Optimal. Leaf size=335 \[ -\frac{7 d^{3/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}-\frac{7 d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.346679, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{7 d^{3/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}-\frac{7 d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{(d x)^{3/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{1}{12} \left (b^2 d^2\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{\left (7 b d^2\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^2} \, dx}{96 a}\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}+\frac{\left (7 d^2\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{128 a^2}\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}+\frac{(7 d) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 a^2}\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}+\frac{7 \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 )}{128 a^{5/2}}+\frac{7 \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 )}{128 a^{5/2}}\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}-\frac{\left (7 d^{3/2}\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 )}{256 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\left (7 d^{3/2}\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 )}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (7 d^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 )}{256 a^{5/2} b^{3/2}}+\frac{\left (7 d^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 )}{256 a^{5/2} b^{3/2}}\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}-\frac{7 d^{3/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (7 d^{3/2}\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 )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\left (7 d^{3/2}\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 )}{128 \sqrt{2} a^{11/4} b^{5/4}}\\ &=-\frac{d \sqrt{d x}}{6 b \left (a+b x^2\right )^3}+\frac{d \sqrt{d x}}{48 a b \left (a+b x^2\right )^2}+\frac{7 d \sqrt{d x}}{192 a^2 b \left (a+b x^2\right )}-\frac{7 d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{7 d^{3/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{11/4} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.132366, size = 260, normalized size = 0.78 \[ \frac{d \sqrt{d x} \left (\frac{56 \sqrt [4]{b}}{a^2 \left (a+b x^2\right )}-\frac{21 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} \sqrt{x}}+\frac{21 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} \sqrt{x}}-\frac{42 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{11/4} \sqrt{x}}+\frac{42 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} \sqrt{x}}+\frac{32 \sqrt [4]{b}}{a \left (a+b x^2\right )^2}-\frac{256 \sqrt [4]{b}}{\left (a+b x^2\right )^3}\right )}{1536 b^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 271, normalized size = 0.8 \begin{align*}{\frac{7\,{d}^{3}b}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{2}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{3\,{d}^{5}}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}a} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{d}^{7}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}b}\sqrt{dx}}+{\frac{7\,d\sqrt{2}}{512\,{a}^{3}b}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\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 ) }+{\frac{7\,d\sqrt{2}}{256\,{a}^{3}b}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{7\,d\sqrt{2}}{256\,{a}^{3}b}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \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 = 1.36485, size = 821, normalized size = 2.45 \begin{align*} \frac{84 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{8} b^{4} d \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{3}{4}} - \sqrt{a^{6} b^{2} \sqrt{-\frac{d^{6}}{a^{11} b^{5}}} + d^{3} x} a^{8} b^{4} \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{3}{4}}}{d^{6}}\right ) + 21 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} b \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} d\right ) - 21 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} b \left (-\frac{d^{6}}{a^{11} b^{5}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} d\right ) + 4 \,{\left (7 \, b^{2} d x^{4} + 18 \, a b d x^{2} - 21 \, a^{2} d\right )} \sqrt{d x}}{768 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25594, size = 409, normalized size = 1.22 \begin{align*} \frac{7 \, \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 )}{256 \, a^{3} b^{2}} + \frac{7 \, \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 )}{256 \, a^{3} b^{2}} + \frac{7 \, \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 )}{512 \, a^{3} b^{2}} - \frac{7 \, \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 )}{512 \, a^{3} b^{2}} + \frac{7 \, \sqrt{d x} b^{2} d^{7} x^{4} + 18 \, \sqrt{d x} a b d^{7} x^{2} - 21 \, \sqrt{d x} a^{2} d^{7}}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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