Optimal. Leaf size=300 \[ -\frac {5 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.32, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {5 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}+\frac {(5 b) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{4 a}\\ &=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (5 b^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{4 a^2 d^2}\\ &=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 a^2 d^3}\\ &=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}+\frac {\left (5 b^{3/2}\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 )}{4 a^2 d^3}-\frac {\left (5 b^{3/2}\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 )}{4 a^2 d^3}\\ &=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (5 \sqrt [4]{b}\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} a^{9/4} d^{3/2}}-\frac {\left (5 \sqrt [4]{b}\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} a^{9/4} d^{3/2}}-\frac {5 \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 a^2 d}-\frac {5 \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 a^2 d}\\ &=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}-\frac {5 \sqrt [4]{b} \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} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \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} a^{9/4} d^{3/2}}-\frac {\left (5 \sqrt [4]{b}\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} a^{9/4} d^{3/2}}+\frac {\left (5 \sqrt [4]{b}\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} a^{9/4} d^{3/2}}\\ &=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \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} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \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} a^{9/4} d^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.10 \[ -\frac {2 x \, _2F_1\left (-\frac {1}{4},2;\frac {3}{4};-\frac {b x^2}{a}\right )}{a^2 (d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 276, normalized size = 0.92 \[ \frac {20 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {125 \, \sqrt {d x} a^{2} b d \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-15625 \, a^{5} b d^{4} \sqrt {-\frac {b}{a^{9} d^{6}}} + 15625 \, b^{2} d x} a^{2} d \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}}}{125 \, b}\right ) - 5 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \log \left (125 \, a^{7} d^{5} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} b\right ) + 5 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{7} d^{5} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} b\right ) - 4 \, {\left (5 \, b x^{2} + 4 \, a\right )} \sqrt {d x}}{8 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 294, normalized size = 0.98 \[ -\frac {\frac {8 \, {\left (5 \, b d^{2} x^{2} + 4 \, a d^{2}\right )}}{{\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )} a^{2}} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{2} d^{2}} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{2} d^{2}} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{2} d^{2}} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{2} d^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 223, normalized size = 0.74 \[ -\frac {\left (d x \right )^{\frac {3}{2}} b}{2 \left (b \,d^{2} x^{2}+d^{2} a \right ) a^{2} d}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} d}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} d}-\frac {5 \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{16 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} d}-\frac {2}{\sqrt {d x}\, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.12, size = 268, normalized size = 0.89 \[ -\frac {\frac {8 \, {\left (5 \, b d^{2} x^{2} + 4 \, a d^{2}\right )}}{\left (d x\right )^{\frac {5}{2}} a^{2} b + \sqrt {d x} a^{3} d^{2}} + \frac {5 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 102, normalized size = 0.34 \[ \frac {5\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{9/4}\,d^{3/2}}-\frac {5\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{9/4}\,d^{3/2}}-\frac {\frac {2\,d}{a}+\frac {5\,b\,d\,x^2}{2\,a^2}}{b\,{\left (d\,x\right )}^{5/2}+a\,d^2\,\sqrt {d\,x}} \]
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 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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