Optimal. Leaf size=370 \[ \frac{663 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.446997, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ \frac{663 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{\left (17 b^3\right ) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{\left (221 b^2\right ) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{96 a^2}\\ &=\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{(663 b) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac{\left (663 b^2\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^4 d^2}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{\left (663 b^3\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{128 a^5 d^4}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{\left (663 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 a^5 d^5}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac{\left (663 b^{5/2}\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 )}{128 a^5 d^5}+\frac{\left (663 b^{5/2}\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 )}{128 a^5 d^5}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{\left (663 b^{5/4}\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^{21/4} d^{7/2}}+\frac{\left (663 b^{5/4}\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^{21/4} d^{7/2}}+\frac{(663 b) \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^5 d^3}+\frac{(663 b) \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^5 d^3}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{663 b^{5/4} \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^{21/4} d^{7/2}}-\frac{663 b^{5/4} \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^{21/4} d^{7/2}}+\frac{\left (663 b^{5/4}\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^{21/4} d^{7/2}}-\frac{\left (663 b^{5/4}\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^{21/4} d^{7/2}}\\ &=-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac{663 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \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^{21/4} d^{7/2}}-\frac{663 b^{5/4} \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^{21/4} d^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0121173, size = 37, normalized size = 0.1 \[ -\frac{2 \sqrt{d x} \, _2F_1\left (-\frac{5}{4},4;-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a^4 d^4 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 304, normalized size = 0.8 \begin{align*} -{\frac{2}{5\,{a}^{4}d} \left ( dx \right ) ^{-{\frac{5}{2}}}}+8\,{\frac{b}{{a}^{5}{d}^{3}\sqrt{dx}}}+{\frac{151\,{b}^{4}}{64\,{a}^{5}{d}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{173\,{b}^{3}}{32\,{a}^{4}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{617\,{b}^{2}d}{192\,{a}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{663\,b\sqrt{2}}{512\,{a}^{5}{d}^{3}}\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{663\,b\sqrt{2}}{256\,{a}^{5}{d}^{3}}\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{663\,b\sqrt{2}}{256\,{a}^{5}{d}^{3}}\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 = 1.54075, size = 1135, normalized size = 3.07 \begin{align*} -\frac{39780 \,{\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} \arctan \left (-\frac{291434247 \, \sqrt{d x} a^{5} b^{4} d^{3} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} - \sqrt{-84933920324457009 \, a^{11} b^{5} d^{8} \sqrt{-\frac{b^{5}}{a^{21} d^{14}}} + 84933920324457009 \, b^{8} d x} a^{5} d^{3} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}}}{291434247 \, b^{5}}\right ) - 9945 \,{\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} \log \left (291434247 \, a^{16} d^{11} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} b^{4}\right ) + 9945 \,{\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} \log \left (-291434247 \, a^{16} d^{11} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} b^{4}\right ) - 4 \,{\left (9945 \, b^{4} x^{8} + 27846 \, a b^{3} x^{6} + 24973 \, a^{2} b^{2} x^{4} + 6528 \, a^{3} b x^{2} - 384 \, a^{4}\right )} \sqrt{d x}}{3840 \,{\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\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{1}{\left (d x\right )^{\frac{7}{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 = 1.30241, size = 471, normalized size = 1.27 \begin{align*} \frac{663 \, \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 )}{256 \, a^{6} b d^{5}} + \frac{663 \, \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 )}{256 \, a^{6} b d^{5}} - \frac{663 \, \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 )}{512 \, a^{6} b d^{5}} + \frac{663 \, \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 )}{512 \, a^{6} b d^{5}} + \frac{453 \, \sqrt{d x} b^{4} d^{5} x^{5} + 1038 \, \sqrt{d x} a b^{3} d^{5} x^{3} + 617 \, \sqrt{d x} a^{2} b^{2} d^{5} x}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{5} d^{3}} + \frac{2 \,{\left (20 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt{d x} a^{5} d^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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