Optimal. Leaf size=331 \[ -\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.29, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {821, 823, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 821
Rule 823
Rule 827
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {\frac {a B}{2}+\frac {5 A c x}{2}}{\sqrt {x} \left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\int \frac {-\frac {3}{4} a^2 B c-\frac {5}{4} a A c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^3 c^2}\\ &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} a^2 B c-\frac {5}{4} a A c^2 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^3 c^2}\\ &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}+\frac {\left (3 \sqrt {a} B-5 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^2 c^{3/2}}+\frac {\left (3 \sqrt {a} B+5 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^2 c^{3/2}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}+\frac {\left (3 \sqrt {a} B+5 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^2 c^{3/2}}+\frac {\left (3 \sqrt {a} B+5 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^2 c^{3/2}}-\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 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^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B+5 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^{9/4} c^{5/4}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 356, normalized size = 1.08 \begin {gather*} \frac {-\frac {3 \sqrt {2} a^{5/4} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}+\frac {3 \sqrt {2} a^{5/4} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}-\frac {6 \sqrt {2} a^{5/4} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{c^{5/4}}+\frac {6 \sqrt {2} a^{5/4} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{c^{5/4}}+\frac {32 a^2 A x^{3/2}}{\left (a+c x^2\right )^2}+\frac {32 a^2 B x^{5/2}}{\left (a+c x^2\right )^2}-\frac {20 (-a)^{3/4} A \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {20 (-a)^{3/4} A \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {40 a A x^{3/2}}{a+c x^2}+\frac {24 a B x^{5/2}}{a+c x^2}-\frac {24 a B \sqrt {x}}{c}}{128 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.89, size = 207, normalized size = 0.63 \begin {gather*} -\frac {\left (3 \sqrt {a} B+5 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^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 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^{9/4} c^{5/4}}+\frac {-3 a^2 B \sqrt {x}+9 a A c x^{3/2}+a B c x^{5/2}+5 A c^2 x^{7/2}}{16 a^2 c \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 1022, normalized size = 3.09 \begin {gather*} -\frac {{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 27 \, B^{3} a^{4} c - 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}}\right ) - {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 27 \, B^{3} a^{4} c - 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}}\right ) - {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 27 \, B^{3} a^{4} c + 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}}\right ) + {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 27 \, B^{3} a^{4} c + 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}}\right ) - 4 \, {\left (5 \, A c^{2} x^{3} + B a c x^{2} + 9 \, A a c x - 3 \, B a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 297, normalized size = 0.90 \begin {gather*} \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \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^{3} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \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^{3} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \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^{3} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \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^{3} c^{3}} + \frac {5 \, A c^{2} x^{\frac {7}{2}} + B a c x^{\frac {5}{2}} + 9 \, A a c x^{\frac {3}{2}} - 3 \, B a^{2} \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 335, normalized size = 1.01 \begin {gather*} \frac {5 \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^{2} c}+\frac {5 \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^{2} c}+\frac {5 \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^{2} c}+\frac {3 \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^{2} c}+\frac {3 \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^{2} c}+\frac {3 \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^{2} c}+\frac {\frac {5 A c \,x^{\frac {7}{2}}}{16 a^{2}}+\frac {B \,x^{\frac {5}{2}}}{16 a}+\frac {9 A \,x^{\frac {3}{2}}}{16 a}-\frac {3 B \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 311, normalized size = 0.94 \begin {gather*} \frac {5 \, A c^{2} x^{\frac {7}{2}} + B a c x^{\frac {5}{2}} + 9 \, A a c x^{\frac {3}{2}} - 3 \, B a^{2} \sqrt {x}}{16 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, 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 (3 \, B a \sqrt {c} + 5 \, 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 (3 \, B a \sqrt {c} - 5 \, 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 (3 \, B a \sqrt {c} - 5 \, 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}}}}{128 \, a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 690, normalized size = 2.08 \begin {gather*} \frac {\frac {9\,A\,x^{3/2}}{16\,a}+\frac {B\,x^{5/2}}{16\,a}-\frac {3\,B\,\sqrt {x}}{16\,c}+\frac {5\,A\,c\,x^{7/2}}{16\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}-\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a^2}-\frac {125\,A^3\,c}{2048\,a^3}+\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^3}-\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^7\,c^2}\right )}-\frac {25\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}-\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a}-\frac {125\,A^3\,c}{2048\,a^2}+\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^5\,c^3}-\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^2}\right )}\right )\,\sqrt {-\frac {9\,B^2\,a\,\sqrt {-a^9\,c^5}-25\,A^2\,c\,\sqrt {-a^9\,c^5}+30\,A\,B\,a^5\,c^3}{4096\,a^9\,c^5}}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}-\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a^2}-\frac {125\,A^3\,c}{2048\,a^3}-\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^3}+\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^7\,c^2}\right )}-\frac {25\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}-\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a}-\frac {125\,A^3\,c}{2048\,a^2}-\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^5\,c^3}+\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^2}\right )}\right )\,\sqrt {-\frac {25\,A^2\,c\,\sqrt {-a^9\,c^5}-9\,B^2\,a\,\sqrt {-a^9\,c^5}+30\,A\,B\,a^5\,c^3}{4096\,a^9\,c^5}} \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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