Optimal. Leaf size=389 \[ -\frac {231 d^{3/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}-\frac {231 d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.50, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {231 d^{3/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}-\frac {231 d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 288
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (b^4 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {\left (3 b^3 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{64 a}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {\left (11 b^2 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^2}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {\left (77 b d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^3}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {\left (231 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^4}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {(231 d) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^4}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {231 \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^{9/2}}+\frac {231 \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^{9/2}}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}-\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}+\frac {\left (231 d^2\right ) \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^{9/2} b^{3/2}}+\frac {\left (231 d^2\right ) \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^{9/2} b^{3/2}}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {231 d^{3/2} \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^{19/4} b^{5/4}}+\frac {231 d^{3/2} \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^{19/4} b^{5/4}}+\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}-\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}\\ &=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {231 d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}-\frac {231 d^{3/2} \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^{19/4} b^{5/4}}+\frac {231 d^{3/2} \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^{19/4} b^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 298, normalized size = 0.77 \[ \frac {d \sqrt {d x} \left (-\frac {1155 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{19/4} \sqrt {x}}+\frac {1155 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{19/4} \sqrt {x}}-\frac {2310 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{19/4} \sqrt {x}}+\frac {2310 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{19/4} \sqrt {x}}+\frac {3080 \sqrt [4]{b}}{a^4 \left (a+b x^2\right )}+\frac {1760 \sqrt [4]{b}}{a^3 \left (a+b x^2\right )^2}+\frac {1280 \sqrt [4]{b}}{a^2 \left (a+b x^2\right )^3}+\frac {1024 \sqrt [4]{b}}{a \left (a+b x^2\right )^4}-\frac {16384 \sqrt [4]{b}}{\left (a+b x^2\right )^5}\right )}{163840 b^{5/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 485, normalized size = 1.25 \[ \frac {4620 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{14} b^{4} d \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {3}{4}} - \sqrt {a^{10} b^{2} \sqrt {-\frac {d^{6}}{a^{19} b^{5}}} + d^{3} x} a^{14} b^{4} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {3}{4}}}{d^{6}}\right ) + 1155 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (231 \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (-231 \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) + 4 \, {\left (385 \, b^{4} d x^{8} + 1760 \, a b^{3} d x^{6} + 3130 \, a^{2} b^{2} d x^{4} + 2648 \, a^{3} b d x^{2} - 1155 \, a^{4} d\right )} \sqrt {d x}}{81920 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 340, normalized size = 0.87 \[ \frac {1}{163840} \, d {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2}} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2}} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2}} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2}} + \frac {8 \, {\left (385 \, \sqrt {d x} b^{4} d^{10} x^{8} + 1760 \, \sqrt {d x} a b^{3} d^{10} x^{6} + 3130 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} + 2648 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 1155 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{4} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 335, normalized size = 0.86 \[ -\frac {231 \sqrt {d x}\, d^{11}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {331 \left (d x \right )^{\frac {5}{2}} d^{9}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {313 \left (d x \right )^{\frac {9}{2}} b \,d^{7}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {11 \left (d x \right )^{\frac {13}{2}} b^{2} d^{5}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {77 \left (d x \right )^{\frac {17}{2}} b^{3} d^{3}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{5} b}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{5} b}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.23, size = 392, normalized size = 1.01 \[ \frac {\frac {8 \, {\left (385 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{4} + 1760 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{6} + 3130 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{8} + 2648 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{10} - 1155 \, \sqrt {d x} a^{4} d^{12}\right )}}{a^{4} b^{6} d^{10} x^{10} + 5 \, a^{5} b^{5} d^{10} x^{8} + 10 \, a^{6} b^{4} d^{10} x^{6} + 10 \, a^{7} b^{3} d^{10} x^{4} + 5 \, a^{8} b^{2} d^{10} x^{2} + a^{9} b d^{10}} + \frac {1155 \, {\left (\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{4} b}}{163840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 209, normalized size = 0.54 \[ \frac {\frac {331\,d^9\,{\left (d\,x\right )}^{5/2}}{2560\,a}-\frac {231\,d^{11}\,\sqrt {d\,x}}{4096\,b}+\frac {11\,b^2\,d^5\,{\left (d\,x\right )}^{13/2}}{128\,a^3}+\frac {77\,b^3\,d^3\,{\left (d\,x\right )}^{17/2}}{4096\,a^4}+\frac {313\,b\,d^7\,{\left (d\,x\right )}^{9/2}}{2048\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {231\,d^{3/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{19/4}\,b^{5/4}}-\frac {231\,d^{3/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{19/4}\,b^{5/4}} \]
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 \[ \int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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