Optimal. Leaf size=458 \[ -\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.333422, antiderivative size = 458, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1112, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
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 )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^3 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (21 d^3 \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 )}{32 b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^3 \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 )}{32 b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt{2} \sqrt [4]{a} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt{2} \sqrt [4]{a} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^5 \left (a b+b^2 x^2\right )\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 )}{64 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^5 \left (a b+b^2 x^2\right )\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 )}{64 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \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 )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \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 )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt{2} \sqrt [4]{a} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt{2} \sqrt [4]{a} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \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 )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \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 )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \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 )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \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 )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0334198, size = 84, normalized size = 0.18 \[ \frac{2 d^3 (d x)^{3/2} \left (7 \left (a+b x^2\right )^2 \, _2F_1\left (\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right )-a \left (7 a+5 b x^2\right )\right )}{5 a b^2 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.238, size = 612, normalized size = 1.3 \begin{align*} \text{result too large to display} \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 = 1.64341, size = 707, normalized size = 1.54 \begin{align*} -\frac{84 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \sqrt{d x} b^{3} d^{13} - \sqrt{d^{27} x - \sqrt{-\frac{d^{18}}{a b^{11}}} a b^{5} d^{18}} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} b^{3}}{d^{18}}\right ) - 21 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \log \left (9261 \, \sqrt{d x} d^{13} + 9261 \, \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{3}{4}} a b^{8}\right ) + 21 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \log \left (9261 \, \sqrt{d x} d^{13} - 9261 \, \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{3}{4}} a b^{8}\right ) + 4 \,{\left (11 \, b d^{4} x^{3} + 7 \, a d^{4} x\right )} \sqrt{d x}}{64 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} 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 = 1.33204, size = 497, normalized size = 1.09 \begin{align*} -\frac{1}{128} \, d^{3}{\left (\frac{8 \,{\left (11 \, \sqrt{d x} b d^{5} x^{3} + 7 \, \sqrt{d x} a d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{42 \, \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^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{42 \, \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^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{21 \, \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^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{21 \, \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^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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