Optimal. Leaf size=649 \[ \frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.53, antiderivative size = 649, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1112
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 )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (119 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1547 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^5 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a^6 d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 b^2 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a^6 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 b^{3/2} \left (a b+b^2 x^2\right )\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 )}{2048 a^6 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 b^{3/2} \left (a b+b^2 x^2\right )\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 )}{2048 a^6 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \sqrt [4]{b} \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \sqrt [4]{b} \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \left (a b+b^2 x^2\right )\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 )}{4096 a^6 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \left (a b+b^2 x^2\right )\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 )}{4096 a^6 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \sqrt [4]{b} \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 \sqrt [4]{b} \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.08 \begin {gather*} -\frac {2 x \left (a+b x^2\right )^5 \, _2F_1\left (-\frac {5}{4},5;-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 a^5 (d x)^{7/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 144.05, size = 297, normalized size = 0.46 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (-\frac {13923 b^{5/4} \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 )}{2048 \sqrt {2} a^{25/4} d^{7/2}}-\frac {13923 b^{5/4} \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 )}{2048 \sqrt {2} a^{25/4} d^{7/2}}+\frac {-2048 a^5 d^{10}+43008 a^4 b d^{10} x^2+220507 a^3 b^2 d^{10} x^4+369733 a^2 b^3 d^{10} x^6+264537 a b^4 d^{10} x^8+69615 b^5 d^{10} x^{10}}{5120 a^6 d^3 (d x)^{5/2} \left (a d^2+b d^2 x^2\right )^4}\right )}{d^2 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.46, size = 524, normalized size = 0.81 \begin {gather*} -\frac {278460 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {2698972561467 \, \sqrt {d x} a^{6} b^{4} d^{3} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-7284452887551739093192089 \, a^{13} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{25} d^{14}}} + 7284452887551739093192089 \, b^{8} d x} a^{6} d^{3} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}}}{2698972561467 \, b^{5}}\right ) - 69615 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (2698972561467 \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) + 69615 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (-2698972561467 \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) - 4 \, {\left (69615 \, b^{5} x^{10} + 264537 \, a b^{4} x^{8} + 369733 \, a^{2} b^{3} x^{6} + 220507 \, a^{3} b^{2} x^{4} + 43008 \, a^{4} b x^{2} - 2048 \, a^{5}\right )} \sqrt {d x}}{20480 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 470, normalized size = 0.72 \begin {gather*} \frac {13923 \, \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 )}{4096 \, a^{7} b d^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {13923 \, \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 )}{4096 \, a^{7} b d^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {13923 \, \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 )}{8192 \, a^{7} b d^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {13923 \, \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 )}{8192 \, a^{7} b d^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {3683 \, \sqrt {d x} b^{5} d^{7} x^{7} + 12357 \, \sqrt {d x} a b^{4} d^{7} x^{5} + 14145 \, \sqrt {d x} a^{2} b^{3} d^{7} x^{3} + 5599 \, \sqrt {d x} a^{3} b^{2} d^{7} x}{1024 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{6} d^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {2 \, {\left (25 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{6} d^{5} x^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1129, normalized size = 1.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -4 \, b \int \frac {1}{{\left (a^{5} b d^{\frac {7}{2}} x^{2} + a^{6} d^{\frac {7}{2}}\right )} x^{\frac {3}{2}}}\,{d x} + \frac {11049 \, b^{5} x^{\frac {15}{2}} + 27135 \, a b^{4} x^{\frac {11}{2}} + 23395 \, a^{2} b^{3} x^{\frac {7}{2}} + 6925 \, a^{3} b^{2} x^{\frac {3}{2}}}{3072 \, {\left (a^{6} b^{4} d^{\frac {7}{2}} x^{8} + 4 \, a^{7} b^{3} d^{\frac {7}{2}} x^{6} + 6 \, a^{8} b^{2} d^{\frac {7}{2}} x^{4} + 4 \, a^{9} b d^{\frac {7}{2}} x^{2} + a^{10} d^{\frac {7}{2}}\right )}} + \frac {{\left (621 \, b^{6} x^{5} + 1042 \, a b^{5} x^{3} + 453 \, a^{2} b^{4} x\right )} x^{\frac {9}{2}} + 2 \, {\left (695 \, a b^{5} x^{5} + 1182 \, a^{2} b^{4} x^{3} + 519 \, a^{3} b^{3} x\right )} x^{\frac {5}{2}} + {\left (801 \, a^{2} b^{4} x^{5} + 1386 \, a^{3} b^{3} x^{3} + 617 \, a^{4} b^{2} x\right )} \sqrt {x}}{192 \, {\left (a^{8} b^{3} d^{\frac {7}{2}} x^{6} + 3 \, a^{9} b^{2} d^{\frac {7}{2}} x^{4} + 3 \, a^{10} b d^{\frac {7}{2}} x^{2} + a^{11} d^{\frac {7}{2}} + {\left (a^{5} b^{6} d^{\frac {7}{2}} x^{6} + 3 \, a^{6} b^{5} d^{\frac {7}{2}} x^{4} + 3 \, a^{7} b^{4} d^{\frac {7}{2}} x^{2} + a^{8} b^{3} d^{\frac {7}{2}}\right )} x^{6} + 3 \, {\left (a^{6} b^{5} d^{\frac {7}{2}} x^{6} + 3 \, a^{7} b^{4} d^{\frac {7}{2}} x^{4} + 3 \, a^{8} b^{3} d^{\frac {7}{2}} x^{2} + a^{9} b^{2} d^{\frac {7}{2}}\right )} x^{4} + 3 \, {\left (a^{7} b^{4} d^{\frac {7}{2}} x^{6} + 3 \, a^{8} b^{3} d^{\frac {7}{2}} x^{4} + 3 \, a^{9} b^{2} d^{\frac {7}{2}} x^{2} + a^{10} b d^{\frac {7}{2}}\right )} x^{2}\right )}} + \frac {3683 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, a^{6} d^{\frac {7}{2}}} + \int \frac {1}{{\left (a^{4} b d^{\frac {7}{2}} x^{2} + a^{5} d^{\frac {7}{2}}\right )} x^{\frac {7}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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