Optimal. Leaf size=306 \[ \frac {c^{5/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}-\frac {c^{5/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}-\frac {c^{5/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4}}+\frac {c^{5/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{11/4}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {829, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {c^{5/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}-\frac {c^{5/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}-\frac {c^{5/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4}}+\frac {c^{5/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{11/4}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 827
Rule 829
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{9/2} \left (a+c x^2\right )} \, dx &=-\frac {2 A}{7 a x^{7/2}}+\frac {\int \frac {a B-A c x}{x^{7/2} \left (a+c x^2\right )} \, dx}{a}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {\int \frac {-a A c-a B c x}{x^{5/2} \left (a+c x^2\right )} \, dx}{a^2}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {\int \frac {-a^2 B c+a A c^2 x}{x^{3/2} \left (a+c x^2\right )} \, dx}{a^3}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}+\frac {\int \frac {a^2 A c^2+a^2 B c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}+\frac {2 \operatorname {Subst}\left (\int \frac {a^2 A c^2+a^2 B c^2 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}-\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^{5/2}}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^{5/2}}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) 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 )}{2 a^{5/2}}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) 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 )}{2 a^{5/2}}+\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) c^{5/4}\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 )}{2 \sqrt {2} a^{11/4}}+\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) c^{5/4}\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 )}{2 \sqrt {2} a^{11/4}}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) c^{5/4}\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 )}{\sqrt {2} a^{11/4}}-\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) c^{5/4}\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 )}{\sqrt {2} a^{11/4}}\\ &=-\frac {2 A}{7 a x^{7/2}}-\frac {2 B}{5 a x^{5/2}}+\frac {2 A c}{3 a^2 x^{3/2}}+\frac {2 B c}{a^2 \sqrt {x}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{11/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.18 \begin {gather*} -\frac {2 \left (5 A \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\frac {c x^2}{a}\right )+7 B x \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\frac {c x^2}{a}\right )\right )}{35 a x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 175, normalized size = 0.57 \begin {gather*} -\frac {\left (\sqrt {a} B c^{5/4}+A c^{7/4}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} a^{11/4}}-\frac {\left (\sqrt {a} B c^{5/4}-A c^{7/4}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} a^{11/4}}-\frac {2 \left (15 a A+21 a B x-35 A c x^2-105 B c x^3\right )}{105 a^2 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 884, normalized size = 2.89 \begin {gather*} \frac {105 \, a^{2} x^{4} \sqrt {-\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} + 2 \, A B c^{3}}{a^{5}}} \log \left (-{\left (B^{4} a^{2} c^{4} - A^{4} c^{6}\right )} \sqrt {x} + {\left (B a^{9} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} - A B^{2} a^{4} c^{3} + A^{3} a^{3} c^{4}\right )} \sqrt {-\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} + 2 \, A B c^{3}}{a^{5}}}\right ) - 105 \, a^{2} x^{4} \sqrt {-\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} + 2 \, A B c^{3}}{a^{5}}} \log \left (-{\left (B^{4} a^{2} c^{4} - A^{4} c^{6}\right )} \sqrt {x} - {\left (B a^{9} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} - A B^{2} a^{4} c^{3} + A^{3} a^{3} c^{4}\right )} \sqrt {-\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} + 2 \, A B c^{3}}{a^{5}}}\right ) - 105 \, a^{2} x^{4} \sqrt {\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} - 2 \, A B c^{3}}{a^{5}}} \log \left (-{\left (B^{4} a^{2} c^{4} - A^{4} c^{6}\right )} \sqrt {x} + {\left (B a^{9} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} + A B^{2} a^{4} c^{3} - A^{3} a^{3} c^{4}\right )} \sqrt {\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} - 2 \, A B c^{3}}{a^{5}}}\right ) + 105 \, a^{2} x^{4} \sqrt {\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} - 2 \, A B c^{3}}{a^{5}}} \log \left (-{\left (B^{4} a^{2} c^{4} - A^{4} c^{6}\right )} \sqrt {x} - {\left (B a^{9} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} + A B^{2} a^{4} c^{3} - A^{3} a^{3} c^{4}\right )} \sqrt {\frac {a^{5} \sqrt {-\frac {B^{4} a^{2} c^{5} - 2 \, A^{2} B^{2} a c^{6} + A^{4} c^{7}}{a^{11}}} - 2 \, A B c^{3}}{a^{5}}}\right ) + 4 \, {\left (105 \, B c x^{3} + 35 \, A c x^{2} - 21 \, B a x - 15 \, A a\right )} \sqrt {x}}{210 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 276, normalized size = 0.90 \begin {gather*} \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \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 )}{2 \, a^{3} c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \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 )}{2 \, a^{3} c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \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 )}{4 \, a^{3} c} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \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 )}{4 \, a^{3} c} + \frac {2 \, {\left (105 \, B c x^{3} + 35 \, A c x^{2} - 21 \, B a x - 15 \, A a\right )}}{105 \, a^{2} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 318, normalized size = 1.04 \begin {gather*} \frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{2 a^{3}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{2 a^{3}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \,c^{2} \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 )}{4 a^{3}}+\frac {\sqrt {2}\, B c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}+\frac {\sqrt {2}\, B c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}+\frac {\sqrt {2}\, B c \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 )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}+\frac {2 B c}{a^{2} \sqrt {x}}+\frac {2 A c}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 B}{5 a \,x^{\frac {5}{2}}}-\frac {2 A}{7 a \,x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 264, normalized size = 0.86 \begin {gather*} \frac {c^{2} {\left (\frac {2 \, \sqrt {2} {\left (B \sqrt {a} + 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 (B \sqrt {a} + 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 (B \sqrt {a} - 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 (B \sqrt {a} - 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}}}\right )}}{4 \, a^{2}} + \frac {2 \, {\left (105 \, B c x^{3} + 35 \, A c x^{2} - 21 \, B a x - 15 \, A a\right )}}{105 \, a^{2} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 678, normalized size = 2.22 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^6\,c^7\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^{11}\,c^5}}{4\,a^{11}}-\frac {B^2\,\sqrt {-a^{11}\,c^5}}{4\,a^{10}}-\frac {A\,B\,c^3}{2\,a^5}}}{16\,B^3\,a^5\,c^7+\frac {16\,A^3\,c^6\,\sqrt {-a^{11}\,c^5}}{a^2}-16\,A^2\,B\,a^4\,c^8-\frac {16\,A\,B^2\,c^5\,\sqrt {-a^{11}\,c^5}}{a}}-\frac {32\,B^2\,a^7\,c^6\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^{11}\,c^5}}{4\,a^{11}}-\frac {B^2\,\sqrt {-a^{11}\,c^5}}{4\,a^{10}}-\frac {A\,B\,c^3}{2\,a^5}}}{16\,B^3\,a^5\,c^7+\frac {16\,A^3\,c^6\,\sqrt {-a^{11}\,c^5}}{a^2}-16\,A^2\,B\,a^4\,c^8-\frac {16\,A\,B^2\,c^5\,\sqrt {-a^{11}\,c^5}}{a}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a^{11}\,c^5}-A^2\,c\,\sqrt {-a^{11}\,c^5}+2\,A\,B\,a^6\,c^3}{4\,a^{11}}}-\frac {\frac {2\,A}{7\,a}+\frac {2\,B\,x}{5\,a}-\frac {2\,A\,c\,x^2}{3\,a^2}-\frac {2\,B\,c\,x^3}{a^2}}{x^{7/2}}+2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^6\,c^7\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^{11}\,c^5}}{4\,a^{10}}-\frac {A^2\,c\,\sqrt {-a^{11}\,c^5}}{4\,a^{11}}-\frac {A\,B\,c^3}{2\,a^5}}}{16\,B^3\,a^5\,c^7-\frac {16\,A^3\,c^6\,\sqrt {-a^{11}\,c^5}}{a^2}-16\,A^2\,B\,a^4\,c^8+\frac {16\,A\,B^2\,c^5\,\sqrt {-a^{11}\,c^5}}{a}}-\frac {32\,B^2\,a^7\,c^6\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^{11}\,c^5}}{4\,a^{10}}-\frac {A^2\,c\,\sqrt {-a^{11}\,c^5}}{4\,a^{11}}-\frac {A\,B\,c^3}{2\,a^5}}}{16\,B^3\,a^5\,c^7-\frac {16\,A^3\,c^6\,\sqrt {-a^{11}\,c^5}}{a^2}-16\,A^2\,B\,a^4\,c^8+\frac {16\,A\,B^2\,c^5\,\sqrt {-a^{11}\,c^5}}{a}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^{11}\,c^5}-B^2\,a\,\sqrt {-a^{11}\,c^5}+2\,A\,B\,a^6\,c^3}{4\,a^{11}}} \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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