Optimal. Leaf size=304 \[ \frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{a} c^{9/4}}+\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {5 B \sqrt {x}}{2 c^2} \]
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Rubi [A] time = 0.31, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {819, 825, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}+\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {5 B \sqrt {x}}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 819
Rule 825
Rule 827
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {\sqrt {x} \left (\frac {3 a A}{2}+\frac {5 a B x}{2}\right )}{a+c x^2} \, dx}{2 a c}\\ &=\frac {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {-\frac {5 a^2 B}{2}+\frac {3}{2} a A c x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{2 a c^2}\\ &=\frac {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {5 a^2 B}{2}+\frac {3}{2} a A c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a c^2}\\ &=\frac {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\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 A+\frac {5 \sqrt {a} B}{\sqrt {c}}\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 {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}+\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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 )}{8 c^2}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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 )}{8 c^2}\\ &=\frac {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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} \sqrt [4]{a} c^{7/4}}-\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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} \sqrt [4]{a} c^{7/4}}\\ &=\frac {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}-\frac {\left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 344, normalized size = 1.13 \begin {gather*} \frac {1}{16} \left (\frac {8 A x^{7/2}}{a^2+a c x^2}+\frac {8 B x^{9/2}}{a^2+a c x^2}+\frac {12 A \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} c^{7/4}}-\frac {12 A \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} c^{7/4}}-\frac {8 A x^{3/2}}{a c}+\frac {5 \sqrt {2} \sqrt [4]{a} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{9/4}}-\frac {5 \sqrt {2} \sqrt [4]{a} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{9/4}}+\frac {10 \sqrt {2} \sqrt [4]{a} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{c^{9/4}}-\frac {10 \sqrt {2} \sqrt [4]{a} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{c^{9/4}}-\frac {8 B x^{5/2}}{a c}+\frac {40 B \sqrt {x}}{c^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.81, size = 190, normalized size = 0.62 \begin {gather*} \frac {\left (5 \sqrt {a} B-3 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} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 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} \sqrt [4]{a} c^{9/4}}+\frac {5 a B \sqrt {x}-A c x^{3/2}+4 B c x^{5/2}}{2 c^2 \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 884, normalized size = 2.91 \begin {gather*} \frac {{\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 125 \, B^{3} a^{2} c^{2} - 45 \, A^{2} B a c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}}\right ) - {\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 125 \, B^{3} a^{2} c^{2} - 45 \, A^{2} B a c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}}\right ) - {\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 125 \, B^{3} a^{2} c^{2} + 45 \, A^{2} B a c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}}\right ) + {\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 125 \, B^{3} a^{2} c^{2} + 45 \, A^{2} B a c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}}\right ) + 4 \, {\left (4 \, B c x^{2} - A c x + 5 \, B a\right )} \sqrt {x}}{8 \, {\left (c^{3} x^{2} + a c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 283, normalized size = 0.93 \begin {gather*} \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} c^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 3 \, \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 )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 3 \, \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 )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 3 \, \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 )}{16 \, a c^{4}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 3 \, \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 )}{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 = 314, normalized size = 1.03 \begin {gather*} -\frac {A \,x^{\frac {3}{2}}}{2 \left (c \,x^{2}+a \right ) c}+\frac {B a \sqrt {x}}{2 \left (c \,x^{2}+a \right ) c^{2}}+\frac {3 \sqrt {2}\, A \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}\, A \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}\, 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 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 c^{2}}-\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 c^{2}}-\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \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 c^{2}}+\frac {2 B \sqrt {x}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 283, normalized size = 0.93 \begin {gather*} -\frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c^{3} x^{2} + a c^{2}\right )}} + \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a \sqrt {c} - 3 \, 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 (5 \, B a \sqrt {c} - 3 \, 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 (5 \, B a \sqrt {c} + 3 \, 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 (5 \, B a \sqrt {c} + 3 \, 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}}}}{16 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 617, normalized size = 2.03 \begin {gather*} \frac {\frac {B\,a\,\sqrt {x}}{2}-\frac {A\,c\,x^{3/2}}{2}}{c^3\,x^2+a\,c^2}+\frac {2\,B\,\sqrt {x}}{c^2}-\mathrm {atan}\left (\frac {B^2\,a^2\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}-\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}+\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,50{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c}+\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^7}-\frac {75\,A\,B^2\,a^2}{4\,c^2}-\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^6}}-\frac {A^2\,a\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}-\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}+\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,18{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c^2}+\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^8}-\frac {75\,A\,B^2\,a^2}{4\,c^3}-\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^7}}\right )\,\sqrt {\frac {9\,A^2\,c\,\sqrt {-a\,c^9}-25\,B^2\,a\,\sqrt {-a\,c^9}+30\,A\,B\,a\,c^5}{64\,a\,c^9}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {B^2\,a^2\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}+\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}-\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,50{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c}-\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^7}-\frac {75\,A\,B^2\,a^2}{4\,c^2}+\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^6}}-\frac {A^2\,a\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}+\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}-\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,18{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c^2}-\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^8}-\frac {75\,A\,B^2\,a^2}{4\,c^3}+\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^7}}\right )\,\sqrt {\frac {25\,B^2\,a\,\sqrt {-a\,c^9}-9\,A^2\,c\,\sqrt {-a\,c^9}+30\,A\,B\,a\,c^5}{64\,a\,c^9}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 170.57, size = 1374, normalized size = 4.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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