Optimal. Leaf size=346 \[ -\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}} \]
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Rubi [A]
time = 0.21, antiderivative size = 346, 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 = {474, 470,
327, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}-\frac {x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{7/2}}{7 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 327
Rule 335
Rule 470
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{5/2} \left (\frac {1}{2} \left (-4 a^2 d^2+7 (b c-a d)^2\right )-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((11 b c-3 a d) (b c-a d)) \int \frac {x^{5/2}}{c+d x^2} \, dx}{4 c d^2}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((11 b c-3 a d) (b c-a d)) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{4 d^3}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((11 b c-3 a d) (b c-a d)) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^3}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((11 b c-3 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 d^{7/2}}+\frac {((11 b c-3 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 d^{7/2}}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((11 b c-3 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 d^4}+\frac {((11 b c-3 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 d^4}+\frac {((11 b c-3 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} \sqrt [4]{c} d^{15/4}}+\frac {((11 b c-3 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} \sqrt [4]{c} d^{15/4}}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}+\frac {((11 b c-3 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} \sqrt [4]{c} d^{15/4}}-\frac {((11 b c-3 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} \sqrt [4]{c} d^{15/4}}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 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} \sqrt [4]{c} d^{15/4}}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 223, normalized size = 0.64 \begin {gather*} \frac {\frac {4 d^{3/4} x^{3/2} \left (-21 a^2 d^2+14 a b d \left (7 c+4 d x^2\right )+b^2 \left (-77 c^2-44 c d x^2+12 d^2 x^4\right )\right )}{c+d x^2}-\frac {21 \sqrt {2} \left (11 b^2 c^2-14 a b c d+3 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 )}{\sqrt [4]{c}}-\frac {21 \sqrt {2} \left (11 b^2 c^2-14 a b c d+3 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 )}{\sqrt [4]{c}}}{168 d^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 200, normalized size = 0.58
method | result | size |
derivativedivides | \(\frac {2 b \left (\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -2 b c \right ) x^{\frac {3}{2}}}{3}\right )}{d^{3}}+\frac {\frac {2 \left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (-\frac {7}{2} a b c d +\frac {11}{4} b^{2} c^{2}+\frac {3}{4} a^{2} d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\) | \(200\) |
default | \(\frac {2 b \left (\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -2 b c \right ) x^{\frac {3}{2}}}{3}\right )}{d^{3}}+\frac {\frac {2 \left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (-\frac {7}{2} a b c d +\frac {11}{4} b^{2} c^{2}+\frac {3}{4} a^{2} d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\) | \(200\) |
risch | \(\frac {2 b \,x^{\frac {3}{2}} \left (3 b d \,x^{2}+14 a d -14 b c \right )}{21 d^{3}}-\frac {x^{\frac {3}{2}} a^{2}}{2 d \left (d \,x^{2}+c \right )}+\frac {x^{\frac {3}{2}} a b c}{d^{2} \left (d \,x^{2}+c \right )}-\frac {x^{\frac {3}{2}} b^{2} c^{2}}{2 d^{3} \left (d \,x^{2}+c \right )}-\frac {7 \sqrt {2}\, b c \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}{8 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {11 \sqrt {2}\, b^{2} c^{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 d^{4} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {7 \sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a}{4 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {11 \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 d^{4} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {7 \sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a}{4 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {11 \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 d^{4} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \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 )}{16 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(515\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 272, normalized size = 0.79 \begin {gather*} -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}}}{2 \, {\left (d^{4} x^{2} + c d^{3}\right )}} + \frac {{\left (11 \, b^{2} c^{2} - 14 \, a b c d + 3 \, 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 \, d^{3}} + \frac {2 \, {\left (3 \, b^{2} d x^{\frac {7}{2}} - 14 \, {\left (b^{2} c - a b d\right )} x^{\frac {3}{2}}\right )}}{21 \, d^{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. 1733 vs.
\(2 (262) = 524\).
time = 1.38, size = 1733, normalized size = 5.01 \begin {gather*} -\frac {84 \, {\left (d^{4} x^{2} + c d^{3}\right )} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (1771561 \, b^{12} c^{12} - 13528284 \, a b^{11} c^{11} d + 45943458 \, a^{2} b^{10} c^{10} d^{2} - 91492940 \, a^{3} b^{9} c^{9} d^{3} + 118659255 \, a^{4} b^{8} c^{8} d^{4} - 105323064 \, a^{5} b^{7} c^{7} d^{5} + 65490076 \, a^{6} b^{6} c^{6} d^{6} - 28724472 \, a^{7} b^{5} c^{5} d^{7} + 8825895 \, a^{8} b^{4} c^{4} d^{8} - 1855980 \, a^{9} b^{3} c^{3} d^{9} + 254178 \, a^{10} b^{2} c^{2} d^{10} - 20412 \, a^{11} b c d^{11} + 729 \, a^{12} d^{12}\right )} x - {\left (14641 \, b^{8} c^{9} d^{7} - 74536 \, a b^{7} c^{8} d^{8} + 158268 \, a^{2} b^{6} c^{7} d^{9} - 181720 \, a^{3} b^{5} c^{6} d^{10} + 122566 \, a^{4} b^{4} c^{5} d^{11} - 49560 \, a^{5} b^{3} c^{4} d^{12} + 11772 \, a^{6} b^{2} c^{3} d^{13} - 1512 \, a^{7} b c^{2} d^{14} + 81 \, a^{8} c d^{15}\right )} \sqrt {-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}}} d^{4} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {1}{4}} - {\left (1331 \, b^{6} c^{6} d^{4} - 5082 \, a b^{5} c^{5} d^{5} + 7557 \, a^{2} b^{4} c^{4} d^{6} - 5516 \, a^{3} b^{3} c^{3} d^{7} + 2061 \, a^{4} b^{2} c^{2} d^{8} - 378 \, a^{5} b c d^{9} + 27 \, a^{6} d^{10}\right )} \sqrt {x} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {1}{4}}}{14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}\right ) - 21 \, {\left (d^{4} x^{2} + c d^{3}\right )} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {1}{4}} \log \left (c d^{11} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {3}{4}} + {\left (1331 \, b^{6} c^{6} - 5082 \, a b^{5} c^{5} d + 7557 \, a^{2} b^{4} c^{4} d^{2} - 5516 \, a^{3} b^{3} c^{3} d^{3} + 2061 \, a^{4} b^{2} c^{2} d^{4} - 378 \, a^{5} b c d^{5} + 27 \, a^{6} d^{6}\right )} \sqrt {x}\right ) + 21 \, {\left (d^{4} x^{2} + c d^{3}\right )} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {1}{4}} \log \left (-c d^{11} \left (-\frac {14641 \, b^{8} c^{8} - 74536 \, a b^{7} c^{7} d + 158268 \, a^{2} b^{6} c^{6} d^{2} - 181720 \, a^{3} b^{5} c^{5} d^{3} + 122566 \, a^{4} b^{4} c^{4} d^{4} - 49560 \, a^{5} b^{3} c^{3} d^{5} + 11772 \, a^{6} b^{2} c^{2} d^{6} - 1512 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c d^{15}}\right )^{\frac {3}{4}} + {\left (1331 \, b^{6} c^{6} - 5082 \, a b^{5} c^{5} d + 7557 \, a^{2} b^{4} c^{4} d^{2} - 5516 \, a^{3} b^{3} c^{3} d^{3} + 2061 \, a^{4} b^{2} c^{2} d^{4} - 378 \, a^{5} b c d^{5} + 27 \, a^{6} d^{6}\right )} \sqrt {x}\right ) - 4 \, {\left (12 \, b^{2} d^{2} x^{5} - 4 \, {\left (11 \, b^{2} c d - 14 \, a b d^{2}\right )} x^{3} - 7 \, {\left (11 \, b^{2} c^{2} - 14 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {x}}{168 \, {\left (d^{4} x^{2} + c d^{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 = 413, normalized size = 1.19 \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 )} d^{3}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \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 d^{6}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \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 d^{6}} - \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \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 d^{6}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \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 d^{6}} + \frac {2 \, {\left (3 \, b^{2} d^{12} x^{\frac {7}{2}} - 14 \, b^{2} c d^{11} x^{\frac {3}{2}} + 14 \, a b d^{12} x^{\frac {3}{2}}\right )}}{21 \, d^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 160, normalized size = 0.46 \begin {gather*} \frac {2\,b^2\,x^{7/2}}{7\,d^2}-\frac {x^{3/2}\,\left (\frac {a^2\,d^2}{2}-a\,b\,c\,d+\frac {b^2\,c^2}{2}\right )}{d^4\,x^2+c\,d^3}-x^{3/2}\,\left (\frac {4\,b^2\,c}{3\,d^3}-\frac {4\,a\,b}{3\,d^2}\right )+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-11\,b\,c\right )}{4\,{\left (-c\right )}^{1/4}\,d^{15/4}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-11\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{1/4}\,d^{15/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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