Optimal. Leaf size=278 \[ -\frac {\sqrt [4]{a} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} c^{7/4}}+\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c} \]
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Rubi [A] time = 0.27, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 12, 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 {\sqrt [4]{a} \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^{7/4}}+\frac {\sqrt [4]{a} \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^{7/4}}+\frac {\sqrt [4]{a} \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^{7/4}}-\frac {\sqrt [4]{a} \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^{7/4}}+\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 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^{3/2} (A+B x)}{a+c x^2} \, dx &=\frac {2 B x^{3/2}}{3 c}+\frac {\int \frac {\sqrt {x} (-a B+A c x)}{a+c x^2} \, dx}{c}\\ &=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}+\frac {\int \frac {-a A c-a B c x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{c^2}\\ &=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}+\frac {2 \operatorname {Subst}\left (\int \frac {-a A c-a B c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}+\frac {\left (\sqrt {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^2}-\frac {\left (\sqrt {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^2}\\ &=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}-\frac {\left (\sqrt {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^2}-\frac {\left (\sqrt {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^2}-\frac {\left (\sqrt [4]{a} \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^{7/4}}-\frac {\left (\sqrt [4]{a} \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^{7/4}}\\ &=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}-\frac {\sqrt [4]{a} \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^{7/4}}+\frac {\sqrt [4]{a} \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^{7/4}}-\frac {\left (\sqrt [4]{a} \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^{7/4}}+\frac {\left (\sqrt [4]{a} \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^{7/4}}\\ &=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}+\frac {\sqrt [4]{a} \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^{7/4}}-\frac {\sqrt [4]{a} \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^{7/4}}-\frac {\sqrt [4]{a} \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^{7/4}}+\frac {\sqrt [4]{a} \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^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 287, normalized size = 1.03 \begin {gather*} \frac {\sqrt [4]{a} A \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} c^{5/4}}-\frac {\sqrt [4]{a} A \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} c^{5/4}}+\frac {\sqrt [4]{a} A \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{5/4}}-\frac {\sqrt [4]{a} A \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} c^{5/4}}+\frac {(-a)^{3/4} B \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}-\frac {(-a)^{3/4} B \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 160, normalized size = 0.58 \begin {gather*} \frac {\left (a^{3/4} B+\sqrt [4]{a} 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^{7/4}}+\frac {\left (a^{3/4} B-\sqrt [4]{a} A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} c^{7/4}}+\frac {2 \left (3 A \sqrt {x}+B x^{3/2}\right )}{3 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 772, normalized size = 2.78 \begin {gather*} -\frac {3 \, c \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - A B^{2} a c^{2} + A^{3} c^{3}\right )} \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}}\right ) - 3 \, c \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - A B^{2} a c^{2} + A^{3} c^{3}\right )} \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}}\right ) - 3 \, c \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + A B^{2} a c^{2} - A^{3} c^{3}\right )} \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}}\right ) + 3 \, c \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + A B^{2} a c^{2} - A^{3} c^{3}\right )} \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}}\right ) - 4 \, {\left (B x + 3 \, A\right )} \sqrt {x}}{6 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 255, normalized size = 0.92 \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 \, c^{4}} - \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 \, c^{4}} - \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 \, c^{4}} + \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 \, c^{4}} + \frac {2 \, {\left (B c^{2} x^{\frac {3}{2}} + 3 \, A c^{2} \sqrt {x}\right )}}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 289, normalized size = 1.04 \begin {gather*} \frac {2 B \,x^{\frac {3}{2}}}{3 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 )}{2 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 )}{2 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 )}{4 c}-\frac {\sqrt {2}\, B 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}\, B 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}\, 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 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {2 A \sqrt {x}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 247, normalized size = 0.89 \begin {gather*} -\frac {a {\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 \, c} + \frac {2 \, {\left (B x^{\frac {3}{2}} + 3 \, A \sqrt {x}\right )}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 601, normalized size = 2.16 \begin {gather*} \frac {2\,A\,\sqrt {x}}{c}+\frac {2\,B\,x^{3/2}}{3\,c}-\mathrm {atan}\left (\frac {B^2\,a^3\,\sqrt {x}\,\sqrt {\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}-\frac {A\,B\,a}{2\,c^3}-\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}-\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}+\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}-\frac {A^2\,a^2\,c\,\sqrt {x}\,\sqrt {\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}-\frac {A\,B\,a}{2\,c^3}-\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}-\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}+\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a\,c^7}-B^2\,a\,\sqrt {-a\,c^7}+2\,A\,B\,a\,c^4}{4\,c^7}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {B^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}-\frac {A\,B\,a}{2\,c^3}-\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}+\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}-\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}-\frac {A^2\,a^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}-\frac {A\,B\,a}{2\,c^3}-\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}+\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}-\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a\,c^7}-A^2\,c\,\sqrt {-a\,c^7}+2\,A\,B\,a\,c^4}{4\,c^7}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.06, size = 379, normalized size = 1.36 \begin {gather*} \begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{a} & \text {for}\: c = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{c} & \text {for}\: a = 0 \\\frac {\sqrt [4]{-1} A \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c} - \frac {\sqrt [4]{-1} A \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c} + \frac {\sqrt [4]{-1} A \sqrt [4]{a} \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} + \frac {2 A \sqrt {x}}{c} + \frac {\left (-1\right )^{\frac {3}{4}} B 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}} B 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}} B 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 B x^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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