Optimal. Leaf size=333 \[ -\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.345098, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {28, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{(d x)^{13/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}+\frac{1}{12} \left (11 b^2 d^2\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{1}{96} \left (77 d^4\right ) \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (77 d^6\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (77 d^5\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 b^2}\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{\left (77 d^5\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 b^{5/2}}+\frac{\left (77 d^5\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 b^{5/2}}\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (77 d^{13/2}\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} \sqrt [4]{a} b^{15/4}}+\frac{\left (77 d^{13/2}\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} \sqrt [4]{a} b^{15/4}}+\frac{\left (77 d^7\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 b^4}+\frac{\left (77 d^7\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 b^4}\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}+\frac{77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}+\frac{\left (77 d^{13/2}\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} \sqrt [4]{a} b^{15/4}}-\frac{\left (77 d^{13/2}\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} \sqrt [4]{a} b^{15/4}}\\ &=-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \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} \sqrt [4]{a} b^{15/4}}\\ \end{align*}
Mathematica [C] time = 0.0256973, size = 83, normalized size = 0.25 \[ \frac{2 d^6 x \sqrt{d x} \left (77 \left (a+b x^2\right )^3 \, _2F_1\left (\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right )-a \left (77 a^2+99 a b x^2+45 b^2 x^4\right )\right )}{45 a b^3 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 271, normalized size = 0.8 \begin{align*} -{\frac{51\,{d}^{7}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}b} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{33\,{d}^{9}a}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{77\,{d}^{11}{a}^{2}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{512\,{b}^{4}}\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{77\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\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{77\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\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 = 1.41557, size = 845, normalized size = 2.54 \begin{align*} -\frac{924 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \sqrt{d x} b^{4} d^{19} - \sqrt{d^{39} x - \sqrt{-\frac{d^{26}}{a b^{15}}} a b^{7} d^{26}} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} b^{4}}{d^{26}}\right ) - 231 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} + 456533 \, \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}\right ) + 231 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} - 456533 \, \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}\right ) + 4 \,{\left (153 \, b^{2} d^{6} x^{5} + 198 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt{d x}}{768 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24082, size = 408, normalized size = 1.23 \begin{align*} \frac{1}{1536} \, d^{5}{\left (\frac{462 \, \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 b^{6}} + \frac{462 \, \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 b^{6}} - \frac{231 \, \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 b^{6}} + \frac{231 \, \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 b^{6}} - \frac{8 \,{\left (153 \, \sqrt{d x} b^{2} d^{7} x^{5} + 198 \, \sqrt{d x} a b d^{7} x^{3} + 77 \, \sqrt{d x} a^{2} d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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