Optimal. Leaf size=504 \[ \frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.368843, antiderivative size = 504, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
Rule 321
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 )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{13/2}}{\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)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (11 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (77 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{5/2}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a d^6 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{32 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a d^5 \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^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (77 a d^5 \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^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a d^5 \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^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a^{3/4} d^{13/2} \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} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a^{3/4} d^{13/2} \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} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a d^7 \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^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a d^7 \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^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \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} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \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} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (77 a^{3/4} d^{13/2} \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} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (77 a^{3/4} d^{13/2} \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} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \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} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \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} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \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} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \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} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0352888, size = 88, normalized size = 0.17 \[ -\frac{2 d^5 (d x)^{3/2} \left (-77 a^2+77 \left (a+b x^2\right )^2 \, _2F_1\left (\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right )-55 a b x^2-5 b^2 x^4\right )}{15 b^3 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.236, size = 679, normalized size = 1.4 \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 = 1.66543, size = 783, normalized size = 1.55 \begin{align*} \frac{924 \, \left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \arctan \left (-\frac{\left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} b^{4} d^{19} - \sqrt{a^{4} d^{39} x - \sqrt{-\frac{a^{3} d^{26}}{b^{15}}} a^{3} b^{7} d^{26}} \left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{1}{4}} b^{4}}{a^{3} d^{26}}\right ) - 231 \, \left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (456533 \, \sqrt{d x} a^{2} d^{19} + 456533 \, \left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{3}{4}} b^{11}\right ) + 231 \, \left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (456533 \, \sqrt{d x} a^{2} d^{19} - 456533 \, \left (-\frac{a^{3} d^{26}}{b^{15}}\right )^{\frac{3}{4}} b^{11}\right ) + 4 \,{\left (32 \, b^{2} d^{6} x^{5} + 121 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt{d x}}{192 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} 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 = 1.31281, size = 524, normalized size = 1.04 \begin{align*} \frac{1}{384} \, d^{5}{\left (\frac{256 \, \sqrt{d x} d x}{b^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{24 \,{\left (19 \, \sqrt{d x} a b d^{5} x^{3} + 15 \, \sqrt{d x} a^{2} d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{3} \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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