Optimal. Leaf size=304 \[ -\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )} \]
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Rubi [A] time = 0.30, antiderivative size = 304, 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 = {823, 829, 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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\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 )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\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 )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )} \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 )^2} \, dx &=\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )}-\frac {\int \frac {-\frac {5}{2} a A c-\frac {3}{2} a B c x}{x^{3/2} \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )}-\frac {\int \frac {-\frac {3}{2} a^2 B c+\frac {5}{2} a A c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{2 a^3 c}\\ &=-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{2} a^2 B c+\frac {5}{2} a A c^2 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^3 c}\\ &=-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )}-\frac {\left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^2}+\frac {\left (5 A+\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )}-\frac {\left (5 A-\frac {3 \sqrt {a} B}{\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 )}{8 a^2}-\frac {\left (5 A-\frac {3 \sqrt {a} B}{\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 )}{8 a^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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}\\ &=-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {\left (\left (5 A-\frac {3 \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 )}{4 \sqrt {2} a^{9/4}}+\frac {\left (\left (5 A-\frac {3 \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 )}{4 \sqrt {2} a^{9/4}}\\ &=-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )}+\frac {\left (5 A-\frac {3 \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 )}{4 \sqrt {2} a^{9/4}}-\frac {\left (5 A-\frac {3 \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 )}{4 \sqrt {2} a^{9/4}}-\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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\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 )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}\\ \end {align*}
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Mathematica [C] time = 0.46, size = 256, normalized size = 0.84 \begin {gather*} \frac {\sqrt [4]{a} \left (\frac {8 a^{3/4} A}{\sqrt {x} \left (a+c x^2\right )}+\frac {8 a^{3/4} B \sqrt {x}}{a+c x^2}-\frac {3 \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 {3 \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 {6 \sqrt {2} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {6 \sqrt {2} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}\right )-\frac {40 A \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {c x^2}{a}\right )}{\sqrt {x}}}{16 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.00, size = 183, normalized size = 0.60 \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 )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\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 )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {-4 a A+a B x-5 A c x^2}{2 a^2 \sqrt {x} \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 877, normalized size = 2.88 \begin {gather*} -\frac {{\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 27 \, B^{3} a^{4} - 75 \, A^{2} B a^{3} c\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}}\right ) - {\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 27 \, B^{3} a^{4} - 75 \, A^{2} B a^{3} c\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}}\right ) - {\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 27 \, B^{3} a^{4} + 75 \, A^{2} B a^{3} c\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}}\right ) + {\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 27 \, B^{3} a^{4} + 75 \, A^{2} B a^{3} c\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}}\right ) + 4 \, {\left (5 \, A c x^{2} - B a x + 4 \, A a\right )} \sqrt {x}}{8 \, {\left (a^{2} c x^{3} + a^{3} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 281, normalized size = 0.92 \begin {gather*} -\frac {5 \, A c x^{2} - B a x + 4 \, A a}{2 \, {\left (c x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2}} + \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 )}{8 \, a^{3} c^{2}} + \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 )}{8 \, a^{3} c^{2}} + \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 )}{16 \, a^{3} c^{2}} - \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 )}{16 \, a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 314, normalized size = 1.03 \begin {gather*} -\frac {A c \,x^{\frac {3}{2}}}{2 \left (c \,x^{2}+a \right ) a^{2}}+\frac {B \sqrt {x}}{2 \left (c \,x^{2}+a \right ) a}-\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}-\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 )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}+\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 )}{8 a^{2}}+\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 )}{8 a^{2}}+\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 )}{16 a^{2}}-\frac {2 A}{a^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 280, normalized size = 0.92 \begin {gather*} -\frac {5 \, A c x^{2} - B a x + 4 \, A a}{2 \, {\left (a^{2} c x^{\frac {5}{2}} + a^{3} \sqrt {x}\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}}}}{16 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 634, normalized size = 2.09 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {1600\,A^2\,a^7\,c^4\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,a^4}-\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4-216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3+600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}-\frac {576\,B^2\,a^8\,c^3\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,a^4}-\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4-216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3+600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}\right )\,\sqrt {\frac {9\,B^2\,a\,\sqrt {-a^9\,c}-25\,A^2\,c\,\sqrt {-a^9\,c}+30\,A\,B\,a^5\,c}{64\,a^9\,c}}+2\,\mathrm {atanh}\left (\frac {1600\,A^2\,a^7\,c^4\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {15\,A\,B}{32\,a^4}-\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4+216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3-600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}-\frac {576\,B^2\,a^8\,c^3\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {15\,A\,B}{32\,a^4}-\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4+216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3-600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}\right )\,\sqrt {\frac {25\,A^2\,c\,\sqrt {-a^9\,c}-9\,B^2\,a\,\sqrt {-a^9\,c}+30\,A\,B\,a^5\,c}{64\,a^9\,c}}-\frac {\frac {2\,A}{a}-\frac {B\,x}{2\,a}+\frac {5\,A\,c\,x^2}{2\,a^2}}{a\,\sqrt {x}+c\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 149.46, size = 1435, normalized size = 4.72
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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