Optimal. Leaf size=350 \[ \frac {195 \sqrt [4]{a} d^{15/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{17/4}}+\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} b^{17/4}}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac {195 d^7 \sqrt {d x}}{64 b^4} \]
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Rubi [A] time = 0.38, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}+\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} b^{17/4}}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac {195 d^7 \sqrt {d x}}{64 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 288
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{12} \left (13 b^2 d^2\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac {1}{32} \left (39 d^4\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {\left (195 d^6\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {\left (195 a d^8\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 b^3}\\ &=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {\left (195 a d^7\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 b^3}\\ &=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {\left (195 \sqrt {a} d^6\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 b^3}-\frac {\left (195 \sqrt {a} d^6\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 b^3}\\ &=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {\left (195 \sqrt [4]{a} d^{15/2}\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 )}{256 \sqrt {2} b^{17/4}}+\frac {\left (195 \sqrt [4]{a} d^{15/2}\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 )}{256 \sqrt {2} b^{17/4}}-\frac {\left (195 \sqrt {a} d^8\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 )}{256 b^{9/2}}-\frac {\left (195 \sqrt {a} d^8\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 )}{256 b^{9/2}}\\ &=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {\left (195 \sqrt [4]{a} d^{15/2}\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 )}{128 \sqrt {2} b^{17/4}}+\frac {\left (195 \sqrt [4]{a} d^{15/2}\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 )}{128 \sqrt {2} b^{17/4}}\\ &=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}+\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 324, normalized size = 0.93 \begin {gather*} \frac {d^7 \sqrt {d x} \left (\frac {49920 a^3 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac {119808 a^2 b^{5/4} x^2}{\left (a+b x^2\right )^3}-\frac {6240 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^2}+\frac {21504 b^{13/4} x^6}{\left (a+b x^2\right )^3}+\frac {93184 a b^{9/4} x^4}{\left (a+b x^2\right )^3}-\frac {10920 a \sqrt [4]{b}}{a+b x^2}+\frac {4095 \sqrt {2} \sqrt [4]{a} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {x}}-\frac {4095 \sqrt {2} \sqrt [4]{a} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {x}}+\frac {8190 \sqrt {2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {x}}-\frac {8190 \sqrt {2} \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {x}}\right )}{10752 b^{17/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.87, size = 227, normalized size = 0.65 \begin {gather*} \frac {d^7 \sqrt {d x} \left (585 a^3 d^6+1638 a^2 b d^6 x^2+1469 a b^2 d^6 x^4+384 b^3 d^6 x^6\right )}{192 b^4 \left (a d^2+b d^2 x^2\right )^3}+\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {d}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} \sqrt {d} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {d x}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}}{\sqrt {a} d+\sqrt {b} d x}\right )}{128 \sqrt {2} b^{17/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 363, normalized size = 1.04 \begin {gather*} -\frac {2340 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \arctan \left (-\frac {\left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {3}{4}} \sqrt {d x} b^{13} d^{7} - \sqrt {d^{15} x + \sqrt {-\frac {a d^{30}}{b^{17}}} b^{8}} \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {3}{4}} b^{13}}{a d^{30}}\right ) + 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} + 195 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} - 195 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 4 \, {\left (384 \, b^{3} d^{7} x^{6} + 1469 \, a b^{2} d^{7} x^{4} + 1638 \, a^{2} b d^{7} x^{2} + 585 \, a^{3} d^{7}\right )} \sqrt {d x}}{768 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 302, normalized size = 0.86 \begin {gather*} -\frac {1}{1536} \, d^{7} {\left (\frac {1170 \, \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 )}{b^{5}} + \frac {1170 \, \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 )}{b^{5}} + \frac {585 \, \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 )}{b^{5}} - \frac {585 \, \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 )}{b^{5}} - \frac {3072 \, \sqrt {d x}}{b^{4}} - \frac {8 \, {\left (317 \, \sqrt {d x} a b^{2} d^{6} x^{4} + 486 \, \sqrt {d x} a^{2} b d^{6} x^{2} + 201 \, \sqrt {d x} a^{3} d^{6}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 287, normalized size = 0.82 \begin {gather*} \frac {67 \sqrt {d x}\, a^{3} d^{13}}{64 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b^{4}}+\frac {81 \left (d x \right )^{\frac {5}{2}} a^{2} d^{11}}{32 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b^{3}}+\frac {317 \left (d x \right )^{\frac {9}{2}} a \,d^{9}}{192 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b^{2}}-\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{7} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 b^{4}}-\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{7} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 b^{4}}-\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{7} \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 b^{4}}+\frac {2 \sqrt {d x}\, d^{7}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.14, size = 343, normalized size = 0.98 \begin {gather*} \frac {\frac {3072 \, \sqrt {d x} d^{8}}{b^{4}} + \frac {8 \, {\left (317 \, \left (d x\right )^{\frac {9}{2}} a b^{2} d^{10} + 486 \, \left (d x\right )^{\frac {5}{2}} a^{2} b d^{12} + 201 \, \sqrt {d x} a^{3} d^{14}\right )}}{b^{7} d^{6} x^{6} + 3 \, a b^{6} d^{6} x^{4} + 3 \, a^{2} b^{5} d^{6} x^{2} + a^{3} b^{4} d^{6}} - \frac {585 \, {\left (\frac {\sqrt {2} d^{10} \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^{10} \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^{9} \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^{9} \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}{b^{4}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.30, size = 171, normalized size = 0.49 \begin {gather*} \frac {\frac {67\,a^3\,d^{13}\,\sqrt {d\,x}}{64}+\frac {81\,a^2\,b\,d^{11}\,{\left (d\,x\right )}^{5/2}}{32}+\frac {317\,a\,b^2\,d^9\,{\left (d\,x\right )}^{9/2}}{192}}{a^3\,b^4\,d^6+3\,a^2\,b^5\,d^6\,x^2+3\,a\,b^6\,d^6\,x^4+b^7\,d^6\,x^6}+\frac {2\,d^7\,\sqrt {d\,x}}{b^4}-\frac {195\,{\left (-a\right )}^{1/4}\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,b^{17/4}}+\frac {{\left (-a\right )}^{1/4}\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,195{}\mathrm {i}}{128\,b^{17/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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