Optimal. Leaf size=414 \[ -\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.28, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1112, 325, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (a b+b^2 x^2\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{a d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (2 b \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{a d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (a b+b^2 x^2\right )\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 )}{a^{3/2} d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (a b+b^2 x^2\right )\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 )}{a^{3/2} d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^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 )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^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 )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a b+b^2 x^2\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 )}{2 a^{3/2} \sqrt {b} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a b+b^2 x^2\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 )}{2 a^{3/2} \sqrt {b} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a b+b^2 x^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 )}{\sqrt {2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^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 )}{\sqrt {2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{7/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.13 \begin {gather*} -\frac {2 x \left (a+b x^2\right ) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\frac {b x^2}{a}\right )}{3 a (d x)^{5/2} \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 44.93, size = 204, normalized size = 0.49 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (\frac {b^{3/4} \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 )}{\sqrt {2} a^{7/4} d^{5/2}}-\frac {b^{3/4} \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 )}{\sqrt {2} a^{7/4} d^{5/2}}-\frac {2}{3 a d (d x)^{3/2}}\right )}{d^2 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 227, normalized size = 0.55 \begin {gather*} -\frac {12 \, a d^{3} x^{2} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{5} b d^{7} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {3}{4}} - \sqrt {a^{4} d^{6} \sqrt {-\frac {b^{3}}{a^{7} d^{10}}} + b^{2} d x} a^{5} d^{7} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 3 \, a d^{3} x^{2} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {1}{4}} \log \left (a^{2} d^{3} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {1}{4}} + \sqrt {d x} b\right ) - 3 \, a d^{3} x^{2} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {1}{4}} \log \left (-a^{2} d^{3} \left (-\frac {b^{3}}{a^{7} d^{10}}\right )^{\frac {1}{4}} + \sqrt {d x} b\right ) + 4 \, \sqrt {d x}}{6 \, a d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 256, normalized size = 0.62 \begin {gather*} -\frac {1}{12} \, {\left (\frac {6 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{2} d^{3}} + \frac {6 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{2} d^{3}} + \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{2} d^{3}} - \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{2} d^{3}} + \frac {8}{\sqrt {d x} a d^{2} x}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 239, normalized size = 0.58 \begin {gather*} -\frac {\left (b \,x^{2}+a \right ) \left (8 a \,d^{2}+6 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (d x \right )^{\frac {3}{2}} b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+6 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (d x \right )^{\frac {3}{2}} b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+3 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (d x \right )^{\frac {3}{2}} b \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 )\right )}{12 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (d x \right )^{\frac {3}{2}} a^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 242, normalized size = 0.58 \begin {gather*} -\frac {\frac {3 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \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 {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \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 {3}{4}}} + \frac {2 \, \sqrt {2} b \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 {a} d} + \frac {2 \, \sqrt {2} b \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 {a} d}\right )}}{a} + \frac {8}{\left (d x\right )^{\frac {3}{2}} a}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{5/2}\,\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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