Optimal. Leaf size=387 \[ -\frac {4389 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}-\frac {4389 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 204
Rule 211
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {\left (19 b^5\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\left (57 b^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{64 a^2}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {\left (209 b^3\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {\left (1463 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^5 d}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \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 )}{8192 a^{11/2} d^2}+\frac {(4389 b) \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 )}{8192 a^{11/2} d^2}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {4389 \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 )}{16384 a^{11/2} \sqrt {b}}+\frac {4389 \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 )}{16384 a^{11/2} \sqrt {b}}-\frac {4389 \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 )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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 )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \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 )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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 )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 295, normalized size = 0.76 \[ \frac {\sqrt {x} \left (\frac {16384 a^{19/4} \sqrt {x}}{\left (a+b x^2\right )^5}+\frac {19456 a^{15/4} \sqrt {x}}{\left (a+b x^2\right )^4}+\frac {24320 a^{11/4} \sqrt {x}}{\left (a+b x^2\right )^3}+\frac {33440 a^{7/4} \sqrt {x}}{\left (a+b x^2\right )^2}+\frac {58520 a^{3/4} \sqrt {x}}{a+b x^2}-\frac {21945 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}+\frac {21945 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}-\frac {43890 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac {43890 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{163840 a^{23/4} \sqrt {d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 475, normalized size = 1.23 \[ \frac {87780 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{12} d^{2} \sqrt {-\frac {1}{a^{23} b d^{2}}} + d x} a^{17} b d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{17} b d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {3}{4}}\right ) + 21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (7315 \, b^{4} x^{8} + 33440 \, a b^{3} x^{6} + 59470 \, a^{2} b^{2} x^{4} + 50312 \, a^{3} b x^{2} + 19015 \, a^{4}\right )} \sqrt {d x}}{81920 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 346, normalized size = 0.89 \[ \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{6} b d} + \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{6} b d} + \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{6} b d} - \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{6} b d} + \frac {7315 \, \sqrt {d x} b^{4} d^{9} x^{8} + 33440 \, \sqrt {d x} a b^{3} d^{9} x^{6} + 59470 \, \sqrt {d x} a^{2} b^{2} d^{9} x^{4} + 50312 \, \sqrt {d x} a^{3} b d^{9} x^{2} + 19015 \, \sqrt {d x} a^{4} d^{9}}{20480 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 333, normalized size = 0.86 \[ \frac {3803 \sqrt {d x}\, d^{9}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {6289 \left (d x \right )^{\frac {5}{2}} b \,d^{7}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {5947 \left (d x \right )^{\frac {9}{2}} b^{2} d^{5}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {209 \left (d x \right )^{\frac {13}{2}} b^{3} d^{3}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}+\frac {1463 \left (d x \right )^{\frac {17}{2}} b^{4} d}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{6} d}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{6} d}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 a^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.11, size = 382, normalized size = 0.99 \[ \frac {\frac {8 \, {\left (7315 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{2} + 33440 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{4} + 59470 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{6} + 50312 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{8} + 19015 \, \sqrt {d x} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {21945 \, {\left (\frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{5}}}{163840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.29, size = 210, normalized size = 0.54 \[ \frac {\frac {3803\,d^9\,\sqrt {d\,x}}{4096\,a}+\frac {5947\,b^2\,d^5\,{\left (d\,x\right )}^{9/2}}{2048\,a^3}+\frac {209\,b^3\,d^3\,{\left (d\,x\right )}^{13/2}}{128\,a^4}+\frac {6289\,b\,d^7\,{\left (d\,x\right )}^{5/2}}{2560\,a^2}+\frac {1463\,b^4\,d\,{\left (d\,x\right )}^{17/2}}{4096\,a^5}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {4389\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}}+\frac {4389\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} \left (a + b x^{2}\right )^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________