Optimal. Leaf size=333 \[ -\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac {77 d^{13/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {77 d^{13/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} \sqrt [4]{a} b^{15/4}} \]
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Rubi [A]
time = 0.23, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 294, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {77 d^{13/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}-\frac {77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 294
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{12} \left (11 b^2 d^2\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac {1}{96} \left (77 d^4\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (77 d^6\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (77 d^5\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 b^2}\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac {\left (77 d^5\right ) \text {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^{5/2}}+\frac {\left (77 d^5\right ) \text {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^{5/2}}\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (77 d^{13/2}\right ) \text {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} \sqrt [4]{a} b^{15/4}}+\frac {\left (77 d^{13/2}\right ) \text {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} \sqrt [4]{a} b^{15/4}}+\frac {\left (77 d^7\right ) \text {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^4}+\frac {\left (77 d^7\right ) \text {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^4}\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac {77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}-\frac {77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}+\frac {\left (77 d^{13/2}\right ) \text {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} \sqrt [4]{a} b^{15/4}}-\frac {\left (77 d^{13/2}\right ) \text {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} \sqrt [4]{a} b^{15/4}}\\ &=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac {77 d^{13/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}-\frac {77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 164, normalized size = 0.49 \begin {gather*} \frac {d^6 \sqrt {d x} \left (-\frac {4 b^{3/4} x^{3/2} \left (77 a^2+198 a b x^2+153 b^2 x^4\right )}{\left (a+b x^2\right )^3}-\frac {231 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}-\frac {231 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}\right )}{768 b^{15/4} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 203, normalized size = 0.61
method | result | size |
derivativedivides | \(2 d^{7} \left (\frac {-\frac {51 \left (d x \right )^{\frac {11}{2}}}{128 b}-\frac {33 d^{2} a \left (d x \right )^{\frac {7}{2}}}{64 b^{2}}-\frac {77 d^{4} a^{2} \left (d x \right )^{\frac {3}{2}}}{384 b^{3}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {77 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(203\) |
default | \(2 d^{7} \left (\frac {-\frac {51 \left (d x \right )^{\frac {11}{2}}}{128 b}-\frac {33 d^{2} a \left (d x \right )^{\frac {7}{2}}}{64 b^{2}}-\frac {77 d^{4} a^{2} \left (d x \right )^{\frac {3}{2}}}{384 b^{3}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {77 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(203\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 317, normalized size = 0.95 \begin {gather*} \frac {\frac {231 \, d^{8} {\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 )}}{b^{3}} - \frac {8 \, {\left (153 \, \left (d x\right )^{\frac {11}{2}} b^{2} d^{8} + 198 \, \left (d x\right )^{\frac {7}{2}} a b d^{10} + 77 \, \left (d x\right )^{\frac {3}{2}} a^{2} d^{12}\right )}}{b^{6} d^{6} x^{6} + 3 \, a b^{5} d^{6} x^{4} + 3 \, a^{2} b^{4} d^{6} x^{2} + a^{3} b^{3} d^{6}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 370, normalized size = 1.11 \begin {gather*} -\frac {924 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \sqrt {d x} b^{4} d^{19} - \sqrt {d^{39} x - \sqrt {-\frac {d^{26}}{a b^{15}}} a b^{7} d^{26}} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} b^{4}}{d^{26}}\right ) - 231 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} + 456533 \, \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {3}{4}} a b^{11}\right ) + 231 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} - 456533 \, \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {3}{4}} a b^{11}\right ) + 4 \, {\left (153 \, b^{2} d^{6} x^{5} + 198 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt {d x}}{768 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \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 {\left (d x\right )^{\frac {13}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.42, size = 314, normalized size = 0.94 \begin {gather*} -\frac {1}{1536} \, d^{6} {\left (\frac {8 \, {\left (153 \, \sqrt {d x} b^{2} d^{6} x^{5} + 198 \, \sqrt {d x} a b d^{6} x^{3} + 77 \, \sqrt {d x} a^{2} d^{6} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{3}} - \frac {462 \, \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 b^{6} d} - \frac {462 \, \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 b^{6} d} + \frac {231 \, \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 b^{6} d} - \frac {231 \, \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 b^{6} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 153, normalized size = 0.46 \begin {gather*} \frac {77\,d^{13/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{1/4}\,b^{15/4}}-\frac {\frac {51\,d^7\,{\left (d\,x\right )}^{11/2}}{64\,b}+\frac {77\,a^2\,d^{11}\,{\left (d\,x\right )}^{3/2}}{192\,b^3}+\frac {33\,a\,d^9\,{\left (d\,x\right )}^{7/2}}{32\,b^2}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}-\frac {77\,d^{13/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{1/4}\,b^{15/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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