Optimal. Leaf size=320 \[ \frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.31, antiderivative size = 320, 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 = {819, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \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^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\int \frac {x^{3/2} \left (\frac {5 a A}{2}+\frac {7 a B x}{2}\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac {\int \frac {\frac {5 a^2 A}{4}+\frac {21}{4} a^2 B x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {5 a^2 A}{4}+\frac {21}{4} a^2 B x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 c^2}\\ &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac {\left (21 B-\frac {5 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 )}{32 c^3}+\frac {\left (21 B+\frac {5 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 )}{32 c^3}\\ &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac {\left (21 B+\frac {5 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 )}{64 c^3}+\frac {\left (21 B+\frac {5 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 )}{64 c^3}+\frac {\left (21 \sqrt {a} B-5 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 )}{64 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B-5 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 )}{64 \sqrt {2} a^{3/4} c^{11/4}}\\ &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B+5 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 )}{32 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B+5 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 )}{32 \sqrt {2} a^{3/4} c^{11/4}}\\ &=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 385, normalized size = 1.20 \begin {gather*} \frac {-\frac {5 \sqrt {2} a^{5/4} A \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} a^{5/4} A \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} a^{5/4} A \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{c^{9/4}}+\frac {10 \sqrt {2} a^{5/4} A \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{c^{9/4}}-\frac {40 a A \sqrt {x}}{c^2}-\frac {8 A x^{9/2}}{a+c x^2}+\frac {32 a A x^{9/2}}{\left (a+c x^2\right )^2}+\frac {84 (-a)^{7/4} B \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{11/4}}+\frac {84 (-a)^{3/4} a B \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{11/4}}-\frac {56 a B x^{3/2}}{c^2}-\frac {24 B x^{11/2}}{a+c x^2}+\frac {32 a B x^{11/2}}{\left (a+c x^2\right )^2}+\frac {8 A x^{5/2}}{c}+\frac {24 B x^{7/2}}{c}}{128 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.19, size = 191, normalized size = 0.60 \begin {gather*} -\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\sqrt {x} \left (5 a A+7 a B x+9 A c x^2+11 B c x^3\right )}{16 c^2 \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 986, normalized size = 3.08 \begin {gather*} \frac {{\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt {-\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} + 210 \, A B}{a c^{5}}} \log \left (-{\left (194481 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (21 \, B a^{3} c^{8} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} - 2205 \, A B^{2} a^{2} c^{3} + 125 \, A^{3} a c^{4}\right )} \sqrt {-\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} + 210 \, A B}{a c^{5}}}\right ) - {\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt {-\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} + 210 \, A B}{a c^{5}}} \log \left (-{\left (194481 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (21 \, B a^{3} c^{8} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} - 2205 \, A B^{2} a^{2} c^{3} + 125 \, A^{3} a c^{4}\right )} \sqrt {-\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} + 210 \, A B}{a c^{5}}}\right ) - {\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt {\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} - 210 \, A B}{a c^{5}}} \log \left (-{\left (194481 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (21 \, B a^{3} c^{8} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} + 2205 \, A B^{2} a^{2} c^{3} - 125 \, A^{3} a c^{4}\right )} \sqrt {\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} - 210 \, A B}{a c^{5}}}\right ) + {\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt {\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} - 210 \, A B}{a c^{5}}} \log \left (-{\left (194481 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (21 \, B a^{3} c^{8} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} + 2205 \, A B^{2} a^{2} c^{3} - 125 \, A^{3} a c^{4}\right )} \sqrt {\frac {a c^{5} \sqrt {-\frac {194481 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{3} c^{11}}} - 210 \, A B}{a c^{5}}}\right ) - 4 \, {\left (11 \, B c x^{3} + 9 \, A c x^{2} + 7 \, B a x + 5 \, A a\right )} \sqrt {x}}{64 \, {\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 293, normalized size = 0.92 \begin {gather*} -\frac {11 \, B c x^{\frac {7}{2}} + 9 \, A c x^{\frac {5}{2}} + 7 \, B a x^{\frac {3}{2}} + 5 \, A a \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} c^{2}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 21 \, \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 )}{64 \, a c^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 21 \, \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 )}{64 \, a c^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 21 \, \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 )}{128 \, a c^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 21 \, \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 )}{128 \, a c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 327, normalized size = 1.02 \begin {gather*} \frac {5 \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 )}{64 a \,c^{2}}+\frac {5 \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 )}{64 a \,c^{2}}+\frac {5 \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 )}{128 a \,c^{2}}+\frac {21 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{3}}+\frac {21 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{3}}+\frac {21 \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 )}{128 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{3}}+\frac {-\frac {11 B \,x^{\frac {7}{2}}}{16 c}-\frac {9 A \,x^{\frac {5}{2}}}{16 c}-\frac {7 B a \,x^{\frac {3}{2}}}{16 c^{2}}-\frac {5 A a \sqrt {x}}{16 c^{2}}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 292, normalized size = 0.91 \begin {gather*} -\frac {11 \, B c x^{\frac {7}{2}} + 9 \, A c x^{\frac {5}{2}} + 7 \, B a x^{\frac {3}{2}} + 5 \, A a \sqrt {x}}{16 \, {\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, B \sqrt {a} + 5 \, 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 (21 \, B \sqrt {a} + 5 \, 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 (21 \, B \sqrt {a} - 5 \, 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 (21 \, B \sqrt {a} - 5 \, 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}}}}{128 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 686, normalized size = 2.14 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {25\,A^2\,\sqrt {x}\,\sqrt {\frac {441\,B^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^2\,c^{11}}-\frac {25\,A^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^3\,c^{10}}-\frac {105\,A\,B}{2048\,a\,c^5}}}{32\,\left (\frac {525\,A^2\,B}{2048\,c^3}-\frac {9261\,B^3\,a}{2048\,c^4}+\frac {125\,A^3\,\sqrt {-a^3\,c^{11}}}{2048\,a^2\,c^8}-\frac {2205\,A\,B^2\,\sqrt {-a^3\,c^{11}}}{2048\,a\,c^9}\right )}-\frac {441\,B^2\,a\,\sqrt {x}\,\sqrt {\frac {441\,B^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^2\,c^{11}}-\frac {25\,A^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^3\,c^{10}}-\frac {105\,A\,B}{2048\,a\,c^5}}}{32\,\left (\frac {525\,A^2\,B}{2048\,c^2}-\frac {9261\,B^3\,a}{2048\,c^3}+\frac {125\,A^3\,\sqrt {-a^3\,c^{11}}}{2048\,a^2\,c^7}-\frac {2205\,A\,B^2\,\sqrt {-a^3\,c^{11}}}{2048\,a\,c^8}\right )}\right )\,\sqrt {-\frac {25\,A^2\,c\,\sqrt {-a^3\,c^{11}}-441\,B^2\,a\,\sqrt {-a^3\,c^{11}}+210\,A\,B\,a^2\,c^6}{4096\,a^3\,c^{11}}}-2\,\mathrm {atanh}\left (\frac {25\,A^2\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^3\,c^{10}}-\frac {105\,A\,B}{2048\,a\,c^5}-\frac {441\,B^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^2\,c^{11}}}}{32\,\left (\frac {525\,A^2\,B}{2048\,c^3}-\frac {9261\,B^3\,a}{2048\,c^4}-\frac {125\,A^3\,\sqrt {-a^3\,c^{11}}}{2048\,a^2\,c^8}+\frac {2205\,A\,B^2\,\sqrt {-a^3\,c^{11}}}{2048\,a\,c^9}\right )}-\frac {441\,B^2\,a\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^3\,c^{10}}-\frac {105\,A\,B}{2048\,a\,c^5}-\frac {441\,B^2\,\sqrt {-a^3\,c^{11}}}{4096\,a^2\,c^{11}}}}{32\,\left (\frac {525\,A^2\,B}{2048\,c^2}-\frac {9261\,B^3\,a}{2048\,c^3}-\frac {125\,A^3\,\sqrt {-a^3\,c^{11}}}{2048\,a^2\,c^7}+\frac {2205\,A\,B^2\,\sqrt {-a^3\,c^{11}}}{2048\,a\,c^8}\right )}\right )\,\sqrt {-\frac {441\,B^2\,a\,\sqrt {-a^3\,c^{11}}-25\,A^2\,c\,\sqrt {-a^3\,c^{11}}+210\,A\,B\,a^2\,c^6}{4096\,a^3\,c^{11}}}-\frac {\frac {9\,A\,x^{5/2}}{16\,c}+\frac {11\,B\,x^{7/2}}{16\,c}+\frac {5\,A\,a\,\sqrt {x}}{16\,c^2}+\frac {7\,B\,a\,x^{3/2}}{16\,c^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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