Optimal. Leaf size=553 \[ \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.27, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 296,
331, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{32 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 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}{32 a^3 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 a^4 d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{16 a^4 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 b^{3/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{32 a^4 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 b^{3/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{32 a^4 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \left (a b+b^2 x^2\right )\right ) \text {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 )}{64 a^4 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \left (a b+b^2 x^2\right )\right ) \text {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 )}{64 a^4 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 204, normalized size = 0.37 \begin {gather*} \frac {x \left (4 \sqrt [4]{a} \left (-32 a^3+416 a^2 b x^2+1053 a b^2 x^4+585 b^3 x^6\right )-585 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-585 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{320 a^{17/4} (d x)^{7/2} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 687, normalized size = 1.24
method | result | size |
risch | \(-\frac {2 \left (-15 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 a^{4} \sqrt {d x}\, x^{2} d^{3} \left (b \,x^{2}+a \right )}+\frac {\left (\frac {21 b^{3} \left (d x \right )^{\frac {7}{2}}}{16 a^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{2}}+\frac {25 b^{2} \left (d x \right )^{\frac {3}{2}} d^{2}}{16 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{2}}+\frac {117 b \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{128 a^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {117 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{64 a^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {117 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{64 a^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{d^{3} \left (b \,x^{2}+a \right )}\) | \(304\) |
default | \(-\frac {\left (-585 \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) \left (d x \right )^{\frac {5}{2}} b^{3} x^{4}-1170 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (d x \right )^{\frac {5}{2}} b^{3} x^{4}-1170 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (d x \right )^{\frac {5}{2}} b^{3} x^{4}-4680 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{3} d^{2} x^{6}-1170 \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) \left (d x \right )^{\frac {5}{2}} a \,b^{2} x^{2}-2340 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (d x \right )^{\frac {5}{2}} a \,b^{2} x^{2}-2340 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (d x \right )^{\frac {5}{2}} a \,b^{2} x^{2}-8424 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a \,b^{2} d^{2} x^{4}-585 \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) \left (d x \right )^{\frac {5}{2}} a^{2} b -1170 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (d x \right )^{\frac {5}{2}} a^{2} b -1170 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (d x \right )^{\frac {5}{2}} a^{2} b -3328 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b \,d^{2} x^{2}+256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{3} d^{2}\right ) \left (b \,x^{2}+a \right )}{640 d^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {5}{2}} a^{4} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(687\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 390, normalized size = 0.71 \begin {gather*} -\frac {2340 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {1601613 \, \sqrt {d x} a^{4} b^{4} d^{3} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-2565164201769 \, a^{9} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{17} d^{14}}} + 2565164201769 \, b^{8} d x} a^{4} d^{3} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}}}{1601613 \, b^{5}}\right ) - 585 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \log \left (1601613 \, a^{13} d^{11} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} b^{4}\right ) + 585 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \log \left (-1601613 \, a^{13} d^{11} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} b^{4}\right ) - 4 \, {\left (585 \, b^{3} x^{6} + 1053 \, a b^{2} x^{4} + 416 \, a^{2} b x^{2} - 32 \, a^{3}\right )} \sqrt {d x}}{320 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.08, size = 390, normalized size = 0.71 \begin {gather*} \frac {21 \, \sqrt {d x} b^{3} d^{3} x^{3} + 25 \, \sqrt {d x} a b^{2} d^{3} x}{16 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{4} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {117 \, \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 )}{64 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {117 \, \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 )}{64 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {117 \, \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 )}{128 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {117 \, \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 )}{128 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {2 \, {\left (15 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{4} d^{5} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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