Optimal. Leaf size=350 \[ -\frac{385 a^{3/4} d^{17/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{19/4}}-\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{19/4}}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}+\frac{385 d^7 (d x)^{3/2}}{192 b^4} \]
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Rubi [A] time = 0.392945, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{385 a^{3/4} d^{17/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{19/4}}-\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{19/4}}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}+\frac{385 d^7 (d x)^{3/2}}{192 b^4} \]
Antiderivative was successfully verified.
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Rule 28
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)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{(d x)^{17/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}+\frac{1}{4} \left (5 b^2 d^2\right ) \int \frac{(d x)^{13/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}+\frac{1}{32} \left (55 d^4\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}+\frac{\left (385 d^6\right ) \int \frac{(d x)^{5/2}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=\frac{385 d^7 (d x)^{3/2}}{192 b^4}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}-\frac{\left (385 a d^8\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{128 b^3}\\ &=\frac{385 d^7 (d x)^{3/2}}{192 b^4}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}-\frac{\left (385 a d^7\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 b^3}\\ &=\frac{385 d^7 (d x)^{3/2}}{192 b^4}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}+\frac{\left (385 a d^7\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 )}{128 b^{7/2}}-\frac{\left (385 a d^7\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 )}{128 b^{7/2}}\\ &=\frac{385 d^7 (d x)^{3/2}}{192 b^4}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}-\frac{\left (385 a^{3/4} d^{17/2}\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} b^{19/4}}-\frac{\left (385 a^{3/4} d^{17/2}\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} b^{19/4}}-\frac{\left (385 a d^9\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 b^5}-\frac{\left (385 a d^9\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 b^5}\\ &=\frac{385 d^7 (d x)^{3/2}}{192 b^4}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}-\frac{385 a^{3/4} d^{17/2} \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} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \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} b^{19/4}}-\frac{\left (385 a^{3/4} d^{17/2}\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} b^{19/4}}+\frac{\left (385 a^{3/4} d^{17/2}\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} b^{19/4}}\\ &=\frac{385 d^7 (d x)^{3/2}}{192 b^4}-\frac{d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}+\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{19/4}}-\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{19/4}}-\frac{385 a^{3/4} d^{17/2} \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} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \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} b^{19/4}}\\ \end{align*}
Mathematica [C] time = 0.0289215, size = 87, normalized size = 0.25 \[ -\frac{2 d^8 x \sqrt{d x} \left (-99 a^2 b x^2-77 a^3-45 a b^2 x^4+77 \left (a+b x^2\right )^3 \, _2F_1\left (\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right )-3 b^3 x^6\right )}{9 b^4 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 290, normalized size = 0.8 \begin{align*}{\frac{2\,{d}^{7}}{3\,{b}^{4}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{127\,{d}^{9}a}{64\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{101\,{d}^{11}{a}^{2}}{32\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{257\,{d}^{13}{a}^{3}}{192\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{385\,{d}^{9}a\sqrt{2}}{512\,{b}^{5}}\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{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{385\,{d}^{9}a\sqrt{2}}{256\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{385\,{d}^{9}a\sqrt{2}}{256\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\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(-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 = 1.45145, size = 917, normalized size = 2.62 \begin{align*} \frac{4620 \, \left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{1}{4}}{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \arctan \left (-\frac{\left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} b^{5} d^{25} - \sqrt{a^{4} d^{51} x - \sqrt{-\frac{a^{3} d^{34}}{b^{19}}} a^{3} b^{9} d^{34}} \left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{1}{4}} b^{5}}{a^{3} d^{34}}\right ) - 1155 \, \left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{1}{4}}{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (57066625 \, \sqrt{d x} a^{2} d^{25} + 57066625 \, \left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{3}{4}} b^{14}\right ) + 1155 \, \left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{1}{4}}{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (57066625 \, \sqrt{d x} a^{2} d^{25} - 57066625 \, \left (-\frac{a^{3} d^{34}}{b^{19}}\right )^{\frac{3}{4}} b^{14}\right ) + 4 \,{\left (128 \, b^{3} d^{8} x^{7} + 765 \, a b^{2} d^{8} x^{5} + 990 \, a^{2} b d^{8} x^{3} + 385 \, a^{3} d^{8} x\right )} \sqrt{d x}}{768 \,{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\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 = 1.31688, size = 412, normalized size = 1.18 \begin{align*} \frac{1}{1536} \, d^{7}{\left (\frac{1024 \, \sqrt{d x} d x}{b^{4}} - \frac{2310 \, \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^{7}} - \frac{2310 \, \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^{7}} + \frac{1155 \, \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^{7}} - \frac{1155 \, \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^{7}} + \frac{8 \,{\left (381 \, \sqrt{d x} a b^{2} d^{7} x^{5} + 606 \, \sqrt{d x} a^{2} b d^{7} x^{3} + 257 \, \sqrt{d x} a^{3} d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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