Optimal. Leaf size=387 \[ \frac {663 \sqrt {d} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.49, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {663 \sqrt {d} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 290
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {\left (17 b^5\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\left (221 b^4\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{320 a^2}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {\left (663 b^3\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a^3}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {\left (663 b^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(663 b) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^5}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(663 b) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^5 d}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac {\left (663 \sqrt {b}\right ) \operatorname {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^5 d}+\frac {\left (663 \sqrt {b}\right ) \operatorname {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^5 d}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {\left (663 \sqrt {d}\right ) \operatorname {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^{21/4} b^{3/4}}+\frac {\left (663 \sqrt {d}\right ) \operatorname {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^{21/4} b^{3/4}}+\frac {(663 d) \operatorname {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^5 b}+\frac {(663 d) \operatorname {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^5 b}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}+\frac {\left (663 \sqrt {d}\right ) \operatorname {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^{21/4} b^{3/4}}-\frac {\left (663 \sqrt {d}\right ) \operatorname {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^{21/4} b^{3/4}}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac {663 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.08 \[ \frac {2 x \sqrt {d x} \, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )}{3 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 469, normalized size = 1.21 \[ -\frac {39780 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {291434247 \, \sqrt {d x} a^{5} b d \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} - \sqrt {-84933920324457009 \, a^{11} b d^{2} \sqrt {-\frac {d^{2}}{a^{21} b^{3}}} + 84933920324457009 \, d^{3} x} a^{5} b \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}}}{291434247 \, d^{2}}\right ) - 9945 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (291434247 \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) + 9945 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (-291434247 \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) - 4 \, {\left (9945 \, b^{4} x^{9} + 47736 \, a b^{3} x^{7} + 90610 \, a^{2} b^{2} x^{5} + 84320 \, a^{3} b x^{3} + 37645 \, a^{4} x\right )} \sqrt {d x}}{245760 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 340, normalized size = 0.88 \[ \frac {\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{3}} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{3}} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{3}} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{3}} + \frac {8 \, {\left (9945 \, \sqrt {d x} b^{4} d^{11} x^{9} + 47736 \, \sqrt {d x} a b^{3} d^{11} x^{7} + 90610 \, \sqrt {d x} a^{2} b^{2} d^{11} x^{5} + 84320 \, \sqrt {d x} a^{3} b d^{11} x^{3} + 37645 \, \sqrt {d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}}}{491520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 336, normalized size = 0.87 \[ \frac {7529 \left (d x \right )^{\frac {3}{2}} d^{9}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {527 \left (d x \right )^{\frac {7}{2}} b \,d^{7}}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {9061 \left (d x \right )^{\frac {11}{2}} b^{2} d^{5}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {1989 \left (d x \right )^{\frac {15}{2}} b^{3} d^{3}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}+\frac {663 \left (d x \right )^{\frac {19}{2}} b^{4} d}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{5}}+\frac {663 \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} b}+\frac {663 \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} b}+\frac {663 \sqrt {2}\, d \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.19, size = 377, normalized size = 0.97 \[ \frac {\frac {8 \, {\left (9945 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{2} + 47736 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{4} + 90610 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{6} + 84320 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{8} + 37645 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {9945 \, d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{5}}}{491520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 210, normalized size = 0.54 \[ \frac {\frac {7529\,d^9\,{\left (d\,x\right )}^{3/2}}{12288\,a}+\frac {9061\,b^2\,d^5\,{\left (d\,x\right )}^{11/2}}{6144\,a^3}+\frac {1989\,b^3\,d^3\,{\left (d\,x\right )}^{15/2}}{2560\,a^4}+\frac {527\,b\,d^7\,{\left (d\,x\right )}^{7/2}}{384\,a^2}+\frac {663\,b^4\,d\,{\left (d\,x\right )}^{19/2}}{4096\,a^5}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {663\,\sqrt {d}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{21/4}\,b^{3/4}}+\frac {663\,\sqrt {d}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{21/4}\,b^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 89.24, size = 547, normalized size = 1.41 \[ \frac {75290 a^{4} d^{19} \left (d x\right )^{\frac {3}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {168640 a^{3} b d^{17} \left (d x\right )^{\frac {7}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {181220 a^{2} b^{2} d^{15} \left (d x\right )^{\frac {11}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {95472 a b^{3} d^{13} \left (d x\right )^{\frac {15}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {19890 b^{4} d^{11} \left (d x\right )^{\frac {19}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + 2 d^{11} \operatorname {RootSum} {\left (1152921504606846976 t^{4} a^{21} b^{3} d^{42} + 193220905761, \left (t \mapsto t \log {\left (\frac {35184372088832 t^{3} a^{16} b^{2} d^{32}}{291434247} + \sqrt {d x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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