Optimal. Leaf size=457 \[ -\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.324991, antiderivative size = 457, 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, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{7/2}}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{(d x)^{7/2}}{a b+b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^2 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (2 a^2 d^3 \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 )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^{3/2} d^2 \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 )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^{3/2} d^2 \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 )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a^{5/4} d^{7/2} \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 )}{2 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a^{5/4} d^{7/2} \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 )}{2 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^{3/2} d^4 \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 )}{2 b^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^{3/2} d^4 \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 )}{2 b^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \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 )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \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 )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^{5/4} d^{7/2} \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 )}{\sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a^{5/4} d^{7/2} \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 )}{\sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 a d^3 \sqrt{d x} \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \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 )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \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 )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \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 )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \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 )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.091201, size = 238, normalized size = 0.52 \[ \frac{d^3 \sqrt{d x} \left (a+b x^2\right ) \left (-5 \sqrt{2} a^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} a^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} a^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} a^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-40 a \sqrt [4]{b} \sqrt{x}+8 b^{5/4} x^{5/2}\right )}{20 b^{9/4} \sqrt{x} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.227, size = 239, normalized size = 0.5 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) d}{20\,{b}^{2}} \left ( 5\,a{d}^{2}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2}\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 ) +10\,a{d}^{2}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2}\arctan \left ({ \left ( \sqrt{2}\sqrt{dx}+\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \right ) +10\,a{d}^{2}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2}\arctan \left ({ \left ( \sqrt{2}\sqrt{dx}-\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \right ) +8\, \left ( dx \right ) ^{5/2}b-40\,a{d}^{2}\sqrt{dx} \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{7}{2}}}{\sqrt{{\left (b x^{2} + a\right )}^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59594, size = 498, normalized size = 1.09 \begin{align*} \frac{20 \, \left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \arctan \left (-\frac{\left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{3}{4}} \sqrt{d x} a b^{7} d^{3} - \left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{3}{4}} \sqrt{a^{2} d^{7} x + \sqrt{-\frac{a^{5} d^{14}}{b^{9}}} b^{4}} b^{7}}{a^{5} d^{14}}\right ) + 5 \, \left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \log \left (\sqrt{d x} a d^{3} + \left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2}\right ) - 5 \, \left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \log \left (\sqrt{d x} a d^{3} - \left (-\frac{a^{5} d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2}\right ) + 4 \,{\left (b d^{3} x^{2} - 5 \, a d^{3}\right )} \sqrt{d x}}{10 \, b^{2}} \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 = 1.25957, size = 374, normalized size = 0.82 \begin{align*} \frac{1}{20} \, d^{2}{\left (\frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a 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^{3}} + \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a 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^{3}} + \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a 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^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a 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^{3}} + \frac{8 \,{\left (\sqrt{d x} b^{4} d^{6} x^{2} - 5 \, \sqrt{d x} a b^{3} d^{6}\right )}}{b^{5} d^{5}}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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