Optimal. Leaf size=504 \[ \frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.371432, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{11/2}}{\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)^{11/2}}{\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)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (9 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 a d^6 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{32 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 a d^5 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{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{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \sqrt{a} d^4 \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^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \sqrt{a} d^4 \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^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \sqrt{a} d^6 \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^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \sqrt{a} d^6 \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^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.168992, size = 484, normalized size = 0.96 \[ \frac{15 a^2 (d x)^{11/2} \left (a+b x^2\right )}{4 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{15 a (d x)^{11/2} \left (a+b x^2\right )^2}{16 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{6 a (d x)^{11/2} \left (a+b x^2\right )}{b^2 x^3 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{2 (d x)^{11/2} \left (a+b x^2\right )}{b x \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.235, size = 696, 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 = 1.66702, size = 693, normalized size = 1.38 \begin{align*} -\frac{180 \, \left (-\frac{a d^{22}}{b^{13}}\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 d^{22}}{b^{13}}\right )^{\frac{3}{4}} \sqrt{d x} b^{10} d^{5} - \sqrt{d^{11} x + \sqrt{-\frac{a d^{22}}{b^{13}}} b^{6}} \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{3}{4}} b^{10}}{a d^{22}}\right ) + 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt{d x} d^{5} + 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}\right ) - 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt{d x} d^{5} - 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}\right ) - 4 \,{\left (32 \, b^{2} d^{5} x^{4} + 81 \, a b d^{5} x^{2} + 45 \, a^{2} d^{5}\right )} \sqrt{d x}}{64 \,{\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 = 1.39015, size = 527, normalized size = 1.05 \begin{align*} -\frac{1}{128} \, d^{4}{\left (\frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{256 \, \sqrt{d x} d}{b^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (17 \, \sqrt{d x} a b d^{5} x^{2} + 13 \, \sqrt{d x} a^{2} d^{5}\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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