Optimal. Leaf size=402 \[ -\frac {\sqrt {x} \left (11 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^2 d \left (c+d x^2\right )^2}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}-\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\sqrt {x} \left (7 a d (6 b c-11 a d)+3 b^2 c^2\right )}{48 c^3 d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.41, antiderivative size = 401, 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, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {\sqrt {x} \left (11 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^2 d \left (c+d x^2\right )^2}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (7 a d (6 b c-11 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^{15/4} d^{5/4}}+\frac {\left (7 a d (6 b c-11 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^{15/4} d^{5/4}}-\frac {\left (7 a d (6 b c-11 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^{15/4} d^{5/4}}+\frac {\left (7 a d (6 b c-11 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^{15/4} d^{5/4}}+\frac {\sqrt {x} \left (\frac {7 a (6 b c-11 a d)}{c^2}+\frac {3 b^2}{d}\right )}{48 c \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 290
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^{5/2} \left (c+d x^2\right )^3} \, dx &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {2 \int \frac {\frac {1}{2} a (6 b c-11 a d)+\frac {3}{2} b^2 c x^2}{\sqrt {x} \left (c+d x^2\right )^3} \, dx}{3 c}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {1}{24} \left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )^2} \, dx\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{32 c}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2}}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2}}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+42 a b c d-77 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^{7/2} d^{3/2}}+\frac {\left (3 b^2 c^2+42 a b c d-77 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^{7/2} d^{3/2}}-\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}\\ &=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {11 a^2 d}{c}\right ) \sqrt {x}}{12 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {7 a (6 b c-11 a d)}{c^2}\right ) \sqrt {x}}{48 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 365, normalized size = 0.91 \begin {gather*} \frac {\frac {3 \sqrt {2} \left (77 a^2 d^2-42 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^{5/4}}+\frac {3 \sqrt {2} \left (-77 a^2 d^2+42 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^{5/4}}+\frac {6 \sqrt {2} \left (77 a^2 d^2-42 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^{5/4}}+\frac {6 \sqrt {2} \left (-77 a^2 d^2+42 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^{5/4}}+\frac {24 c^{3/4} \sqrt {x} \left (-15 a^2 d^2+14 a b c d+b^2 c^2\right )}{d \left (c+d x^2\right )}-\frac {256 a^2 c^{3/4}}{x^{3/2}}-\frac {96 c^{7/4} \sqrt {x} (b c-a d)^2}{d \left (c+d x^2\right )^2}}{384 c^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.97, size = 260, normalized size = 0.65 \begin {gather*} -\frac {\left (-77 a^2 d^2+42 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^{15/4} d^{5/4}}+\frac {\left (-77 a^2 d^2+42 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^{15/4} d^{5/4}}+\frac {-32 a^2 c^2 d-121 a^2 c d^2 x^2-77 a^2 d^3 x^4+66 a b c^2 d x^2+42 a b c d^2 x^4-9 b^2 c^3 x^2+3 b^2 c^2 d x^4}{48 c^3 d x^{3/2} \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.58, size = 1433, normalized size = 3.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 426, normalized size = 1.06 \begin {gather*} -\frac {2 \, a^{2}}{3 \, c^{3} x^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \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 )}{64 \, c^{4} d^{2}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \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 )}{64 \, c^{4} d^{2}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \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 )}{128 \, c^{4} d^{2}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \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 )}{128 \, c^{4} d^{2}} + \frac {b^{2} c^{2} d x^{\frac {5}{2}} + 14 \, a b c d^{2} x^{\frac {5}{2}} - 15 \, a^{2} d^{3} x^{\frac {5}{2}} - 3 \, b^{2} c^{3} \sqrt {x} + 22 \, a b c^{2} d \sqrt {x} - 19 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 562, normalized size = 1.40 \begin {gather*} -\frac {15 a^{2} d^{2} x^{\frac {5}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c^{3}}+\frac {7 a b d \,x^{\frac {5}{2}}}{8 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {b^{2} x^{\frac {5}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c}-\frac {19 a^{2} d \sqrt {x}}{16 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {11 a b \sqrt {x}}{8 \left (d \,x^{2}+c \right )^{2} c}-\frac {3 b^{2} \sqrt {x}}{16 \left (d \,x^{2}+c \right )^{2} d}-\frac {77 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 c^{4}}-\frac {77 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 c^{4}}-\frac {77 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d \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 c^{4}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{32 c^{3}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{32 c^{3}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \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 c^{3}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 c^{2} d}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 c^{2} d}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \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 c^{2} d}-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.44, size = 377, normalized size = 0.94 \begin {gather*} -\frac {32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 42 \, a b c d^{2} - 77 \, a^{2} d^{3}\right )} x^{4} + {\left (9 \, b^{2} c^{3} - 66 \, a b c^{2} d + 121 \, a^{2} c d^{2}\right )} x^{2}}{48 \, {\left (c^{3} d^{3} x^{\frac {11}{2}} + 2 \, c^{4} d^{2} x^{\frac {7}{2}} + c^{5} d x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, 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 (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, 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 (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, 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 (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, 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}}}}{128 \, c^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 1508, normalized size = 3.75
result too large to display
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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