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 = 0.348355, antiderivative size = 335, normalized size of antiderivative = 1., 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} \[ \frac{5 d^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{9/4} b^{7/4}}-\frac{5 d^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{9/4} b^{7/4}}-\frac{5 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{5 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac{d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}-\frac{d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 290
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
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*}
Mathematica [C] time = 0.020821, size = 60, normalized size = 0.18 \[ \frac{2 d (d x)^{3/2} \left (\left (a+b x^2\right )^3 \, _2F_1\left (\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right )-a^3\right )}{9 a^3 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 277, normalized size = 0.8 \begin{align*}{\frac{5\,{d}^{3}b}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{2}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{7\,{d}^{5}}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}a} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{5\,{d}^{7}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}b} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{3}\sqrt{2}}{512\,{b}^{2}{a}^{2}}\ln \left ({ \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{5\,{d}^{3}\sqrt{2}}{256\,{b}^{2}{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{5\,{d}^{3}\sqrt{2}}{256\,{b}^{2}{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41831, size = 898, normalized size = 2.68 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{5}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28677, size = 409, normalized size = 1.22 \begin{align*} \frac{1}{1536} \, d{\left (\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}} + \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}} - \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}} + \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}} + \frac{8 \,{\left (15 \, \sqrt{d x} b^{2} d^{7} x^{5} + 42 \, \sqrt{d x} a b d^{7} x^{3} - 5 \, \sqrt{d x} a^{2} d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{2} b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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