Optimal. Leaf size=459 \[ \frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.326733, antiderivative size = 459, 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, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(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{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{a d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{a^2 d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (2 b^2 \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 )}{a^2 d^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b^{3/2} \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 )}{a^2 d^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (b^{3/2} \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 )}{a^2 d^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\sqrt [4]{b} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\sqrt [4]{b} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a b+b^2 x^2\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 a^2 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a b+b^2 x^2\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 a^2 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\sqrt [4]{b} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\sqrt [4]{b} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \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} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0139664, size = 52, normalized size = 0.11 \[ -\frac{2 x \left (a+b x^2\right ) \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a (d x)^{7/2} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.236, size = 251, normalized size = 0.6 \begin{align*}{\frac{b{x}^{2}+a}{20\,{d}^{3}{a}^{2}} \left ( 5\,b\sqrt{2} \left ( dx \right ) ^{5/2}\ln \left ( -{ \left ( \sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}-dx-\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\,b\sqrt{2} \left ( dx \right ) ^{5/2}\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\,b\sqrt{2} \left ( dx \right ) ^{5/2}\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 ) +40\,b{x}^{2}{d}^{2}\sqrt [4]{{\frac{a{d}^{2}}{b}}}-8\,a{d}^{2}\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (b x^{2} + a\right )}^{2}} \left (d x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69109, size = 574, normalized size = 1.25 \begin{align*} -\frac{20 \, a^{2} d^{4} x^{3} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{2} b^{4} d^{3} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{1}{4}} - \sqrt{-a^{5} b^{5} d^{8} \sqrt{-\frac{b^{5}}{a^{9} d^{14}}} + b^{8} d x} a^{2} d^{3} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{1}{4}}}{b^{5}}\right ) - 5 \, a^{2} d^{4} x^{3} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{1}{4}} \log \left (a^{7} d^{11} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{3}{4}} + \sqrt{d x} b^{4}\right ) + 5 \, a^{2} d^{4} x^{3} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{1}{4}} \log \left (-a^{7} d^{11} \left (-\frac{b^{5}}{a^{9} d^{14}}\right )^{\frac{3}{4}} + \sqrt{d x} b^{4}\right ) - 4 \,{\left (5 \, b x^{2} - a\right )} \sqrt{d x}}{10 \, a^{2} d^{4} x^{3}} \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 = 1.2879, size = 383, normalized size = 0.83 \begin{align*} \frac{1}{20} \,{\left (\frac{10 \, \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^{3} b d^{5}} + \frac{10 \, \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^{3} b d^{5}} - \frac{5 \, \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^{3} b d^{5}} + \frac{5 \, \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^{3} b d^{5}} + \frac{8 \,{\left (5 \, b d^{2} x^{2} - a d^{2}\right )}}{\sqrt{d x} a^{2} d^{5} x^{2}}\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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