Optimal. Leaf size=399 \[ -\frac {x^{3/2} \left (9 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {x^{3/2} \left (5 a d (2 b c-9 a d)+3 b^2 c^2\right )}{16 c^3 d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.39, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {462, 457, 290, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {x^{3/2} \left (9 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {x^{3/2} \left (\frac {5 a (2 b c-9 a d)}{c^2}+\frac {3 b^2}{d}\right )}{16 c \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 329
Rule 457
Rule 462
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {2 \int \frac {\sqrt {x} \left (\frac {1}{2} a (2 b c-9 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^3} \, dx}{c}\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {1}{8} \left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) \int \frac {\sqrt {x}}{\left (c+d x^2\right )^2} \, dx\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c}\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c}\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^3 d^{3/2}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^3 d^{3/2}}\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {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 )}{64 c^3 d^2}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {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 )}{64 c^3 d^2}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {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 )}{64 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {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 )}{64 \sqrt {2} c^{13/4} d^{7/4}}\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}\\ &=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 364, normalized size = 0.91 \begin {gather*} \frac {\frac {8 \sqrt [4]{c} x^{3/2} \left (-13 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{d \left (c+d x^2\right )}+\frac {\sqrt {2} \left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{d^{7/4}}+\frac {\sqrt {2} \left (45 a^2 d^2-10 a b c d-3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{d^{7/4}}+\frac {2 \sqrt {2} \left (45 a^2 d^2-10 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{d^{7/4}}+\frac {2 \sqrt {2} \left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{d^{7/4}}-\frac {256 a^2 \sqrt [4]{c}}{\sqrt {x}}-\frac {32 c^{5/4} x^{3/2} (b c-a d)^2}{d \left (c+d x^2\right )^2}}{128 c^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.00, size = 260, normalized size = 0.65 \begin {gather*} -\frac {\left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {-32 a^2 c^2 d-81 a^2 c d^2 x^2-45 a^2 d^3 x^4+18 a b c^2 d x^2+10 a b c d^2 x^4-b^2 c^3 x^2+3 b^2 c^2 d x^4}{16 c^3 d \sqrt {x} \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.84, size = 1819, normalized size = 4.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 427, normalized size = 1.07 \begin {gather*} -\frac {2 \, a^{2}}{c^{3} \sqrt {x}} + \frac {3 \, b^{2} c^{2} d x^{\frac {7}{2}} + 10 \, a b c d^{2} x^{\frac {7}{2}} - 13 \, a^{2} d^{3} x^{\frac {7}{2}} - b^{2} c^{3} x^{\frac {3}{2}} + 18 \, a b c^{2} d x^{\frac {3}{2}} - 17 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{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 )}{64 \, c^{4} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{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 )}{64 \, c^{4} d^{4}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{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 )}{128 \, c^{4} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{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 )}{128 \, c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 568, normalized size = 1.42 \begin {gather*} -\frac {13 a^{2} d^{2} x^{\frac {7}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c^{3}}+\frac {5 a b d \,x^{\frac {7}{2}}}{8 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {3 b^{2} x^{\frac {7}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c}-\frac {17 a^{2} d \,x^{\frac {3}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {9 a b \,x^{\frac {3}{2}}}{8 \left (d \,x^{2}+c \right )^{2} c}-\frac {b^{2} x^{\frac {3}{2}}}{16 \left (d \,x^{2}+c \right )^{2} d}-\frac {45 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{3}}-\frac {45 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{3}}-\frac {45 \sqrt {2}\, a^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{128 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{3}}+\frac {5 \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{32 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2} d}+\frac {5 \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{32 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2} d}+\frac {5 \sqrt {2}\, a b \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2} d}+\frac {3 \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c \,d^{2}}+\frac {3 \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c \,d^{2}}+\frac {3 \sqrt {2}\, b^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{128 \left (\frac {c}{d}\right )^{\frac {1}{4}} c \,d^{2}}-\frac {2 a^{2}}{c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.52, size = 307, normalized size = 0.77 \begin {gather*} -\frac {32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 10 \, a b c d^{2} - 45 \, a^{2} d^{3}\right )} x^{4} + {\left (b^{2} c^{3} - 18 \, a b c^{2} d + 81 \, a^{2} c d^{2}\right )} x^{2}}{16 \, {\left (c^{3} d^{3} x^{\frac {9}{2}} + 2 \, c^{4} d^{2} x^{\frac {5}{2}} + c^{5} d \sqrt {x}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 10 \, a b c d - 45 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 192, normalized size = 0.48 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{13/4}\,d^{7/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{13/4}\,d^{7/4}}-\frac {\frac {2\,a^2}{c}-\frac {x^4\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{16\,c^3}+\frac {x^2\,\left (81\,a^2\,d^2-18\,a\,b\,c\,d+b^2\,c^2\right )}{16\,c^2\,d}}{c^2\,\sqrt {x}+d^2\,x^{9/2}+2\,c\,d\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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