Optimal. Leaf size=556 \[ \frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.29, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 296,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \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 )}{2048 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \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 )}{2048 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \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 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (39 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 b \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 \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 )}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (195 \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 )}{2048 a^4 \sqrt {b} d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 \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 )}{2048 a^4 \sqrt {b} d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 d \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 )}{4096 a^4 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 d \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 )}{4096 a^4 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (195 \sqrt {d} \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 )}{2048 \sqrt {2} a^{17/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (195 \sqrt {d} \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 )}{2048 \sqrt {2} a^{17/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {195 (d x)^{3/2}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \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 )}{2048 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \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 )}{2048 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {195 \sqrt {d} \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 )}{4096 \sqrt {2} a^{17/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 201, normalized size = 0.36 \begin {gather*} \frac {\sqrt {d x} \left (a+b x^2\right ) \left (4 \sqrt [4]{a} x^{3/2} \left (1853 a^3+3107 a^2 b x^2+2223 a b^2 x^4+585 b^3 x^6\right )-\frac {585 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}-\frac {585 \sqrt {2} \left (a+b x^2\right )^4 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}\right )}{12288 a^{17/4} \sqrt {x} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs.
\(2(360)=720\).
time = 0.05, size = 1051, normalized size = 1.89
method | result | size |
default | \(\text {Expression too large to display}\) | \(1051\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 569, normalized size = 1.02 \begin {gather*} \frac {195 \, b^{3} \sqrt {d} x^{\frac {15}{2}} + 117 \, a b^{2} \sqrt {d} x^{\frac {11}{2}} + 65 \, a^{2} b \sqrt {d} x^{\frac {7}{2}} + 15 \, a^{3} \sqrt {d} x^{\frac {3}{2}}}{1024 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} + \frac {{\left (117 \, b^{4} \sqrt {d} x^{5} + 130 \, a b^{3} \sqrt {d} x^{3} + 45 \, a^{2} b^{2} \sqrt {d} x\right )} x^{\frac {9}{2}} + 2 \, {\left (143 \, a b^{3} \sqrt {d} x^{5} + 174 \, a^{2} b^{2} \sqrt {d} x^{3} + 63 \, a^{3} b \sqrt {d} x\right )} x^{\frac {5}{2}} + {\left (201 \, a^{2} b^{2} \sqrt {d} x^{5} + 282 \, a^{3} b \sqrt {d} x^{3} + 113 \, a^{4} \sqrt {d} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{2} + a^{9} + {\left (a^{3} b^{6} x^{6} + 3 \, a^{4} b^{5} x^{4} + 3 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )} x^{6} + 3 \, {\left (a^{4} b^{5} x^{6} + 3 \, a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )} x^{4} + 3 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b\right )} x^{2}\right )}} + \frac {195 \, \sqrt {d} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 414, normalized size = 0.74 \begin {gather*} -\frac {2340 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {7414875 \, \sqrt {d x} a^{4} b d \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {1}{4}} - \sqrt {-54980371265625 \, a^{9} b d^{2} \sqrt {-\frac {d^{2}}{a^{17} b^{3}}} + 54980371265625 \, d^{3} x} a^{4} b \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {1}{4}}}{7414875 \, d^{2}}\right ) - 585 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {1}{4}} \log \left (7414875 \, a^{13} b^{2} \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {3}{4}} + 7414875 \, \sqrt {d x} d\right ) + 585 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {1}{4}} \log \left (-7414875 \, a^{13} b^{2} \left (-\frac {d^{2}}{a^{17} b^{3}}\right )^{\frac {3}{4}} + 7414875 \, \sqrt {d x} d\right ) - 4 \, {\left (585 \, b^{3} x^{7} + 2223 \, a b^{2} x^{5} + 3107 \, a^{2} b x^{3} + 1853 \, a^{3} x\right )} \sqrt {d x}}{12288 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\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 {\sqrt {d x}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.88, size = 371, normalized size = 0.67 \begin {gather*} \frac {\frac {1170 \, \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 )}{a^{5} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1170 \, \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 )}{a^{5} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {585 \, \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 )}{a^{5} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {585 \, \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 )}{a^{5} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (585 \, \sqrt {d x} b^{3} d^{9} x^{7} + 2223 \, \sqrt {d x} a b^{2} d^{9} x^{5} + 3107 \, \sqrt {d x} a^{2} b d^{9} x^{3} + 1853 \, \sqrt {d x} a^{3} d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )}}{24576 \, d} \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 {\sqrt {d\,x}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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