Optimal. Leaf size=281 \[ \frac {3 d^{5/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {3 d^{5/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {3 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {3 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.28, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {3 d^{5/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {3 d^{5/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {3 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {3 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 288
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{5/2}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac {1}{4} \left (3 d^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx\\ &=-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac {1}{2} (3 d) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )\\ &=-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}-\frac {(3 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 )}{4 \sqrt {b}}+\frac {(3 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 )}{4 \sqrt {b}}\\ &=-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac {\left (3 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 )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (3 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 )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (3 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 )}{8 b^2}+\frac {\left (3 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 )}{8 b^2}\\ &=-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac {3 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 )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {3 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 )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (3 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 )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {\left (3 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 )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}\\ &=-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}-\frac {3 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {3 d^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {3 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 )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {3 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 )}{8 \sqrt {2} \sqrt [4]{a} b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.19 \[ \frac {2 d (d x)^{3/2} \left (\left (a+b x^2\right ) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {b x^2}{a}\right )-a\right )}{a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 247, normalized size = 0.88 \[ -\frac {4 \, \sqrt {d x} d^{2} x + 12 \, {\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {1}{4}} \sqrt {d x} b^{2} d^{7} - \sqrt {d^{15} x - \sqrt {-\frac {d^{10}}{a b^{7}}} a b^{3} d^{10}} \left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {1}{4}} b^{2}}{d^{10}}\right ) - 3 \, {\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {1}{4}} \log \left (27 \, \sqrt {d x} d^{7} + 27 \, \left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {3}{4}} a b^{5}\right ) + 3 \, {\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {1}{4}} \log \left (27 \, \sqrt {d x} d^{7} - 27 \, \left (-\frac {d^{10}}{a b^{7}}\right )^{\frac {3}{4}} a b^{5}\right )}{8 \, {\left (b^{2} x^{2} + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 277, normalized size = 0.99 \[ -\frac {1}{16} \, {\left (\frac {8 \, \sqrt {d x} d^{2} x}{{\left (b d^{2} x^{2} + a d^{2}\right )} b} - \frac {6 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4} d} - \frac {6 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4} d} + \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{4} d} - \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{4} d}\right )} d^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 209, normalized size = 0.74 \[ -\frac {\left (d x \right )^{\frac {3}{2}} d^{3}}{2 \left (b \,d^{2} x^{2}+d^{2} a \right ) b}+\frac {3 \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {3 \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {3 \sqrt {2}\, d^{3} \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{16 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.10, size = 256, normalized size = 0.91 \[ -\frac {\frac {8 \, \left (d x\right )^{\frac {3}{2}} d^{4}}{b^{2} d^{2} x^{2} + a b d^{2}} - \frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 92, normalized size = 0.33 \[ \frac {3\,d^{5/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{1/4}\,b^{7/4}}-\frac {3\,d^{5/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{1/4}\,b^{7/4}}-\frac {d^3\,{\left (d\,x\right )}^{3/2}}{2\,b\,\left (b\,d^2\,x^2+a\,d^2\right )} \]
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 \[ \int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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