\,d^{10}\,{\left (d\,x\right )}^{3/2}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{11/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{15/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{7/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{19/2}}+\frac {33649\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{27/4}\,d^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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