Optimal. Leaf size=292 \[ \frac {c^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}-\frac {c^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}+\frac {c^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4}}-\frac {c^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4}}+\frac {2 A c}{a^2 \sqrt {x}}-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}} \]
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Rubi [A] time = 0.32, 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 = {829, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {c^{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} a^{9/4}}-\frac {c^{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} a^{9/4}}+\frac {c^{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} a^{9/4}}-\frac {c^{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} a^{9/4}}+\frac {2 A c}{a^2 \sqrt {x}}-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}} \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^{7/2} \left (a+c x^2\right )} \, dx &=-\frac {2 A}{5 a x^{5/2}}+\frac {\int \frac {a B-A c x}{x^{5/2} \left (a+c x^2\right )} \, dx}{a}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {\int \frac {-a A c-a B c x}{x^{3/2} \left (a+c x^2\right )} \, dx}{a^2}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}+\frac {\int \frac {-a^2 B c+a A c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{a^3}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}+\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 )}{a^3}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}-\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^2}-\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}-\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) \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 )}{2 a^2}-\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) \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 )}{2 a^2}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4}\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^{9/4}}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4}\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^{9/4}}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}-\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) c^{3/4}\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^{9/4}}+\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) c^{3/4}\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^{9/4}}\\ &=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.18 \begin {gather*} -\frac {2 \left (3 A \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\frac {c x^2}{a}\right )+5 B x \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\frac {c x^2}{a}\right )\right )}{15 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 167, normalized size = 0.57 \begin {gather*} \frac {\left (\sqrt {a} B c^{3/4}-A c^{5/4}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B c^{3/4}+A c^{5/4}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} a^{9/4}}-\frac {2 \left (3 a A+5 a B x-15 A c x^2\right )}{15 a^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 869, normalized size = 2.98 \begin {gather*} \frac {15 \, a^{2} x^{3} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} + {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + B^{3} a^{4} c - A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}}\right ) - 15 \, a^{2} x^{3} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} - {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + B^{3} a^{4} c - A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}}\right ) - 15 \, a^{2} x^{3} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} + {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - B^{3} a^{4} c + A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}}\right ) + 15 \, a^{2} x^{3} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} - {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - B^{3} a^{4} c + A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}}\right ) + 4 \, {\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )} \sqrt {x}}{30 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 265, normalized size = 0.91 \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 \, a^{3} c} - \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 \, a^{3} c} - \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 \, a^{3} c} + \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 \, a^{3} c} + \frac {2 \, {\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )}}{15 \, a^{2} x^{\frac {5}{2}}} \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 {\sqrt {2}\, A 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}\, A 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}\, A 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 {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2}}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2}}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \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 a^{2}}+\frac {2 A c}{a^{2} \sqrt {x}}-\frac {2 B}{3 a \,x^{\frac {3}{2}}}-\frac {2 A}{5 a \,x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 263, normalized size = 0.90 \begin {gather*} -\frac {c {\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 \, a^{2}} + \frac {2 \, {\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )}}{15 \, a^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 664, normalized size = 2.27 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^7\,c^6\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}-\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6+16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}-16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}-\frac {32\,B^2\,a^8\,c^5\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}-\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6+16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}-16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}\right )\,\sqrt {\frac {A^2\,c\,\sqrt {-a^9\,c^3}-B^2\,a\,\sqrt {-a^9\,c^3}+2\,A\,B\,a^5\,c^2}{4\,a^9}}-2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^7\,c^6\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}-\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6-16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}+16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}-\frac {32\,B^2\,a^8\,c^5\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}-\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6-16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}+16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}\right )\,\sqrt {\frac {B^2\,a\,\sqrt {-a^9\,c^3}-A^2\,c\,\sqrt {-a^9\,c^3}+2\,A\,B\,a^5\,c^2}{4\,a^9}}-\frac {\frac {2\,A}{5\,a}+\frac {2\,B\,x}{3\,a}-\frac {2\,A\,c\,x^2}{a^2}}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 108.65, size = 398, normalized size = 1.36 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{a} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{c} & \text {for}\: a = 0 \\- \frac {2 A}{5 a x^{\frac {5}{2}}} + \frac {2 A c}{a^{2} \sqrt {x}} - \frac {\left (-1\right )^{\frac {3}{4}} A c \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 a^{\frac {9}{4}} \sqrt [4]{\frac {1}{c}}} + \frac {\left (-1\right )^{\frac {3}{4}} A c \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 a^{\frac {9}{4}} \sqrt [4]{\frac {1}{c}}} + \frac {\left (-1\right )^{\frac {3}{4}} A c \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{c}}} \right )}}{a^{\frac {9}{4}} \sqrt [4]{\frac {1}{c}}} - \frac {2 B}{3 a x^{\frac {3}{2}}} + \frac {\sqrt [4]{-1} B c \sqrt [4]{\frac {1}{c}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 a^{\frac {7}{4}}} - \frac {\sqrt [4]{-1} B c \sqrt [4]{\frac {1}{c}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 a^{\frac {7}{4}}} + \frac {\sqrt [4]{-1} B c \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 )}}{a^{\frac {7}{4}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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