Optimal. Leaf size=394 \[ \frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^{9/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.461055, antiderivative size = 394, 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, 288, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^{9/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5} \]
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)^{9/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (7 b^4 d^2\right ) \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{320} \left (21 b^2 d^4\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{\left (63 b d^4\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{\left (63 d^4\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^2}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{\left (63 d^4\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{8192 a^3 b}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{\left (63 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 a^3 b}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}-\frac{\left (63 d^3\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 )}{8192 a^3 b^{3/2}}+\frac{\left (63 d^3\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 )}{8192 a^3 b^{3/2}}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{\left (63 d^{9/2}\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 )}{16384 \sqrt{2} a^{13/4} b^{11/4}}+\frac{\left (63 d^{9/2}\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 )}{16384 \sqrt{2} a^{13/4} b^{11/4}}+\frac{\left (63 d^5\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 )}{16384 a^3 b^3}+\frac{\left (63 d^5\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 )}{16384 a^3 b^3}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{63 d^{9/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}+\frac{\left (63 d^{9/2}\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 )}{8192 \sqrt{2} a^{13/4} b^{11/4}}-\frac{\left (63 d^{9/2}\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 )}{8192 \sqrt{2} a^{13/4} b^{11/4}}\\ &=-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}-\frac{63 d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.0270838, size = 61, normalized size = 0.15 \[ \frac{2 d^4 x \sqrt{d x} \left (\frac{7 \, _2F_1\left (\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right )}{a^4}+\frac{-7 a-17 b x^2}{\left (a+b x^2\right )^5}\right )}{221 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 339, normalized size = 0.9 \begin{align*} -{\frac{21\,{d}^{13}a}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{11}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{287\,{d}^{9}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{189\,{d}^{7}b}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{63\,{d}^{5}{b}^{2}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{63\,{d}^{5}\sqrt{2}}{32768\,{a}^{3}{b}^{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{63\,{d}^{5}\sqrt{2}}{16384\,{a}^{3}{b}^{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{63\,{d}^{5}\sqrt{2}}{16384\,{a}^{3}{b}^{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 = 1.4876, size = 1242, normalized size = 3.15 \begin{align*} -\frac{1260 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{250047 \, \sqrt{d x} a^{3} b^{3} d^{13} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} - \sqrt{-62523502209 \, a^{7} b^{5} d^{18} \sqrt{-\frac{d^{18}}{a^{13} b^{11}}} + 62523502209 \, d^{27} x} a^{3} b^{3} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}}}{250047 \, d^{18}}\right ) - 315 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} \log \left (250047 \, a^{10} b^{8} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{3}{4}} + 250047 \, \sqrt{d x} d^{13}\right ) + 315 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} \log \left (-250047 \, a^{10} b^{8} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{3}{4}} + 250047 \, \sqrt{d x} d^{13}\right ) - 4 \,{\left (315 \, b^{4} d^{4} x^{9} + 1512 \, a b^{3} d^{4} x^{7} + 2870 \, a^{2} b^{2} d^{4} x^{5} - 480 \, a^{3} b d^{4} x^{3} - 105 \, a^{4} d^{4} x\right )} \sqrt{d x}}{81920 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\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 = 1.26712, size = 463, normalized size = 1.18 \begin{align*} \frac{1}{163840} \, d^{3}{\left (\frac{630 \, \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^{4} b^{5}} + \frac{630 \, \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^{4} b^{5}} - \frac{315 \, \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^{4} b^{5}} + \frac{315 \, \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^{4} b^{5}} + \frac{8 \,{\left (315 \, \sqrt{d x} b^{4} d^{11} x^{9} + 1512 \, \sqrt{d x} a b^{3} d^{11} x^{7} + 2870 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{5} - 480 \, \sqrt{d x} a^{3} b d^{11} x^{3} - 105 \, \sqrt{d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{3} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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