Optimal. Leaf size=363 \[ \frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}-\frac {(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}-\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}} \]
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Rubi [A]
time = 0.26, antiderivative size = 360, normalized size of antiderivative = 0.99, 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 = {473, 468,
331, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {-\frac {9 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{10 c \sqrt {x} \left (c+d x^2\right )}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {(b c-9 a d) (b c-a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}-\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 331
Rule 335
Rule 468
Rule 473
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^2} \, dx &=-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}+\frac {2 \int \frac {\frac {1}{2} a (10 b c-9 a d)+\frac {5}{2} b^2 c x^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx}{5 c}\\ &=-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}-\frac {((b c-9 a d) (b c-a d)) \int \frac {1}{x^{3/2} \left (c+d x^2\right )} \, dx}{4 c^2 d}\\ &=\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}+\frac {((b c-9 a d) (b c-a d)) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{4 c^3}\\ &=\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}+\frac {((b c-9 a d) (b c-a d)) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^3}\\ &=\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}-\frac {((b c-9 a d) (b c-a d)) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 \sqrt {d}}+\frac {((b c-9 a d) (b c-a d)) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 \sqrt {d}}\\ &=\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}+\frac {((b c-9 a d) (b c-a d)) \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 )}{8 c^3 d}+\frac {((b c-9 a d) (b c-a d)) \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 )}{8 c^3 d}+\frac {((b c-9 a d) (b c-a d)) \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 )}{8 \sqrt {2} c^{13/4} d^{3/4}}+\frac {((b c-9 a d) (b c-a d)) \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 )}{8 \sqrt {2} c^{13/4} d^{3/4}}\\ &=\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}+\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}-\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}+\frac {((b c-9 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}-\frac {((b c-9 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}\\ &=\frac {(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac {5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt {x} \left (c+d x^2\right )}-\frac {(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} d^{3/4}}+\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}-\frac {(b c-9 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} d^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 227, normalized size = 0.63 \begin {gather*} \frac {\frac {4 \sqrt [4]{c} \left (5 b^2 c^2 x^4-10 a b c x^2 \left (4 c+5 d x^2\right )+a^2 \left (-4 c^2+36 c d x^2+45 d^2 x^4\right )\right )}{x^{5/2} \left (c+d x^2\right )}-\frac {5 \sqrt {2} \left (b^2 c^2-10 a b c d+9 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{3/4}}-\frac {5 \sqrt {2} \left (b^2 c^2-10 a b c d+9 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{3/4}}}{40 c^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 200, normalized size = 0.55
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {1}{4} a^{2} d^{2}-\frac {1}{2} a b c d +\frac {1}{4} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9}{4} a^{2} d^{2}-\frac {5}{2} a b c d +\frac {1}{4} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{3}}-\frac {2 a^{2}}{5 c^{2} x^{\frac {5}{2}}}+\frac {4 a \left (a d -b c \right )}{c^{3} \sqrt {x}}\) | \(200\) |
default | \(\frac {\frac {2 \left (\frac {1}{4} a^{2} d^{2}-\frac {1}{2} a b c d +\frac {1}{4} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9}{4} a^{2} d^{2}-\frac {5}{2} a b c d +\frac {1}{4} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{3}}-\frac {2 a^{2}}{5 c^{2} x^{\frac {5}{2}}}+\frac {4 a \left (a d -b c \right )}{c^{3} \sqrt {x}}\) | \(200\) |
risch | \(-\frac {2 a \left (-10 a d \,x^{2}+10 c \,x^{2} b +a c \right )}{5 c^{3} x^{\frac {5}{2}}}+\frac {x^{\frac {3}{2}} a^{2} d^{2}}{2 c^{3} \left (d \,x^{2}+c \right )}-\frac {x^{\frac {3}{2}} a d b}{c^{2} \left (d \,x^{2}+c \right )}+\frac {x^{\frac {3}{2}} b^{2}}{2 c \left (d \,x^{2}+c \right )}+\frac {9 \sqrt {2}\, a^{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 ) d}{16 c^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, a \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}{8 c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) d}{8 c^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b}{4 c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) d}{8 c^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b}{4 c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, b^{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 )}{16 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(518\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 275, normalized size = 0.76 \begin {gather*} \frac {5 \, {\left (b^{2} c^{2} - 10 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{4} - 4 \, a^{2} c^{2} - 4 \, {\left (10 \, a b c^{2} - 9 \, a^{2} c d\right )} x^{2}}{10 \, {\left (c^{3} d x^{\frac {9}{2}} + c^{4} x^{\frac {5}{2}}\right )}} + \frac {{\left (b^{2} c^{2} - 10 \, a b c d + 9 \, 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 )}}{16 \, c^{3}} \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. 1737 vs.
\(2 (279) = 558\).
time = 2.32, size = 1737, normalized size = 4.79 \begin {gather*} -\frac {20 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (b^{12} c^{12} - 60 \, a b^{11} c^{11} d + 1554 \, a^{2} b^{10} c^{10} d^{2} - 22700 \, a^{3} b^{9} c^{9} d^{3} + 205215 \, a^{4} b^{8} c^{8} d^{4} - 1188600 \, a^{5} b^{7} c^{7} d^{5} + 4443580 \, a^{6} b^{6} c^{6} d^{6} - 10697400 \, a^{7} b^{5} c^{5} d^{7} + 16622415 \, a^{8} b^{4} c^{4} d^{8} - 16548300 \, a^{9} b^{3} c^{3} d^{9} + 10195794 \, a^{10} b^{2} c^{2} d^{10} - 3542940 \, a^{11} b c d^{11} + 531441 \, a^{12} d^{12}\right )} x - {\left (b^{8} c^{15} d - 40 \, a b^{7} c^{14} d^{2} + 636 \, a^{2} b^{6} c^{13} d^{3} - 5080 \, a^{3} b^{5} c^{12} d^{4} + 21286 \, a^{4} b^{4} c^{11} d^{5} - 45720 \, a^{5} b^{3} c^{10} d^{6} + 51516 \, a^{6} b^{2} c^{9} d^{7} - 29160 \, a^{7} b c^{8} d^{8} + 6561 \, a^{8} c^{7} d^{9}\right )} \sqrt {-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}}} c^{3} d \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {1}{4}} - {\left (b^{6} c^{9} d - 30 \, a b^{5} c^{8} d^{2} + 327 \, a^{2} b^{4} c^{7} d^{3} - 1540 \, a^{3} b^{3} c^{6} d^{4} + 2943 \, a^{4} b^{2} c^{5} d^{5} - 2430 \, a^{5} b c^{4} d^{6} + 729 \, a^{6} c^{3} d^{7}\right )} \sqrt {x} \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {1}{4}}}{b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}\right ) - 5 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {1}{4}} \log \left (c^{10} d^{2} \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {3}{4}} + {\left (b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 327 \, a^{2} b^{4} c^{4} d^{2} - 1540 \, a^{3} b^{3} c^{3} d^{3} + 2943 \, a^{4} b^{2} c^{2} d^{4} - 2430 \, a^{5} b c d^{5} + 729 \, a^{6} d^{6}\right )} \sqrt {x}\right ) + 5 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {1}{4}} \log \left (-c^{10} d^{2} \left (-\frac {b^{8} c^{8} - 40 \, a b^{7} c^{7} d + 636 \, a^{2} b^{6} c^{6} d^{2} - 5080 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 45720 \, a^{5} b^{3} c^{3} d^{5} + 51516 \, a^{6} b^{2} c^{2} d^{6} - 29160 \, a^{7} b c d^{7} + 6561 \, a^{8} d^{8}}{c^{13} d^{3}}\right )^{\frac {3}{4}} + {\left (b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 327 \, a^{2} b^{4} c^{4} d^{2} - 1540 \, a^{3} b^{3} c^{3} d^{3} + 2943 \, a^{4} b^{2} c^{2} d^{4} - 2430 \, a^{5} b c d^{5} + 729 \, a^{6} d^{6}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, {\left (b^{2} c^{2} - 10 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{4} - 4 \, a^{2} c^{2} - 4 \, {\left (10 \, a b c^{2} - 9 \, a^{2} c d\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 0.61, size = 401, normalized size = 1.10 \begin {gather*} \frac {b^{2} c^{2} x^{\frac {3}{2}} - 2 \, a b c d x^{\frac {3}{2}} + a^{2} d^{2} x^{\frac {3}{2}}}{2 \, {\left (d x^{2} + c\right )} c^{3}} - \frac {2 \, {\left (10 \, a b c x^{2} - 10 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{3} x^{\frac {5}{2}}} + \frac {\sqrt {2} {\left (\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 + 9 \, \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 )}{8 \, c^{4} d^{3}} + \frac {\sqrt {2} {\left (\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 + 9 \, \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 )}{8 \, c^{4} d^{3}} - \frac {\sqrt {2} {\left (\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 + 9 \, \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 )}{16 \, c^{4} d^{3}} + \frac {\sqrt {2} {\left (\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 + 9 \, \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 )}{16 \, c^{4} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 152, normalized size = 0.42 \begin {gather*} \frac {\frac {x^4\,\left (9\,a^2\,d^2-10\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3}-\frac {2\,a^2}{5\,c}+\frac {2\,a\,x^2\,\left (9\,a\,d-10\,b\,c\right )}{5\,c^2}}{c\,x^{5/2}+d\,x^{9/2}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (9\,a\,d-b\,c\right )}{4\,{\left (-c\right )}^{13/4}\,d^{3/4}}+\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (9\,a\,d-b\,c\right )}{4\,{\left (-c\right )}^{13/4}\,d^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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