Optimal. Leaf size=350 \[ -\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{195 \sqrt [4]{a} d^{15/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{17/4}}+\frac{195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{17/4}}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac{195 d^7 \sqrt{d x}}{64 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.381998, 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, 211, 1165, 628, 1162, 617, 204} \[ -\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{195 \sqrt [4]{a} d^{15/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{17/4}}+\frac{195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{17/4}}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac{195 d^7 \sqrt{d x}}{64 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
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)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac{1}{12} \left (13 b^2 d^2\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{1}{32} \left (39 d^4\right ) \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac{\left (195 d^6\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=\frac{195 d^7 \sqrt{d x}}{64 b^4}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac{\left (195 a d^8\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{128 b^3}\\ &=\frac{195 d^7 \sqrt{d x}}{64 b^4}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac{\left (195 a d^7\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 b^3}\\ &=\frac{195 d^7 \sqrt{d x}}{64 b^4}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac{\left (195 \sqrt{a} d^6\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^3}-\frac{\left (195 \sqrt{a} d^6\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^3}\\ &=\frac{195 d^7 \sqrt{d x}}{64 b^4}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac{\left (195 \sqrt [4]{a} d^{15/2}\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^{17/4}}+\frac{\left (195 \sqrt [4]{a} d^{15/2}\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^{17/4}}-\frac{\left (195 \sqrt{a} d^8\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^{9/2}}-\frac{\left (195 \sqrt{a} d^8\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^{9/2}}\\ &=\frac{195 d^7 \sqrt{d x}}{64 b^4}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac{195 \sqrt [4]{a} d^{15/2} \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^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \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^{17/4}}-\frac{\left (195 \sqrt [4]{a} d^{15/2}\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^{17/4}}+\frac{\left (195 \sqrt [4]{a} d^{15/2}\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^{17/4}}\\ &=\frac{195 d^7 \sqrt{d x}}{64 b^4}-\frac{d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac{195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{17/4}}+\frac{195 \sqrt [4]{a} d^{15/2} \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^{17/4}}-\frac{195 \sqrt [4]{a} d^{15/2} \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^{17/4}}\\ \end{align*}
Mathematica [A] time = 0.144782, size = 324, normalized size = 0.93 \[ \frac{d^7 \sqrt{d x} \left (\frac{119808 a^2 b^{5/4} x^2}{\left (a+b x^2\right )^3}-\frac{6240 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^2}+\frac{49920 a^3 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac{21504 b^{13/4} x^6}{\left (a+b x^2\right )^3}+\frac{93184 a b^{9/4} x^4}{\left (a+b x^2\right )^3}-\frac{10920 a \sqrt [4]{b}}{a+b x^2}+\frac{4095 \sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt{x}}-\frac{4095 \sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt{x}}+\frac{8190 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{x}}-\frac{8190 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{x}}\right )}{10752 b^{17/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.067, size = 287, normalized size = 0.8 \begin{align*} 2\,{\frac{{d}^{7}\sqrt{dx}}{{b}^{4}}}+{\frac{317\,{d}^{9}a}{192\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{81\,{d}^{11}{a}^{2}}{32\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{67\,{d}^{13}{a}^{3}}{64\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}}\sqrt{dx}}-{\frac{195\,{d}^{7}\sqrt{2}}{512\,{b}^{4}}\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{195\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\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{195\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.40936, size = 826, normalized size = 2.36 \begin{align*} -\frac{2340 \, \left (-\frac{a d^{30}}{b^{17}}\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 d^{30}}{b^{17}}\right )^{\frac{3}{4}} \sqrt{d x} b^{13} d^{7} - \sqrt{d^{15} x + \sqrt{-\frac{a d^{30}}{b^{17}}} b^{8}} \left (-\frac{a d^{30}}{b^{17}}\right )^{\frac{3}{4}} b^{13}}{a d^{30}}\right ) + 585 \, \left (-\frac{a d^{30}}{b^{17}}\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 (195 \, \sqrt{d x} d^{7} + 195 \, \left (-\frac{a d^{30}}{b^{17}}\right )^{\frac{1}{4}} b^{4}\right ) - 585 \, \left (-\frac{a d^{30}}{b^{17}}\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 (195 \, \sqrt{d x} d^{7} - 195 \, \left (-\frac{a d^{30}}{b^{17}}\right )^{\frac{1}{4}} b^{4}\right ) - 4 \,{\left (384 \, b^{3} d^{7} x^{6} + 1469 \, a b^{2} d^{7} x^{4} + 1638 \, a^{2} b d^{7} x^{2} + 585 \, a^{3} d^{7}\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31077, size = 414, normalized size = 1.18 \begin{align*} -\frac{1}{1536} \, d^{6}{\left (\frac{1170 \, \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^{5}} + \frac{1170 \, \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^{5}} + \frac{585 \, \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^{5}} - \frac{585 \, \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^{5}} - \frac{3072 \, \sqrt{d x} d}{b^{4}} - \frac{8 \,{\left (317 \, \sqrt{d x} a b^{2} d^{7} x^{4} + 486 \, \sqrt{d x} a^{2} b d^{7} x^{2} + 201 \, \sqrt{d x} a^{3} d^{7}\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.
[In]
[Out]