Optimal. Leaf size=292 \[ -\frac {a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}-\frac {a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{9/4}}+\frac {a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} c^{9/4}}-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c} \]
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Rubi [A] time = 0.34, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {825, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {a^{3/4} \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} c^{9/4}}+\frac {a^{3/4} \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} c^{9/4}}-\frac {a^{3/4} \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} c^{9/4}}+\frac {a^{3/4} \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} c^{9/4}}-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 825
Rule 827
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx &=\frac {2 B x^{5/2}}{5 c}+\frac {\int \frac {x^{3/2} (-a B+A c x)}{a+c x^2} \, dx}{c}\\ &=\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}+\frac {\int \frac {\sqrt {x} (-a A c-a B c x)}{a+c x^2} \, dx}{c^2}\\ &=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}+\frac {\int \frac {a^2 B c-a A c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{c^3}\\ &=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}+\frac {2 \operatorname {Subst}\left (\int \frac {a^2 B c-a A c^2 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}+\frac {\left (a \left (\sqrt {a} B-A \sqrt {c}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{c^{5/2}}+\frac {\left (a \left (\sqrt {a} B+A \sqrt {c}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{c^{5/2}}\\ &=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}+\frac {\left (a \left (\sqrt {a} B-A \sqrt {c}\right )\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 c^{5/2}}+\frac {\left (a \left (\sqrt {a} B-A \sqrt {c}\right )\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 c^{5/2}}-\frac {\left (a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right )\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} c^{9/4}}-\frac {\left (a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right )\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} c^{9/4}}\\ &=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}-\frac {a^{3/4} \left (\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 )}{2 \sqrt {2} c^{9/4}}+\frac {a^{3/4} \left (\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 )}{2 \sqrt {2} c^{9/4}}+\frac {\left (a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right )\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} c^{9/4}}-\frac {\left (a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right )\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} c^{9/4}}\\ &=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}-\frac {a^{3/4} \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} c^{9/4}}+\frac {a^{3/4} \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} c^{9/4}}-\frac {a^{3/4} \left (\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 )}{2 \sqrt {2} c^{9/4}}+\frac {a^{3/4} \left (\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 )}{2 \sqrt {2} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 290, normalized size = 0.99 \begin {gather*} \frac {-15 \sqrt {2} a^{5/4} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )+15 \sqrt {2} a^{5/4} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )-30 \sqrt {2} a^{5/4} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )+30 \sqrt {2} a^{5/4} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )+60 (-a)^{3/4} A \sqrt {c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )-60 (-a)^{3/4} A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )-120 a B \sqrt [4]{c} \sqrt {x}+40 A c^{5/4} x^{3/2}+24 B c^{5/4} x^{5/2}}{60 c^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 173, normalized size = 0.59 \begin {gather*} -\frac {\left (a^{5/4} B-a^{3/4} A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} c^{9/4}}+\frac {\left (a^{3/4} A \sqrt {c}+a^{5/4} B\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} c^{9/4}}+\frac {2 \left (-15 a B \sqrt {x}+5 A c x^{3/2}+3 B c x^{5/2}\right )}{15 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 862, normalized size = 2.95 \begin {gather*} -\frac {15 \, c^{2} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} + {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + B^{3} a^{3} c^{2} - A^{2} B a^{2} c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}}\right ) - 15 \, c^{2} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} - {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + B^{3} a^{3} c^{2} - A^{2} B a^{2} c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}}\right ) - 15 \, c^{2} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} + {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - B^{3} a^{3} c^{2} + A^{2} B a^{2} c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}}\right ) + 15 \, c^{2} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} - {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - B^{3} a^{3} c^{2} + A^{2} B a^{2} c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}}\right ) - 4 \, {\left (3 \, B c x^{2} + 5 \, A c x - 15 \, B a\right )} \sqrt {x}}{30 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 262, normalized size = 0.90 \begin {gather*} \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \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 )}{2 \, c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \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 )}{2 \, c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \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 )}{4 \, c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \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 )}{4 \, c^{4}} + \frac {2 \, {\left (3 \, B c^{4} x^{\frac {5}{2}} + 5 \, A c^{4} x^{\frac {3}{2}} - 15 \, B a c^{3} \sqrt {x}\right )}}{15 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 302, normalized size = 1.03 \begin {gather*} \frac {2 B \,x^{\frac {5}{2}}}{5 c}+\frac {2 A \,x^{\frac {3}{2}}}{3 c}-\frac {\sqrt {2}\, A a \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}} c^{2}}-\frac {\sqrt {2}\, A a \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}} c^{2}}-\frac {\sqrt {2}\, A 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 )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{2 c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{2 c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B 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 )}{4 c^{2}}-\frac {2 B a \sqrt {x}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 265, normalized size = 0.91 \begin {gather*} \frac {a {\left (\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - 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 (B a \sqrt {c} - 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 (B a \sqrt {c} + 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 (B a \sqrt {c} + 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}}}\right )}}{4 \, c^{2}} + \frac {2 \, {\left (3 \, B c x^{\frac {5}{2}} + 5 \, A c x^{\frac {3}{2}} - 15 \, B a \sqrt {x}\right )}}{15 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 665, normalized size = 2.28 \begin {gather*} \frac {2\,A\,x^{3/2}}{3\,c}+\frac {2\,B\,x^{5/2}}{5\,c}-\frac {2\,B\,a\,\sqrt {x}}{c^2}-\mathrm {atan}\left (\frac {A^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A\,B\,a^2}{2\,c^4}-\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c^2}-\frac {16\,A\,B^2\,a^5}{c^3}-\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^8}+\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^7}}-\frac {B^2\,a^4\,\sqrt {x}\,\sqrt {\frac {A\,B\,a^2}{2\,c^4}-\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c}-\frac {16\,A\,B^2\,a^5}{c^2}-\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^7}+\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^6}}\right )\,\sqrt {\frac {B^2\,a\,\sqrt {-a^3\,c^9}-A^2\,c\,\sqrt {-a^3\,c^9}+2\,A\,B\,a^2\,c^5}{4\,c^9}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {A^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {A\,B\,a^2}{2\,c^4}-\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c^2}-\frac {16\,A\,B^2\,a^5}{c^3}+\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^8}-\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^7}}-\frac {B^2\,a^4\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {A\,B\,a^2}{2\,c^4}-\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c}-\frac {16\,A\,B^2\,a^5}{c^2}+\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^7}-\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^6}}\right )\,\sqrt {\frac {A^2\,c\,\sqrt {-a^3\,c^9}-B^2\,a\,\sqrt {-a^3\,c^9}+2\,A\,B\,a^2\,c^5}{4\,c^9}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 33.96, size = 403, normalized size = 1.38 \begin {gather*} \begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{a} & \text {for}\: c = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{c} & \text {for}\: a = 0 \\\frac {\left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c^{2} \sqrt [4]{\frac {1}{c}}} - \frac {\left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c^{2} \sqrt [4]{\frac {1}{c}}} - \frac {\left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{c}}} \right )}}{c^{2} \sqrt [4]{\frac {1}{c}}} + \frac {2 A x^{\frac {3}{2}}}{3 c} - \frac {\sqrt [4]{-1} B a^{\frac {5}{4}} \sqrt [4]{\frac {1}{c}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c^{2}} + \frac {\sqrt [4]{-1} B a^{\frac {5}{4}} \sqrt [4]{\frac {1}{c}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c^{2}} - \frac {\sqrt [4]{-1} B a^{\frac {5}{4}} \sqrt [4]{\frac {1}{c}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{c}}} \right )}}{c^{2}} - \frac {2 B a \sqrt {x}}{c^{2}} + \frac {2 B x^{\frac {5}{2}}}{5 c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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