Optimal. Leaf size=333 \[ -\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.35, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {823, 829, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}-\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 823
Rule 827
Rule 829
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx &=\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}-\frac {\int \frac {-\frac {9}{2} a A c-\frac {7}{2} a B c x}{x^{3/2} \left (a+c x^2\right )^2} \, dx}{4 a^2 c}\\ &=\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}+\frac {\int \frac {\frac {45}{4} a^2 A c^2+\frac {21}{4} a^2 B c^2 x}{x^{3/2} \left (a+c x^2\right )} \, dx}{8 a^4 c^2}\\ &=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}+\frac {\int \frac {\frac {21}{4} a^3 B c^2-\frac {45}{4} a^2 A c^3 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^5 c^2}\\ &=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {21}{4} a^3 B c^2-\frac {45}{4} a^2 A c^3 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^5 c^2}\\ &=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}-\frac {\left (3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^3}+\frac {\left (3 \left (15 A+\frac {7 \sqrt {a} B}{\sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^3}\\ &=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}-\frac {\left (3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a^3}-\frac {\left (3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a^3}-\frac {\left (3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {\left (3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}\\ &=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}-\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {\left (3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \sqrt [4]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}+\frac {\left (3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \sqrt [4]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}\\ &=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}+\frac {3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {3 \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 300, normalized size = 0.90 \begin {gather*} \frac {\sqrt [4]{a} \left (\frac {32 a^{7/4} A}{\sqrt {x} \left (a+c x^2\right )^2}+\frac {72 a^{3/4} A}{\sqrt {x} \left (a+c x^2\right )}+\frac {32 a^{7/4} B \sqrt {x}}{\left (a+c x^2\right )^2}+\frac {56 a^{3/4} B \sqrt {x}}{a+c x^2}-\frac {21 \sqrt {2} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{\sqrt [4]{c}}+\frac {21 \sqrt {2} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{\sqrt [4]{c}}-\frac {42 \sqrt {2} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {42 \sqrt {2} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}\right )-\frac {360 A \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {c x^2}{a}\right )}{\sqrt {x}}}{128 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.92, size = 206, normalized size = 0.62 \begin {gather*} -\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {-32 a^2 A+11 a^2 B x-81 a A c x^2+7 a B c x^3-45 A c^2 x^4}{16 a^3 \sqrt {x} \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 958, normalized size = 2.88 \begin {gather*} -\frac {3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} + 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 343 \, B^{3} a^{5} - 1575 \, A^{2} B a^{4} c\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}}\right ) - 3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} - 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 343 \, B^{3} a^{5} - 1575 \, A^{2} B a^{4} c\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}}\right ) - 3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} + 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 343 \, B^{3} a^{5} + 1575 \, A^{2} B a^{4} c\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}}\right ) + 3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} - 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 343 \, B^{3} a^{5} + 1575 \, A^{2} B a^{4} c\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}}\right ) + 4 \, {\left (45 \, A c^{2} x^{4} - 7 \, B a c x^{3} + 81 \, A a c x^{2} - 11 \, B a^{2} x + 32 \, A a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 304, normalized size = 0.91 \begin {gather*} -\frac {2 \, A}{a^{3} \sqrt {x}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac {13 \, A c^{2} x^{\frac {7}{2}} - 7 \, B a c x^{\frac {5}{2}} + 17 \, A a c x^{\frac {3}{2}} - 11 \, B a^{2} \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 354, normalized size = 1.06 \begin {gather*} -\frac {13 A \,c^{2} x^{\frac {7}{2}}}{16 \left (c \,x^{2}+a \right )^{2} a^{3}}+\frac {7 B c \,x^{\frac {5}{2}}}{16 \left (c \,x^{2}+a \right )^{2} a^{2}}-\frac {17 A c \,x^{\frac {3}{2}}}{16 \left (c \,x^{2}+a \right )^{2} a^{2}}+\frac {11 B \sqrt {x}}{16 \left (c \,x^{2}+a \right )^{2} a}-\frac {45 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{3}}-\frac {45 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{3}}-\frac {45 \sqrt {2}\, A \ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{128 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{128 a^{3}}-\frac {2 A}{a^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 313, normalized size = 0.94 \begin {gather*} -\frac {45 \, A c^{2} x^{4} - 7 \, B a c x^{3} + 81 \, A a c x^{2} - 11 \, B a^{2} x + 32 \, A a^{2}}{16 \, {\left (a^{3} c^{2} x^{\frac {9}{2}} + 2 \, a^{4} c x^{\frac {5}{2}} + a^{5} \sqrt {x}\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (7 \, B a \sqrt {c} - 15 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (7 \, B a \sqrt {c} - 15 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (7 \, B a \sqrt {c} + 15 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, B a \sqrt {c} + 15 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{128 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 673, normalized size = 2.02 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {66355200\,A^2\,a^{10}\,c^4\,\sqrt {x}\,\sqrt {\frac {945\,A\,B}{2048\,a^6}-\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4-4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3+21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}-\frac {14450688\,B^2\,a^{11}\,c^3\,\sqrt {x}\,\sqrt {\frac {945\,A\,B}{2048\,a^6}-\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4-4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3+21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}\right )\,\sqrt {\frac {9\,\left (49\,B^2\,a\,\sqrt {-a^{13}\,c}-225\,A^2\,c\,\sqrt {-a^{13}\,c}+210\,A\,B\,a^7\,c\right )}{4096\,a^{13}\,c}}+2\,\mathrm {atanh}\left (\frac {66355200\,A^2\,a^{10}\,c^4\,\sqrt {x}\,\sqrt {\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {945\,A\,B}{2048\,a^6}-\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4+4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3-21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}-\frac {14450688\,B^2\,a^{11}\,c^3\,\sqrt {x}\,\sqrt {\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {945\,A\,B}{2048\,a^6}-\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4+4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3-21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}\right )\,\sqrt {\frac {9\,\left (225\,A^2\,c\,\sqrt {-a^{13}\,c}-49\,B^2\,a\,\sqrt {-a^{13}\,c}+210\,A\,B\,a^7\,c\right )}{4096\,a^{13}\,c}}-\frac {\frac {2\,A}{a}-\frac {11\,B\,x}{16\,a}+\frac {81\,A\,c\,x^2}{16\,a^2}-\frac {7\,B\,c\,x^3}{16\,a^2}+\frac {45\,A\,c^2\,x^4}{16\,a^3}}{a^2\,\sqrt {x}+c^2\,x^{9/2}+2\,a\,c\,x^{5/2}} \end {gather*}
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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