Optimal. Leaf size=300 \[ -\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{9/4} d^{3/2}} \]
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Rubi [A]
time = 0.20, 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, 296, 331,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {5 \sqrt [4]{b} \text {ArcTan}\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} \text {ArcTan}\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 \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}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
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 ) \text {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 ) \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 )}{4 a^2 d^3}-\frac {\left (5 b^{3/2}\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 )}{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 ) \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 )}{8 \sqrt {2} a^{9/4} d^{3/2}}-\frac {\left (5 \sqrt [4]{b}\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 )}{8 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \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 )}{8 a^2 d}-\frac {5 \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 )}{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 ) \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 )}{4 \sqrt {2} a^{9/4} d^{3/2}}+\frac {\left (5 \sqrt [4]{b}\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 )}{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 [A]
time = 0.28, size = 165, normalized size = 0.55 \begin {gather*} \frac {x \left (-4 \sqrt [4]{a} \left (4 a+5 b x^2\right )+5 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{9/4} (d x)^{3/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 191, normalized size = 0.64
method | result | size |
derivativedivides | \(2 d^{3} \left (-\frac {1}{a^{2} d^{4} \sqrt {d x}}-\frac {b \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 d^{2} x^{2} b +4 a \,d^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} d^{4}}\right )\) | \(191\) |
default | \(2 d^{3} \left (-\frac {1}{a^{2} d^{4} \sqrt {d x}}-\frac {b \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 d^{2} x^{2} b +4 a \,d^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} d^{4}}\right )\) | \(191\) |
risch | \(-\frac {2}{a^{2} d \sqrt {d x}}+\frac {-\frac {b \left (d x \right )^{\frac {3}{2}}}{2 a^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )}-\frac {5 \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{16 a^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}}{d}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 268, normalized size = 0.89 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 276, normalized size = 0.92 \begin {gather*} \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 )}} \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 {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.08, size = 294, normalized size = 0.98 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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