Optimal. Leaf size=320 \[ \frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.31, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {823, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac {\sqrt {x} (A+B x)}{4 a \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 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}-\frac {\int \frac {-\frac {7}{2} a A c-\frac {5}{2} a B c x}{\sqrt {x} \left (a+c x^2\right )^2} \, dx}{4 a^2 c}\\ &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac {\int \frac {\frac {21}{4} a^2 A c^2+\frac {5}{4} a^2 B c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^4 c^2}\\ &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {21}{4} a^2 A c^2+\frac {5}{4} a^2 B c^2 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^4 c^2}\\ &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}-\frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^{5/2} c}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^{5/2} c}\\ &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\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^{5/2} c}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\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^{5/2} c}+\frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\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^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\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^{11/4} c^{3/4}}\\ &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {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^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B+21 A \sqrt {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^{11/4} c^{3/4}}\\ &=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}-\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 344, normalized size = 1.08 \begin {gather*} \frac {\frac {32 a^2 A \sqrt {x}}{\left (a+c x^2\right )^2}+\frac {32 a^2 B x^{3/2}}{\left (a+c x^2\right )^2}+\frac {56 a A \sqrt {x}}{a+c x^2}-\frac {21 \sqrt {2} \sqrt [4]{a} A \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} \sqrt [4]{a} A \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} \sqrt [4]{a} A \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {42 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}-\frac {20 (-a)^{3/4} B \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {20 (-a)^{3/4} B \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {40 a B x^{3/2}}{a+c x^2}}{128 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.58, size = 199, normalized size = 0.62 \begin {gather*} -\frac {\left (5 \sqrt {a} B+21 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^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B-21 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^{11/4} c^{3/4}}+\frac {11 a A \sqrt {x}+9 a B x^{3/2}+7 A c x^{5/2}+5 B c x^{7/2}}{16 a^2 \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 981, normalized size = 3.07 \begin {gather*} \frac {{\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 525 \, A B^{2} a^{4} c + 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}}\right ) - {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 525 \, A B^{2} a^{4} c + 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}}\right ) - {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 525 \, A B^{2} a^{4} c - 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}}\right ) + {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 525 \, A B^{2} a^{4} c - 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}}\right ) + 4 \, {\left (5 \, B c x^{3} + 7 \, A c x^{2} + 9 \, B a x + 11 \, A a\right )} \sqrt {x}}{64 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 293, normalized size = 0.92 \begin {gather*} \frac {5 \, B c x^{\frac {7}{2}} + 7 \, A c x^{\frac {5}{2}} + 9 \, B a x^{\frac {3}{2}} + 11 \, A a \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{2}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\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^{3} c^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\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^{3} c^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 349, normalized size = 1.09 \begin {gather*} \frac {B \,x^{\frac {3}{2}}}{4 \left (c \,x^{2}+a \right )^{2} a}+\frac {A \sqrt {x}}{4 \left (c \,x^{2}+a \right )^{2} a}+\frac {5 B \,x^{\frac {3}{2}}}{16 \left (c \,x^{2}+a \right ) a^{2}}+\frac {7 A \sqrt {x}}{16 \left (c \,x^{2}+a \right ) a^{2}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \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}\, A \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}\, 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 a^{3}}+\frac {5 \sqrt {2}\, B \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^{2} c}+\frac {5 \sqrt {2}\, B \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^{2} c}+\frac {5 \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 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 291, normalized size = 0.91 \begin {gather*} \frac {5 \, B c x^{\frac {7}{2}} + 7 \, A c x^{\frac {5}{2}} + 9 \, B a x^{\frac {3}{2}} + 11 \, A a \sqrt {x}}{16 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B \sqrt {a} + 21 \, A \sqrt {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 (5 \, B \sqrt {a} + 21 \, A \sqrt {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 (5 \, B \sqrt {a} - 21 \, A \sqrt {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 (5 \, B \sqrt {a} - 21 \, A \sqrt {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}}}}{128 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 687, normalized size = 2.15 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {441\,A^2\,c^3\,\sqrt {x}\,\sqrt {\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}-\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a}+\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^7}-\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}-\frac {2205\,A^2\,B\,c^2}{2048\,a^2}\right )}-\frac {25\,B^2\,c^2\,\sqrt {x}\,\sqrt {\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}-\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a^2}+\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}-\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^9}-\frac {2205\,A^2\,B\,c^2}{2048\,a^3}\right )}\right )\,\sqrt {-\frac {441\,A^2\,c\,\sqrt {-a^{11}\,c^3}-25\,B^2\,a\,\sqrt {-a^{11}\,c^3}+210\,A\,B\,a^6\,c^2}{4096\,a^{11}\,c^3}}+2\,\mathrm {atanh}\left (\frac {441\,A^2\,c^3\,\sqrt {x}\,\sqrt {\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}-\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a}-\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^7}+\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}-\frac {2205\,A^2\,B\,c^2}{2048\,a^2}\right )}-\frac {25\,B^2\,c^2\,\sqrt {x}\,\sqrt {\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}-\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a^2}-\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}+\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^9}-\frac {2205\,A^2\,B\,c^2}{2048\,a^3}\right )}\right )\,\sqrt {-\frac {25\,B^2\,a\,\sqrt {-a^{11}\,c^3}-441\,A^2\,c\,\sqrt {-a^{11}\,c^3}+210\,A\,B\,a^6\,c^2}{4096\,a^{11}\,c^3}}+\frac {\frac {11\,A\,\sqrt {x}}{16\,a}+\frac {9\,B\,x^{3/2}}{16\,a}+\frac {7\,A\,c\,x^{5/2}}{16\,a^2}+\frac {5\,B\,c\,x^{7/2}}{16\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \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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