Optimal. Leaf size=459 \[ \frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294,
296, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \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^{5/4} b^{7/4} \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 294
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {(d x)^{5/2}}{\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 {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d \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 b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3 d \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 b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d \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 b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^{5/2} \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^{5/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^{5/2} \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^{5/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^3 \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 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^3 \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 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 d^{5/2} \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^{5/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3 d^{5/2} \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^{5/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 d (d x)^{3/2}}{16 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 d^{5/2} \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^{5/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 179, normalized size = 0.39 \begin {gather*} -\frac {(d x)^{5/2} \left (4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (a-3 b x^2\right )+3 \sqrt {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 )+3 \sqrt {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 )}{64 a^{5/4} b^{7/4} x^{5/2} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(616\) vs.
\(2(293)=586\).
time = 0.04, size = 617, normalized size = 1.34
method | result | size |
default | \(\frac {\left (3 \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 ) b^{2} d^{4} x^{4}+6 \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 ) b^{2} d^{4} x^{4}+6 \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 ) b^{2} d^{4} x^{4}+24 \left (d x \right )^{\frac {7}{2}} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2}+6 \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 ) a b \,d^{4} x^{2}+12 \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 ) a b \,d^{4} x^{2}+12 \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 ) a b \,d^{4} x^{2}-8 \left (d x \right )^{\frac {3}{2}} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a b \,d^{2}+3 \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 ) a^{2} d^{4}+6 \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 ) a^{2} d^{4}+6 \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 ) a^{2} d^{4}\right ) \left (b \,x^{2}+a \right )}{128 d \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2} a \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(617\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 272, normalized size = 0.59 \begin {gather*} -\frac {d^{\frac {5}{2}} x^{\frac {3}{2}}}{2 \, {\left (a b^{2} x^{2} + a^{2} b + {\left (b^{3} x^{2} + a b^{2}\right )} x^{2}\right )}} + \frac {3 \, d^{\frac {5}{2}} {\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 )}}{128 \, a b} + \frac {3 \, b d^{\frac {5}{2}} x^{\frac {7}{2}} + 7 \, a d^{\frac {5}{2}} x^{\frac {3}{2}}}{16 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 326, normalized size = 0.71 \begin {gather*} -\frac {12 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {27 \, \sqrt {d x} a b^{2} d^{7} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {1}{4}} - \sqrt {-729 \, a^{3} b^{3} d^{10} \sqrt {-\frac {d^{10}}{a^{5} b^{7}}} + 729 \, d^{15} x} a b^{2} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {1}{4}}}{27 \, d^{10}}\right ) - 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (27 \, a^{4} b^{5} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + 27 \, \sqrt {d x} d^{7}\right ) + 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (-27 \, a^{4} b^{5} \left (-\frac {d^{10}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + 27 \, \sqrt {d x} d^{7}\right ) - 4 \, {\left (3 \, b d^{2} x^{3} - a d^{2} x\right )} \sqrt {d x}}{64 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\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 {\left (d x\right )^{\frac {5}{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 = 3.55, size = 348, normalized size = 0.76 \begin {gather*} \frac {1}{128} \, d^{2} {\left (\frac {8 \, {\left (3 \, \sqrt {d x} b d^{4} x^{3} - \sqrt {d x} a d^{4} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a b \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {6 \, \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^{2} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {6 \, \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^{2} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {3 \, \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^{2} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, \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^{2} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )}\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 {{\left (d\,x\right )}^{5/2}}{{\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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