Optimal. Leaf size=557 \[ -\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.43, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \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 211
Rule 288
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{11/2}}{\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 {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (9 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (15 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (15 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{2048 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^5 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^4 \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^{3/2} b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^4 \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^{3/2} b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 d^{11/2} \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^{7/4} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 d^{11/2} \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^{7/4} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^6 \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^{3/2} b^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^6 \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^{3/2} b^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^{11/2} \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^{7/4} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 d^{11/2} \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^{7/4} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {15 d^5 \sqrt {d x}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^3 (d x)^{5/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {15 d^5 \sqrt {d x}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{11/2} \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^{7/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 352, normalized size = 0.63 \begin {gather*} \frac {d (d x)^{9/2} \left (a+b x^2\right ) \left (-\frac {3465 \sqrt {2} \left (a+b x^2\right )^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{7/4}}+\frac {3465 \sqrt {2} \left (a+b x^2\right )^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{7/4}}-\frac {6930 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {6930 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-46080 a^2 \sqrt [4]{b} \sqrt {x}-147456 a b^{5/4} x^{5/2}+\frac {9240 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^3}{a}+5280 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2+3840 a \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )-180224 b^{9/4} x^{9/2}\right )}{630784 b^{13/4} x^{9/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 115.50, size = 272, normalized size = 0.49 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (-\frac {45 d^{11/2} \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 )}{2048 \sqrt {2} a^{7/4} b^{13/4}}+\frac {45 d^{11/2} \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 )}{2048 \sqrt {2} a^{7/4} b^{13/4}}-\frac {d^7 \sqrt {d x} \left (45 a^3 d^6+171 a^2 b d^6 x^2+239 a b^2 d^6 x^4-15 b^3 d^6 x^6\right )}{1024 a b^3 \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 = 1.20, size = 447, normalized size = 0.80 \begin {gather*} \frac {180 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {3}{4}} \sqrt {d x} a^{5} b^{10} d^{5} - \sqrt {d^{11} x + \sqrt {-\frac {d^{22}}{a^{7} b^{13}}} a^{4} b^{6}} \left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {3}{4}} a^{5} b^{10}}{d^{22}}\right ) + 45 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (45 \, \sqrt {d x} d^{5} + 45 \, \left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {1}{4}} a^{2} b^{3}\right ) - 45 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (45 \, \sqrt {d x} d^{5} - 45 \, \left (-\frac {d^{22}}{a^{7} b^{13}}\right )^{\frac {1}{4}} a^{2} b^{3}\right ) + 4 \, {\left (15 \, b^{3} d^{5} x^{6} - 239 \, a b^{2} d^{5} x^{4} - 171 \, a^{2} b d^{5} x^{2} - 45 \, a^{3} d^{5}\right )} \sqrt {d x}}{4096 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 408, normalized size = 0.73 \begin {gather*} \frac {1}{8192} \, d^{5} {\left (\frac {90 \, \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^{2} b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {90 \, \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^{2} b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {45 \, \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^{2} b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {45 \, \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^{2} b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (15 \, \sqrt {d x} b^{3} d^{8} x^{6} - 239 \, \sqrt {d x} a b^{2} d^{8} x^{4} - 171 \, \sqrt {d x} a^{2} b d^{8} x^{2} - 45 \, \sqrt {d x} a^{3} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1136, normalized size = 2.04
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.71, size = 595, normalized size = 1.07 \begin {gather*} \frac {45 \, d^{5} {\left (\frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{8192 \, a b^{3}} - \frac {35 \, b^{3} d^{\frac {11}{2}} x^{\frac {13}{2}} + 173 \, a b^{2} d^{\frac {11}{2}} x^{\frac {9}{2}} + 657 \, a^{2} b d^{\frac {11}{2}} x^{\frac {5}{2}} + 135 \, a^{3} d^{\frac {11}{2}} \sqrt {x}}{3072 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} + \frac {{\left (5 \, b^{4} d^{\frac {11}{2}} x^{5} + 18 \, a b^{3} d^{\frac {11}{2}} x^{3} + 45 \, a^{2} b^{2} d^{\frac {11}{2}} x\right )} x^{\frac {11}{2}} - 2 \, {\left (21 \, a b^{3} d^{\frac {11}{2}} x^{5} + 42 \, a^{2} b^{2} d^{\frac {11}{2}} x^{3} - 11 \, a^{3} b d^{\frac {11}{2}} x\right )} x^{\frac {7}{2}} - {\left (15 \, a^{2} b^{2} d^{\frac {11}{2}} x^{5} + 38 \, a^{3} b d^{\frac {11}{2}} x^{3} - 9 \, a^{4} d^{\frac {11}{2}} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{4} b^{5} x^{6} + 3 \, a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{2} + a^{7} b^{2} + {\left (a b^{8} x^{6} + 3 \, a^{2} b^{7} x^{4} + 3 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} x^{6} + 3 \, {\left (a^{2} b^{7} x^{6} + 3 \, a^{3} b^{6} x^{4} + 3 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )} x^{4} + 3 \, {\left (a^{3} b^{6} x^{6} + 3 \, a^{4} b^{5} x^{4} + 3 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )} x^{2}\right )}} \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 {{\left (d\,x\right )}^{11/2}}{{\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(-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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