Optimal. Leaf size=335 \[ \frac {5 d^{5/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.35, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \begin {gather*} \frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3} \end {gather*}
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)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{4} \left (b^2 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {\left (5 b d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{32 a}\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {\left (5 d^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 a^2}\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^2}\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}-\frac {(5 d) \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 a^2 \sqrt {b}}+\frac {(5 d) \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 a^2 \sqrt {b}}\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {\left (5 d^{5/2}\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} a^{9/4} b^{7/4}}+\frac {\left (5 d^{5/2}\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} a^{9/4} b^{7/4}}+\frac {\left (5 d^3\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 a^2 b^2}+\frac {\left (5 d^3\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 a^2 b^2}\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}+\frac {\left (5 d^{5/2}\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} a^{9/4} b^{7/4}}-\frac {\left (5 d^{5/2}\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} a^{9/4} b^{7/4}}\\ &=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}-\frac {5 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 60, normalized size = 0.18 \begin {gather*} \frac {2 d (d x)^{3/2} \left (\left (a+b x^2\right )^3 \, _2F_1\left (\frac {3}{4},4;\frac {7}{4};-\frac {b x^2}{a}\right )-a^3\right )}{9 a^3 b \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.79, size = 213, normalized size = 0.64 \begin {gather*} -\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \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} a^{9/4} b^{7/4}}+\frac {(d x)^{3/2} \left (-5 a^2 d^7+42 a b d^7 x^2+15 b^2 d^7 x^4\right )}{192 a^2 b \left (a d^2+b d^2 x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 396, normalized size = 1.18 \begin {gather*} -\frac {60 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {125 \, \sqrt {d x} a^{2} b^{2} d^{7} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} - \sqrt {-15625 \, a^{5} b^{3} d^{10} \sqrt {-\frac {d^{10}}{a^{9} b^{7}}} + 15625 \, d^{15} x} a^{2} b^{2} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}}}{125 \, d^{10}}\right ) - 15 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (125 \, a^{7} b^{5} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d^{7}\right ) + 15 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{7} b^{5} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d^{7}\right ) - 4 \, {\left (15 \, b^{2} d^{2} x^{5} + 42 \, a b d^{2} x^{3} - 5 \, a^{2} d^{2} x\right )} \sqrt {d x}}{768 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 317, normalized size = 0.95 \begin {gather*} \frac {1}{1536} \, d^{2} {\left (\frac {8 \, {\left (15 \, \sqrt {d x} b^{2} d^{6} x^{5} + 42 \, \sqrt {d x} a b d^{6} x^{3} - 5 \, \sqrt {d x} a^{2} d^{6} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{2} b} + \frac {30 \, \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^{3} b^{4} d} + \frac {30 \, \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^{3} b^{4} d} - \frac {15 \, \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^{3} b^{4} d} + \frac {15 \, \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^{3} b^{4} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 277, normalized size = 0.83 \begin {gather*} -\frac {5 \left (d x \right )^{\frac {3}{2}} d^{7}}{192 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b}+\frac {7 \left (d x \right )^{\frac {7}{2}} d^{5}}{32 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a}+\frac {5 \left (d x \right )^{\frac {11}{2}} b \,d^{3}}{64 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{2}}+\frac {5 \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}+\frac {5 \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}+\frac {5 \sqrt {2}\, d^{3} \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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 323, normalized size = 0.96 \begin {gather*} \frac {\frac {15 \, d^{4} {\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^{2} b} + \frac {8 \, {\left (15 \, \left (d x\right )^{\frac {11}{2}} b^{2} d^{4} + 42 \, \left (d x\right )^{\frac {7}{2}} a b d^{6} - 5 \, \left (d x\right )^{\frac {3}{2}} a^{2} d^{8}\right )}}{a^{2} b^{4} d^{6} x^{6} + 3 \, a^{3} b^{3} d^{6} x^{4} + 3 \, a^{4} b^{2} d^{6} x^{2} + a^{5} b d^{6}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 149, normalized size = 0.44 \begin {gather*} \frac {\frac {7\,d^5\,{\left (d\,x\right )}^{7/2}}{32\,a}-\frac {5\,d^7\,{\left (d\,x\right )}^{3/2}}{192\,b}+\frac {5\,b\,d^3\,{\left (d\,x\right )}^{11/2}}{64\,a^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 {5\,d^{5/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{9/4}\,b^{7/4}}-\frac {5\,d^{5/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{9/4}\,b^{7/4}} \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 {5}{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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