Optimal. Leaf size=287 \[ \frac {\left (\sqrt {a} B-3 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (\sqrt {a} B-3 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (\sqrt {a} B+3 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (\sqrt {a} B+3 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {823, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {\left (\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} a^{7/4} c^{3/4}}-\frac {\left (\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} a^{7/4} c^{3/4}}-\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 617
Rule 628
Rule 823
Rule 827
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^2} \, dx &=\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac {\int \frac {-\frac {3}{2} a A c-\frac {1}{2} a B c x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{2} a A c-\frac {1}{2} a B c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a^2 c}\\ &=\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac {\left (\sqrt {a} B-3 A \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 a^{3/2} c}+\frac {\left (\sqrt {a} B+3 A \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 a^{3/2} c}\\ &=\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )}+\frac {\left (\sqrt {a} B+3 A \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 a^{3/2} c}+\frac {\left (\sqrt {a} B+3 A \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 a^{3/2} c}+\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {\left (\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} a^{7/4} c^{3/4}}\\ &=\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )}+\frac {\left (\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} a^{7/4} c^{3/4}}-\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {\left (\sqrt {a} B+3 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 )}{4 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (\sqrt {a} B+3 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 )}{4 \sqrt {2} a^{7/4} c^{3/4}}\\ &=\frac {\sqrt {x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {\left (\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} a^{7/4} c^{3/4}}-\frac {\left (\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} a^{7/4} c^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 304, normalized size = 1.06 \begin {gather*} \frac {\frac {8 a A \sqrt {x}}{a+c x^2}-\frac {3 \sqrt {2} \sqrt [4]{a} A \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{\sqrt [4]{c}}+\frac {3 \sqrt {2} \sqrt [4]{a} A \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{\sqrt [4]{c}}-\frac {6 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {6 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}-\frac {4 (-a)^{3/4} B \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {4 (-a)^{3/4} B \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {8 a B x^{3/2}}{a+c x^2}}{16 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.68, size = 175, normalized size = 0.61 \begin {gather*} -\frac {\left (\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} a^{7/4} c^{3/4}}-\frac {\left (\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} a^{7/4} c^{3/4}}+\frac {A \sqrt {x}+B x^{3/2}}{2 a \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 877, normalized size = 3.06 \begin {gather*} \frac {{\left (a c x^{2} + a^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} + 6 \, A B}{a^{3} c}} \log \left (-{\left (B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (B a^{6} c^{2} \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} - 3 \, A B^{2} a^{3} c + 27 \, A^{3} a^{2} c^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} + 6 \, A B}{a^{3} c}}\right ) - {\left (a c x^{2} + a^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} + 6 \, A B}{a^{3} c}} \log \left (-{\left (B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (B a^{6} c^{2} \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} - 3 \, A B^{2} a^{3} c + 27 \, A^{3} a^{2} c^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} + 6 \, A B}{a^{3} c}}\right ) - {\left (a c x^{2} + a^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} - 6 \, A B}{a^{3} c}} \log \left (-{\left (B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (B a^{6} c^{2} \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} + 3 \, A B^{2} a^{3} c - 27 \, A^{3} a^{2} c^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} - 6 \, A B}{a^{3} c}}\right ) + {\left (a c x^{2} + a^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} - 6 \, A B}{a^{3} c}} \log \left (-{\left (B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (B a^{6} c^{2} \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} + 3 \, A B^{2} a^{3} c - 27 \, A^{3} a^{2} c^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {-\frac {B^{4} a^{2} - 18 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{7} c^{3}}} - 6 \, A B}{a^{3} c}}\right ) + 4 \, {\left (B x + A\right )} \sqrt {x}}{8 \, {\left (a c x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 273, normalized size = 0.95 \begin {gather*} \frac {B x^{\frac {3}{2}} + A \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} a} + \frac {\sqrt {2} {\left (3 \, \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 )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \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 )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \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 )}{16 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (3 \, \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 )}{16 \, a^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 313, normalized size = 1.09 \begin {gather*} \frac {B \,x^{\frac {3}{2}}}{2 \left (c \,x^{2}+a \right ) a}+\frac {A \sqrt {x}}{2 \left (c \,x^{2}+a \right ) a}+\frac {3 \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 )}{8 a^{2}}+\frac {3 \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 )}{8 a^{2}}+\frac {3 \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 )}{16 a^{2}}+\frac {\sqrt {2}\, B \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}} a c}+\frac {\sqrt {2}\, B \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}} a c}+\frac {\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 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.14, size = 256, normalized size = 0.89 \begin {gather*} \frac {B x^{\frac {3}{2}} + A \sqrt {x}}{2 \, {\left (a c x^{2} + a^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B \sqrt {a} + 3 \, 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} + 3 \, 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} - 3 \, 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} - 3 \, 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}}}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.28, size = 649, normalized size = 2.26 \begin {gather*} \frac {\frac {A\,\sqrt {x}}{2\,a}+\frac {B\,x^{3/2}}{2\,a}}{c\,x^2+a}-2\,\mathrm {atanh}\left (\frac {2\,B^2\,c^2\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^7\,c^3}}{64\,a^6\,c^3}-\frac {9\,A^2\,\sqrt {-a^7\,c^3}}{64\,a^7\,c^2}-\frac {3\,A\,B}{32\,a^3\,c}}}{\frac {B^3\,c}{4\,a}+\frac {3\,A\,B^2\,\sqrt {-a^7\,c^3}}{4\,a^5}-\frac {27\,A^3\,c\,\sqrt {-a^7\,c^3}}{4\,a^6}-\frac {9\,A^2\,B\,c^2}{4\,a^2}}-\frac {18\,A^2\,c^3\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^7\,c^3}}{64\,a^6\,c^3}-\frac {9\,A^2\,\sqrt {-a^7\,c^3}}{64\,a^7\,c^2}-\frac {3\,A\,B}{32\,a^3\,c}}}{\frac {B^3\,c}{4}+\frac {3\,A\,B^2\,\sqrt {-a^7\,c^3}}{4\,a^4}-\frac {27\,A^3\,c\,\sqrt {-a^7\,c^3}}{4\,a^5}-\frac {9\,A^2\,B\,c^2}{4\,a}}\right )\,\sqrt {-\frac {9\,A^2\,c\,\sqrt {-a^7\,c^3}-B^2\,a\,\sqrt {-a^7\,c^3}+6\,A\,B\,a^4\,c^2}{64\,a^7\,c^3}}-2\,\mathrm {atanh}\left (\frac {2\,B^2\,c^2\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^7\,c^3}}{64\,a^7\,c^2}-\frac {3\,A\,B}{32\,a^3\,c}-\frac {B^2\,\sqrt {-a^7\,c^3}}{64\,a^6\,c^3}}}{\frac {B^3\,c}{4\,a}-\frac {3\,A\,B^2\,\sqrt {-a^7\,c^3}}{4\,a^5}+\frac {27\,A^3\,c\,\sqrt {-a^7\,c^3}}{4\,a^6}-\frac {9\,A^2\,B\,c^2}{4\,a^2}}-\frac {18\,A^2\,c^3\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^7\,c^3}}{64\,a^7\,c^2}-\frac {3\,A\,B}{32\,a^3\,c}-\frac {B^2\,\sqrt {-a^7\,c^3}}{64\,a^6\,c^3}}}{\frac {B^3\,c}{4}-\frac {3\,A\,B^2\,\sqrt {-a^7\,c^3}}{4\,a^4}+\frac {27\,A^3\,c\,\sqrt {-a^7\,c^3}}{4\,a^5}-\frac {9\,A^2\,B\,c^2}{4\,a}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a^7\,c^3}-9\,A^2\,c\,\sqrt {-a^7\,c^3}+6\,A\,B\,a^4\,c^2}{64\,a^7\,c^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 77.78, size = 1294, normalized size = 4.51
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________