Optimal. Leaf size=310 \[ \frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (7 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (7 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (7 b c+a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} d^{11/4}}+\frac {(b c-a d) (7 b c+a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} d^{11/4}} \]
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Rubi [A]
time = 0.19, antiderivative size = 310, 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 = {474, 470, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {(b c-a d) (a d+7 b c) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (a d+7 b c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} d^{11/4}}+\frac {(b c-a d) (a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} d^{11/4}}+\frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{3/2}}{3 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 470
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {\sqrt {x} \left (\frac {1}{2} \left (-4 a^2 d^2+3 (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^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (7 b c+a d)) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{4 c d^2}\\ &=\frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (7 b c+a d)) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c d^2}\\ &=\frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((b c-a d) (7 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 d^{5/2}}-\frac {((b c-a d) (7 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 d^{5/2}}\\ &=\frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (7 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 d^3}-\frac {((b c-a d) (7 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 d^3}-\frac {((b c-a d) (7 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^{5/4} d^{11/4}}-\frac {((b c-a d) (7 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^{5/4} d^{11/4}}\\ &=\frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(b c-a d) (7 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^{5/4} d^{11/4}}+\frac {(b c-a d) (7 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^{5/4} d^{11/4}}-\frac {((b c-a d) (7 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^{5/4} d^{11/4}}+\frac {((b c-a d) (7 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^{5/4} d^{11/4}}\\ &=\frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (7 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (7 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (7 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^{5/4} d^{11/4}}+\frac {(b c-a d) (7 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^{5/4} d^{11/4}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 204, normalized size = 0.66 \begin {gather*} \frac {\frac {4 \sqrt [4]{c} d^{3/4} x^{3/2} \left (-6 a b c d+3 a^2 d^2+b^2 c \left (7 c+4 d x^2\right )\right )}{c+d x^2}+3 \sqrt {2} \left (7 b^2 c^2-6 a b c d-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 )+3 \sqrt {2} \left (7 b^2 c^2-6 a b c d-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 )}{24 c^{5/4} d^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 187, normalized size = 0.60
method | result | size |
derivativedivides | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{2}}\) | \(187\) |
default | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{2}}\) | \(187\) |
risch | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}}+\frac {x^{\frac {3}{2}} a^{2}}{2 c \left (d \,x^{2}+c \right )}-\frac {x^{\frac {3}{2}} a b}{d \left (d \,x^{2}+c \right )}+\frac {x^{\frac {3}{2}} b^{2} c}{2 d^{2} \left (d \,x^{2}+c \right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{8 d c \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{4 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {7 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2} c}{8 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{8 d c \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{4 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {7 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2} c}{8 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {\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}}{16 d c \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \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}{8 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {7 \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} c}{16 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(499\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 258, normalized size = 0.83 \begin {gather*} \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{2}} - \frac {{\left (7 \, b^{2} c^{2} - 6 \, a b c d - 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 d^{2}} \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. 1723 vs.
\(2 (230) = 460\).
time = 1.23, size = 1723, normalized size = 5.56 \begin {gather*} -\frac {12 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (117649 \, b^{12} c^{12} - 605052 \, a b^{11} c^{11} d + 1195698 \, a^{2} b^{10} c^{10} d^{2} - 1049580 \, a^{3} b^{9} c^{9} d^{3} + 247695 \, a^{4} b^{8} c^{8} d^{4} + 184968 \, a^{5} b^{7} c^{7} d^{5} - 73604 \, a^{6} b^{6} c^{6} d^{6} - 26424 \, a^{7} b^{5} c^{5} d^{7} + 5055 \, a^{8} b^{4} c^{4} d^{8} + 3060 \, a^{9} b^{3} c^{3} d^{9} + 498 \, a^{10} b^{2} c^{2} d^{10} + 36 \, a^{11} b c d^{11} + a^{12} d^{12}\right )} x - {\left (2401 \, b^{8} c^{11} d^{5} - 8232 \, a b^{7} c^{10} d^{6} + 9212 \, a^{2} b^{6} c^{9} d^{7} - 2520 \, a^{3} b^{5} c^{8} d^{8} - 1434 \, a^{4} b^{4} c^{7} d^{9} + 360 \, a^{5} b^{3} c^{6} d^{10} + 188 \, a^{6} b^{2} c^{5} d^{11} + 24 \, a^{7} b c^{4} d^{12} + a^{8} c^{3} d^{13}\right )} \sqrt {-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}}} c d^{3} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {1}{4}} + {\left (343 \, b^{6} c^{7} d^{3} - 882 \, a b^{5} c^{6} d^{4} + 609 \, a^{2} b^{4} c^{5} d^{5} + 36 \, a^{3} b^{3} c^{4} d^{6} - 87 \, a^{4} b^{2} c^{3} d^{7} - 18 \, a^{5} b c^{2} d^{8} - a^{6} c d^{9}\right )} \sqrt {x} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {1}{4}}}{2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}\right ) - 3 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {1}{4}} \log \left (c^{4} d^{8} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {3}{4}} - {\left (343 \, b^{6} c^{6} - 882 \, a b^{5} c^{5} d + 609 \, a^{2} b^{4} c^{4} d^{2} + 36 \, a^{3} b^{3} c^{3} d^{3} - 87 \, a^{4} b^{2} c^{2} d^{4} - 18 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} \sqrt {x}\right ) + 3 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {1}{4}} \log \left (-c^{4} d^{8} \left (-\frac {2401 \, b^{8} c^{8} - 8232 \, a b^{7} c^{7} d + 9212 \, a^{2} b^{6} c^{6} d^{2} - 2520 \, a^{3} b^{5} c^{5} d^{3} - 1434 \, a^{4} b^{4} c^{4} d^{4} + 360 \, a^{5} b^{3} c^{3} d^{5} + 188 \, a^{6} b^{2} c^{2} d^{6} + 24 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{5} d^{11}}\right )^{\frac {3}{4}} - {\left (343 \, b^{6} c^{6} - 882 \, a b^{5} c^{5} d + 609 \, a^{2} b^{4} c^{4} d^{2} + 36 \, a^{3} b^{3} c^{3} d^{3} - 87 \, a^{4} b^{2} c^{2} d^{4} - 18 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} \sqrt {x}\right ) - 4 \, {\left (4 \, b^{2} c d x^{3} + {\left (7 \, b^{2} c^{2} - 6 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {x}}{24 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.35, size = 173, normalized size = 0.56 \begin {gather*} \frac {4 a b \operatorname {RootSum} {\left (256 t^{4} c d^{3} + 1, \left ( t \mapsto t \log {\left (64 t^{3} c d^{2} + \sqrt {x} \right )} \right )\right )}}{d} - \frac {4 b^{2} c \operatorname {RootSum} {\left (256 t^{4} c d^{3} + 1, \left ( t \mapsto t \log {\left (64 t^{3} c d^{2} + \sqrt {x} \right )} \right )\right )}}{d^{2}} + \frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}} + \frac {2 x^{\frac {3}{2}} \left (a d - b c\right )^{2}}{4 c^{2} d^{2} + 4 c d^{3} x^{2}} + \frac {2 \left (a d - b c\right )^{2} \operatorname {RootSum} {\left (65536 t^{4} c^{5} d^{3} + 1, \left ( t \mapsto t \log {\left (4096 t^{3} c^{4} d^{2} + \sqrt {x} \right )} \right )\right )}}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 388, normalized size = 1.25 \begin {gather*} \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{2}} + \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 d^{2}} - \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \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^{2} d^{5}} - \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \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^{2} d^{5}} + \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \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^{2} d^{5}} - \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \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^{2} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 137, normalized size = 0.44 \begin {gather*} \frac {2\,b^2\,x^{3/2}}{3\,d^2}+\frac {x^{3/2}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c\,\left (d^3\,x^2+c\,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 (a\,d+7\,b\,c\right )}{4\,{\left (-c\right )}^{5/4}\,d^{11/4}}-\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 (a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{5/4}\,d^{11/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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