Optimal. Leaf size=260 \[ -\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}+\frac {(b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}} \]
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Rubi [A]
time = 0.19, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {473, 470, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {2 a^2}{3 c x^{3/2}}+\frac {(b c-a d)^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}+\frac {2 b^2 \sqrt {x}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 470
Rule 473
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 \int \frac {\frac {3}{2} a (2 b c-a d)+\frac {3}{2} b^2 c x^2}{\sqrt {x} \left (c+d x^2\right )} \, dx}{3 c}\\ &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}-\frac {(b c-a d)^2 \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{c d}\\ &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}-\frac {\left (2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c d}\\ &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^{3/2} d}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^{3/2} d}\\ &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 c^{3/2} d^{3/2}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 c^{3/2} d^{3/2}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}\\ &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}+\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}\\ &=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}+\frac {(b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} d^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 155, normalized size = 0.60 \begin {gather*} \frac {\frac {4 c^{3/4} \sqrt [4]{d} \left (-a^2 d+3 b^2 c x^2\right )}{x^{3/2}}+3 \sqrt {2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-3 \sqrt {2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{6 c^{7/4} d^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 155, normalized size = 0.60
method | result | size |
derivativedivides | \(\frac {2 b^{2} \sqrt {x}}{d}-\frac {2 a^{2}}{3 c \,x^{\frac {3}{2}}}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} d}\) | \(155\) |
default | \(\frac {2 b^{2} \sqrt {x}}{d}-\frac {2 a^{2}}{3 c \,x^{\frac {3}{2}}}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} d}\) | \(155\) |
risch | \(-\frac {2 \left (-3 b^{2} c \,x^{2}+a^{2} d \right )}{3 d \,x^{\frac {3}{2}} c}-\frac {d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 d}-\frac {d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{2 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2}}{2 d}-\frac {d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a^{2}}{4 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a b}{2 c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) b^{2}}{4 d}\) | \(443\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 286, normalized size = 1.10 \begin {gather*} \frac {2 \, b^{2} \sqrt {x}}{d} - \frac {2 \, a^{2}}{3 \, c x^{\frac {3}{2}}} - \frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{4 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1253 vs.
\(2 (187) = 374\).
time = 0.49, size = 1253, normalized size = 4.82 \begin {gather*} -\frac {12 \, c d x^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{4} d^{2} \sqrt {-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x} c^{5} d^{4} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {3}{4}} - {\left (b^{2} c^{7} d^{4} - 2 \, a b c^{6} d^{5} + a^{2} c^{5} d^{6}\right )} \sqrt {x} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {3}{4}}}{b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}\right ) + 3 \, c d x^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (c^{2} d \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 3 \, c d x^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (-c^{2} d \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{7} d^{5}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (3 \, b^{2} c x^{2} - a^{2} d\right )} \sqrt {x}}{6 \, c d x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 9.48, size = 408, normalized size = 1.57 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}}{d} & \text {for}\: c = 0 \\\frac {- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}}{c} & \text {for}\: d = 0 \\- \frac {2 a^{2}}{3 c x^{\frac {3}{2}}} + \frac {a^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 c^{2}} - \frac {a^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 c^{2}} - \frac {a^{2} d \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c^{2}} - \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{c} + \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{c} + \frac {2 a b \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c} + \frac {2 b^{2} \sqrt {x}}{d} + \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d} - \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d} - \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 344, normalized size = 1.32 \begin {gather*} \frac {2 \, b^{2} \sqrt {x}}{d} - \frac {2 \, a^{2}}{3 \, c x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c^{2} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c^{2} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c^{2} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 1201, normalized size = 4.62 \begin {gather*} \frac {2\,b^2\,\sqrt {x}}{d}-\frac {2\,a^2}{3\,c\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{7/4}\,d^{5/4}}+\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{7/4}\,d^{5/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{7/4}\,d^{5/4}}-\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{7/4}\,d^{5/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{7/4}\,d^{5/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{7/4}\,d^{5/4}}+\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{7/4}\,d^{5/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{7/4}\,d^{5/4}}-\frac {\left (\frac {\sqrt {x}\,\left (16\,a^4\,c^3\,d^{10}-64\,a^3\,b\,c^4\,d^9+96\,a^2\,b^2\,c^5\,d^8-64\,a\,b^3\,c^6\,d^7+16\,b^4\,c^7\,d^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c^5\,d^9-32\,a\,b\,c^6\,d^8+16\,b^2\,c^7\,d^7\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{7/4}\,d^{5/4}}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{7/4}\,d^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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