Optimal. Leaf size=289 \[ \frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.24, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {819, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {\left (3 \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 )}{8 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (3 \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 )}{8 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 819
Rule 827
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {\frac {a A}{2}+\frac {3 a B x}{2}}{\sqrt {x} \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {a A}{2}+\frac {3}{2} a B x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a c}\\ &=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 c^2}+\frac {\left (3 B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 c^2}\\ &=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\left (3 B+\frac {A \sqrt {c}}{\sqrt {a}}\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 c^2}+\frac {\left (3 B+\frac {A \sqrt {c}}{\sqrt {a}}\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 c^2}+\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}\\ &=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B+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 )}{4 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B+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 )}{4 \sqrt {2} a^{3/4} c^{7/4}}\\ &=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}-\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 323, normalized size = 1.12 \begin {gather*} \frac {-\frac {\sqrt {2} \sqrt [4]{a} A \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}+\frac {\sqrt {2} \sqrt [4]{a} A \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}-\frac {2 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{c^{5/4}}+\frac {2 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{c^{5/4}}+\frac {8 A x^{5/2}}{a+c x^2}-\frac {12 (-a)^{3/4} B \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac {12 (-a)^{3/4} B \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac {8 B x^{7/2}}{a+c x^2}-\frac {8 A \sqrt {x}}{c}-\frac {8 B x^{3/2}}{c}}{16 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.81, size = 171, normalized size = 0.59 \begin {gather*} -\frac {\left (3 \sqrt {a} B+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^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B-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^{3/4} c^{7/4}}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 888, normalized size = 3.07 \begin {gather*} \frac {{\left (c^{2} x^{2} + a c\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 9 \, A B^{2} a^{2} c^{2} + A^{3} a c^{3}\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}}\right ) - {\left (c^{2} x^{2} + a c\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 9 \, A B^{2} a^{2} c^{2} + A^{3} a c^{3}\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}}\right ) - {\left (c^{2} x^{2} + a c\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 9 \, A B^{2} a^{2} c^{2} - A^{3} a c^{3}\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}}\right ) + {\left (c^{2} x^{2} + a c\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 9 \, A B^{2} a^{2} c^{2} - A^{3} a c^{3}\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}}\right ) - 4 \, {\left (B x + A\right )} \sqrt {x}}{8 \, {\left (c^{2} x^{2} + a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 271, normalized size = 0.94 \begin {gather*} -\frac {B x^{\frac {3}{2}} + A \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 3 \, \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 )}{8 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 3 \, \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 )}{8 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 3 \, \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 )}{16 \, a c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 3 \, \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 )}{16 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 307, normalized size = 1.06 \begin {gather*} \frac {\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 )}{8 a c}+\frac {\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 )}{8 a c}+\frac {\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 )}{16 a c}+\frac {3 \sqrt {2}\, B \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}} c^{2}}+\frac {3 \sqrt {2}\, B \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}} c^{2}}+\frac {3 \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 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {-\frac {B \,x^{\frac {3}{2}}}{2 c}-\frac {A \sqrt {x}}{2 c}}{c \,x^{2}+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 259, normalized size = 0.90 \begin {gather*} -\frac {B x^{\frac {3}{2}} + A \sqrt {x}}{2 \, {\left (c^{2} x^{2} + a c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, 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 (3 \, 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 (3 \, 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 (3 \, 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}}}}{16 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 656, normalized size = 2.27 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {18\,B^2\,a\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}-\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}+\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}-\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}-\frac {2\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}-\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}+\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}-\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^3\,c^7}-9\,B^2\,a\,\sqrt {-a^3\,c^7}+6\,A\,B\,a^2\,c^4}{64\,a^3\,c^7}}+2\,\mathrm {atanh}\left (\frac {18\,B^2\,a\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}-\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}-\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}+\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}-\frac {2\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}-\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}-\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}+\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}\right )\,\sqrt {-\frac {9\,B^2\,a\,\sqrt {-a^3\,c^7}-A^2\,c\,\sqrt {-a^3\,c^7}+6\,A\,B\,a^2\,c^4}{64\,a^3\,c^7}}-\frac {\frac {A\,\sqrt {x}}{2\,c}+\frac {B\,x^{3/2}}{2\,c}}{c\,x^2+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 90.99, size = 1316, normalized size = 4.55
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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