Optimal. Leaf size=376 \[ \frac {(b c-a d)^2 (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}-\frac {d x^{3/2} \left (11 a^2 d^2-21 a b c d+6 b^2 c^2\right )}{6 a b^3}-\frac {d^2 x^{7/2} (7 b c-11 a d)}{14 a b^2}+\frac {x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.43, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 468, 570, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {d x^{3/2} \left (11 a^2 d^2-21 a b c d+6 b^2 c^2\right )}{6 a b^3}+\frac {(b c-a d)^2 (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}-\frac {d^2 x^{7/2} (7 b c-11 a d)}{14 a b^2}+\frac {x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 466
Rule 468
Rule 570
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (c+d x^4\right ) \left (-c (b c+3 a d)+d (7 b c-11 a d) x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^2}{b^2}+\frac {d^2 (7 b c-11 a d) x^6}{b}-\frac {\left (b^3 c^3+9 a b^2 c^2 d-21 a^2 b c d^2+11 a^3 d^3\right ) x^2}{b^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a b^3}\\ &=-\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a b^{7/2}}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a b^{7/2}}\\ &=-\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b^4}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b^4}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}\\ &=-\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}-\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}\\ &=-\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {(b c-a d)^2 (b c+11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (b c+11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}\\ \end {align*}
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Mathematica [C] time = 2.63, size = 355, normalized size = 0.94 \begin {gather*} \frac {95 a \left (a \left (77 a^2 \left (16875 c^3+50625 c^2 d x^2+50625 c d^2 x^4+15467 d^3 x^6\right )+22 a b x^2 \left (25931 c^3+77793 c^2 d x^2+87201 c d^2 x^4+28043 d^3 x^6\right )+b^2 x^4 \left (56099 c^3+79593 c^2 d x^2+79593 c d^2 x^4+26531 d^3 x^6\right )\right )-77 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b x^2}{a}\right ) \left (a^3 \left (16875 c^3+50625 c^2 d x^2+50625 c d^2 x^4+15467 d^3 x^6\right )+a^2 b x^2 \left (14641 c^3+43923 c^2 d x^2+46611 c d^2 x^4+14641 d^3 x^6\right )+a b^2 x^4 \left (2401 c^3+6051 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right )+b^3 x^6 \left (-101 c^3+81 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right )\right )\right )-32768 b^4 x^8 \left (c+d x^2\right )^3 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {23}{4};-\frac {b x^2}{a}\right )}{5617920 a^3 b^3 x^{9/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.52, size = 246, normalized size = 0.65 \begin {gather*} -\frac {(11 a d+b c) (a d-b c)^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(11 a d+b c) (a d-b c)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}-\frac {x^{3/2} \left (77 a^3 d^3-147 a^2 b c d^2+44 a^2 b d^3 x^2+63 a b^2 c^2 d-84 a b^2 c d^2 x^2-12 a b^2 d^3 x^4-21 b^3 c^3\right )}{42 a b^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.39, size = 2531, normalized size = 6.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 516, normalized size = 1.37 \begin {gather*} \frac {b^{3} c^{3} x^{\frac {3}{2}} - 3 \, a b^{2} c^{2} d x^{\frac {3}{2}} + 3 \, a^{2} b c d^{2} x^{\frac {3}{2}} - a^{3} d^{3} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{6}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{6}} + \frac {2 \, {\left (3 \, b^{12} d^{3} x^{\frac {7}{2}} + 21 \, b^{12} c d^{2} x^{\frac {3}{2}} - 14 \, a b^{11} d^{3} x^{\frac {3}{2}}\right )}}{21 \, b^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 706, normalized size = 1.88 \begin {gather*} \frac {2 d^{3} x^{\frac {7}{2}}}{7 b^{2}}-\frac {a^{2} d^{3} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {3 a c \,d^{2} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {c^{3} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) a}-\frac {3 c^{2} d \,x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) b}-\frac {4 a \,d^{3} x^{\frac {3}{2}}}{3 b^{3}}+\frac {2 c \,d^{2} x^{\frac {3}{2}}}{b^{2}}+\frac {11 \sqrt {2}\, a^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}+\frac {11 \sqrt {2}\, a^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}+\frac {11 \sqrt {2}\, a^{2} d^{3} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}-\frac {21 \sqrt {2}\, a c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}-\frac {21 \sqrt {2}\, a c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}-\frac {21 \sqrt {2}\, a c \,d^{2} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {\sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {\sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {\sqrt {2}\, c^{3} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {9 \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {9 \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {9 \sqrt {2}\, c^{2} d \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.51, size = 309, normalized size = 0.82 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{\frac {3}{2}}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, {\left (3 \, b d^{3} x^{\frac {7}{2}} + 7 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x^{\frac {3}{2}}\right )}}{21 \, b^{3}} + \frac {{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 21 \, a^{2} b c d^{2} + 11 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 616, normalized size = 1.64 \begin {gather*} \frac {2\,d^3\,x^{7/2}}{7\,b^2}-x^{3/2}\,\left (\frac {4\,a\,d^3}{3\,b^3}-\frac {2\,c\,d^2}{b^2}\right )-\frac {x^{3/2}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )\,\left (121\,a^6\,d^6-462\,a^5\,b\,c\,d^5+639\,a^4\,b^2\,c^2\,d^4-356\,a^3\,b^3\,c^3\,d^3+39\,a^2\,b^4\,c^4\,d^2+18\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (1331\,a^9\,d^9-7623\,a^8\,b\,c\,d^8+17820\,a^7\,b^2\,c^2\,d^7-21372\,a^6\,b^3\,c^3\,d^6+13194\,a^5\,b^4\,c^4\,d^5-3186\,a^4\,b^5\,c^5\,d^4-372\,a^3\,b^6\,c^6\,d^3+180\,a^2\,b^7\,c^7\,d^2+27\,a\,b^8\,c^8\,d+b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )}{4\,{\left (-a\right )}^{5/4}\,b^{15/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )\,\left (121\,a^6\,d^6-462\,a^5\,b\,c\,d^5+639\,a^4\,b^2\,c^2\,d^4-356\,a^3\,b^3\,c^3\,d^3+39\,a^2\,b^4\,c^4\,d^2+18\,a\,b^5\,c^5\,d+b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (1331\,a^9\,d^9-7623\,a^8\,b\,c\,d^8+17820\,a^7\,b^2\,c^2\,d^7-21372\,a^6\,b^3\,c^3\,d^6+13194\,a^5\,b^4\,c^4\,d^5-3186\,a^4\,b^5\,c^5\,d^4-372\,a^3\,b^6\,c^6\,d^3+180\,a^2\,b^7\,c^7\,d^2+27\,a\,b^8\,c^8\,d+b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{5/4}\,b^{15/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 136.46, size = 173, normalized size = 0.46 \begin {gather*} - \frac {4 a d^{3} x^{\frac {3}{2}}}{3 b^{3}} - \frac {2 x^{\frac {3}{2}} \left (a d - b c\right )^{3}}{4 a^{2} b^{3} + 4 a b^{4} x^{2}} + \frac {2 c d^{2} x^{\frac {3}{2}}}{b^{2}} + \frac {2 d^{3} x^{\frac {7}{2}}}{7 b^{2}} + \frac {6 d \left (a d - b c\right )^{2} \operatorname {RootSum} {\left (256 t^{4} a b^{3} + 1, \left (t \mapsto t \log {\left (64 t^{3} a b^{2} + \sqrt {x} \right )} \right )\right )}}{b^{3}} - \frac {2 \left (a d - b c\right )^{3} \operatorname {RootSum} {\left (65536 t^{4} a^{5} b^{3} + 1, \left (t \mapsto t \log {\left (4096 t^{3} a^{4} b^{2} + \sqrt {x} \right )} \right )\right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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