Optimal. Leaf size=370 \[ \frac {663 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.45, antiderivative size = 370, 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, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {663 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {\left (17 b^3\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {\left (221 b^2\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{96 a^2}\\ &=\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {(663 b) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (663 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^4 d^2}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (663 b^3\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 a^5 d^4}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (663 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^5 d^5}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (663 b^{5/2}\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 )}{128 a^5 d^5}+\frac {\left (663 b^{5/2}\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 )}{128 a^5 d^5}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (663 b^{5/4}\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 )}{256 \sqrt {2} a^{21/4} d^{7/2}}+\frac {\left (663 b^{5/4}\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 )}{256 \sqrt {2} a^{21/4} d^{7/2}}+\frac {(663 b) \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 )}{256 a^5 d^3}+\frac {(663 b) \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 )}{256 a^5 d^3}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {663 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}+\frac {\left (663 b^{5/4}\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 )}{128 \sqrt {2} a^{21/4} d^{7/2}}-\frac {\left (663 b^{5/4}\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 )}{128 \sqrt {2} a^{21/4} d^{7/2}}\\ &=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {663 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.10 \[ -\frac {2 \sqrt {d x} \, _2F_1\left (-\frac {5}{4},4;-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 a^4 d^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 457, normalized size = 1.24 \[ -\frac {39780 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {291434247 \, \sqrt {d x} a^{5} b^{4} d^{3} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-84933920324457009 \, a^{11} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{21} d^{14}}} + 84933920324457009 \, b^{8} d x} a^{5} d^{3} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}}}{291434247 \, b^{5}}\right ) - 9945 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (291434247 \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) + 9945 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (-291434247 \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 4 \, {\left (9945 \, b^{4} x^{8} + 27846 \, a b^{3} x^{6} + 24973 \, a^{2} b^{2} x^{4} + 6528 \, a^{3} b x^{2} - 384 \, a^{4}\right )} \sqrt {d x}}{3840 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 349, normalized size = 0.94 \[ \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{6} b d^{5}} + \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{6} b d^{5}} - \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac {453 \, \sqrt {d x} b^{4} d^{5} x^{5} + 1038 \, \sqrt {d x} a b^{3} d^{5} x^{3} + 617 \, \sqrt {d x} a^{2} b^{2} d^{5} x}{192 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{5} d^{3}} + \frac {2 \, {\left (20 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{5} d^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 304, normalized size = 0.82 \[ \frac {617 \left (d x \right )^{\frac {3}{2}} b^{2} d}{192 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{3}}+\frac {173 \left (d x \right )^{\frac {7}{2}} b^{3}}{32 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{4} d}+\frac {151 \left (d x \right )^{\frac {11}{2}} b^{4}}{64 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{5} d^{3}}-\frac {2}{5 \left (d x \right )^{\frac {5}{2}} a^{4} d}+\frac {663 \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} d^{3}}+\frac {663 \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} d^{3}}+\frac {663 \sqrt {2}\, b \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{512 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} d^{3}}+\frac {8 b}{\sqrt {d x}\, a^{5} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.17, size = 350, normalized size = 0.95 \[ \frac {\frac {8 \, {\left (9945 \, b^{4} d^{8} x^{8} + 27846 \, a b^{3} d^{8} x^{6} + 24973 \, a^{2} b^{2} d^{8} x^{4} + 6528 \, a^{3} b d^{8} x^{2} - 384 \, a^{4} d^{8}\right )}}{\left (d x\right )^{\frac {17}{2}} a^{5} b^{3} d^{2} + 3 \, \left (d x\right )^{\frac {13}{2}} a^{6} b^{2} d^{4} + 3 \, \left (d x\right )^{\frac {9}{2}} a^{7} b d^{6} + \left (d x\right )^{\frac {5}{2}} a^{8} d^{8}} + \frac {9945 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{5} d^{2}}}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.33, size = 179, normalized size = 0.48 \[ \frac {\frac {34\,b\,d^5\,x^2}{5\,a^2}-\frac {2\,d^5}{5\,a}+\frac {24973\,b^2\,d^5\,x^4}{960\,a^3}+\frac {4641\,b^3\,d^5\,x^6}{160\,a^4}+\frac {663\,b^4\,d^5\,x^8}{64\,a^5}}{b^3\,{\left (d\,x\right )}^{17/2}+a^3\,d^6\,{\left (d\,x\right )}^{5/2}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{9/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{13/2}}-\frac {663\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{21/4}\,d^{7/2}}+\frac {663\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{21/4}\,d^{7/2}} \]
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 \[ \int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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