Optimal. Leaf size=394 \[ \frac {63 d^{9/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{13/4} b^{11/4}}-\frac {63 d^{9/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{13/4} b^{11/4}}-\frac {63 d^{9/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{13/4} b^{11/4}}+\frac {63 d^{9/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{13/4} b^{11/4}}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.46, antiderivative size = 394, 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, 297, 1162, 617, 204, 1165, 628} \[ \frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^{9/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{13/4} b^{11/4}}-\frac {63 d^{9/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{13/4} b^{11/4}}-\frac {63 d^{9/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{13/4} b^{11/4}}+\frac {63 d^{9/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{13/4} b^{11/4}}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 288
Rule 290
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{9/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (7 b^4 d^2\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (21 b^2 d^4\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {\left (63 b d^4\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {\left (63 d^4\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^2}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac {\left (63 d^4\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^3 b}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac {\left (63 d^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^3 b}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}-\frac {\left (63 d^3\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^3 b^{3/2}}+\frac {\left (63 d^3\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^3 b^{3/2}}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac {\left (63 d^{9/2}\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^{13/4} b^{11/4}}+\frac {\left (63 d^{9/2}\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^{13/4} b^{11/4}}+\frac {\left (63 d^5\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^3 b^3}+\frac {\left (63 d^5\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^3 b^3}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac {63 d^{9/2} \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^{13/4} b^{11/4}}-\frac {63 d^{9/2} \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^{13/4} b^{11/4}}+\frac {\left (63 d^{9/2}\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^{13/4} b^{11/4}}-\frac {\left (63 d^{9/2}\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^{13/4} b^{11/4}}\\ &=-\frac {d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac {7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac {63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac {63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}-\frac {63 d^{9/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{13/4} b^{11/4}}+\frac {63 d^{9/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{13/4} b^{11/4}}+\frac {63 d^{9/2} \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^{13/4} b^{11/4}}-\frac {63 d^{9/2} \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^{13/4} b^{11/4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 61, normalized size = 0.15 \[ \frac {2 d^4 x \sqrt {d x} \left (\frac {7 \, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )}{a^4}+\frac {-7 a-17 b x^2}{\left (a+b x^2\right )^5}\right )}{221 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 520, normalized size = 1.32 \[ -\frac {1260 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {250047 \, \sqrt {d x} a^{3} b^{3} d^{13} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {1}{4}} - \sqrt {-62523502209 \, a^{7} b^{5} d^{18} \sqrt {-\frac {d^{18}}{a^{13} b^{11}}} + 62523502209 \, d^{27} x} a^{3} b^{3} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {1}{4}}}{250047 \, d^{18}}\right ) - 315 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {1}{4}} \log \left (250047 \, a^{10} b^{8} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {3}{4}} + 250047 \, \sqrt {d x} d^{13}\right ) + 315 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {1}{4}} \log \left (-250047 \, a^{10} b^{8} \left (-\frac {d^{18}}{a^{13} b^{11}}\right )^{\frac {3}{4}} + 250047 \, \sqrt {d x} d^{13}\right ) - 4 \, {\left (315 \, b^{4} d^{4} x^{9} + 1512 \, a b^{3} d^{4} x^{7} + 2870 \, a^{2} b^{2} d^{4} x^{5} - 480 \, a^{3} b d^{4} x^{3} - 105 \, a^{4} d^{4} x\right )} \sqrt {d x}}{81920 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 355, normalized size = 0.90 \[ \frac {1}{163840} \, d^{4} {\left (\frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{5} d} + \frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{5} d} - \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{5} d} + \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{5} d} + \frac {8 \, {\left (315 \, \sqrt {d x} b^{4} d^{10} x^{9} + 1512 \, \sqrt {d x} a b^{3} d^{10} x^{7} + 2870 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} - 480 \, \sqrt {d x} a^{3} b d^{10} x^{3} - 105 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{3} b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 339, normalized size = 0.86 \[ -\frac {21 \left (d x \right )^{\frac {3}{2}} a \,d^{13}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {3 \left (d x \right )^{\frac {7}{2}} d^{11}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {287 \left (d x \right )^{\frac {11}{2}} d^{9}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {189 \left (d x \right )^{\frac {15}{2}} b \,d^{7}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {63 \left (d x \right )^{\frac {19}{2}} b^{2} d^{5}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {63 \sqrt {2}\, d^{5} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{3} b^{3}}+\frac {63 \sqrt {2}\, d^{5} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{3} b^{3}}+\frac {63 \sqrt {2}\, d^{5} \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.17, size = 385, normalized size = 0.98 \[ \frac {\frac {315 \, d^{6} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{3} b^{2}} + \frac {8 \, {\left (315 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{6} + 1512 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{8} + 2870 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{10} - 480 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{12} - 105 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{14}\right )}}{a^{3} b^{7} d^{10} x^{10} + 5 \, a^{4} b^{6} d^{10} x^{8} + 10 \, a^{5} b^{5} d^{10} x^{6} + 10 \, a^{6} b^{4} d^{10} x^{4} + 5 \, a^{7} b^{3} d^{10} x^{2} + a^{8} b^{2} d^{10}}}{163840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 207, normalized size = 0.53 \[ \frac {\frac {287\,d^9\,{\left (d\,x\right )}^{11/2}}{2048\,a}-\frac {3\,d^{11}\,{\left (d\,x\right )}^{7/2}}{128\,b}+\frac {63\,b^2\,d^5\,{\left (d\,x\right )}^{19/2}}{4096\,a^3}-\frac {21\,a\,d^{13}\,{\left (d\,x\right )}^{3/2}}{4096\,b^2}+\frac {189\,b\,d^7\,{\left (d\,x\right )}^{15/2}}{2560\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {63\,d^{9/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{13/4}\,b^{11/4}}+\frac {63\,d^{9/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{13/4}\,b^{11/4}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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