Integrand size = 20, antiderivative size = 331 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {835, 837, 841, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]
[In]
[Out]
Rule 210
Rule 631
Rule 642
Rule 835
Rule 837
Rule 841
Rule 1176
Rule 1179
Rule 1182
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {\frac {a B}{2}+\frac {5 A c x}{2}}{\sqrt {x} \left (a+c x^2\right )^2} \, dx}{4 a c} \\ & = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\int \frac {-\frac {3}{4} a^2 B c-\frac {5}{4} a A c^2 x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^3 c^2} \\ & = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} a^2 B c-\frac {5}{4} a A c^2 x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^3 c^2} \\ & = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^2 c^{3/2}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^2 c^{3/2}} \\ & = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \text {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 a^2 c^{3/2}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \text {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 a^2 c^{3/2}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \text {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^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \text {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^{9/4} c^{5/4}} \\ & = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 \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^{9/4} c^{5/4}}+\frac {\left (3 \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^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \text {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^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \text {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^{9/4} c^{5/4}} \\ & = -\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 \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^{9/4} c^{5/4}}+\frac {\left (3 \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^{9/4} c^{5/4}}-\frac {\left (3 \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^{9/4} c^{5/4}}+\frac {\left (3 \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^{9/4} c^{5/4}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {x} \left (-3 a^2 B+5 A c^2 x^3+a c x (9 A+B x)\right )}{\left (a+c x^2\right )^2}-\sqrt {2} \left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+\sqrt {2} \left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{64 a^{9/4} c^{5/4}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {5 A c \,x^{\frac {7}{2}}}{16 a^{2}}+\frac {B \,x^{\frac {5}{2}}}{16 a}+\frac {9 A \,x^{\frac {3}{2}}}{16 a}-\frac {3 B \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {5 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 a^{2} c}\) | \(266\) |
default | \(\frac {\frac {5 A c \,x^{\frac {7}{2}}}{16 a^{2}}+\frac {B \,x^{\frac {5}{2}}}{16 a}+\frac {9 A \,x^{\frac {3}{2}}}{16 a}-\frac {3 B \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {5 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 a^{2} c}\) | \(266\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (235) = 470\).
Time = 0.36 (sec) , antiderivative size = 1022, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 27 \, B^{3} a^{4} c - 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}}\right ) - {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 27 \, B^{3} a^{4} c - 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}}\right ) - {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 27 \, B^{3} a^{4} c + 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}}\right ) + {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 27 \, B^{3} a^{4} c + 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}}\right ) - 4 \, {\left (5 \, A c^{2} x^{3} + B a c x^{2} + 9 \, A a c x - 3 \, B a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {5 \, A c^{2} x^{\frac {7}{2}} + B a c x^{\frac {5}{2}} + 9 \, A a c x^{\frac {3}{2}} - 3 \, B a^{2} \sqrt {x}}{16 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, B a \sqrt {c} - 5 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, B a \sqrt {c} - 5 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{128 \, a^{2} c} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} + \frac {5 \, A c^{2} x^{\frac {7}{2}} + B a c x^{\frac {5}{2}} + 9 \, A a c x^{\frac {3}{2}} - 3 \, B a^{2} \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{2} c} \]
[In]
[Out]
Time = 10.04 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {9\,A\,x^{3/2}}{16\,a}+\frac {B\,x^{5/2}}{16\,a}-\frac {3\,B\,\sqrt {x}}{16\,c}+\frac {5\,A\,c\,x^{7/2}}{16\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}-\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a^2}-\frac {125\,A^3\,c}{2048\,a^3}+\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^3}-\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^7\,c^2}\right )}-\frac {25\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}-\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a}-\frac {125\,A^3\,c}{2048\,a^2}+\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^5\,c^3}-\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^2}\right )}\right )\,\sqrt {-\frac {9\,B^2\,a\,\sqrt {-a^9\,c^5}-25\,A^2\,c\,\sqrt {-a^9\,c^5}+30\,A\,B\,a^5\,c^3}{4096\,a^9\,c^5}}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}-\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a^2}-\frac {125\,A^3\,c}{2048\,a^3}-\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^3}+\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^7\,c^2}\right )}-\frac {25\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}-\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a}-\frac {125\,A^3\,c}{2048\,a^2}-\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^5\,c^3}+\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^2}\right )}\right )\,\sqrt {-\frac {25\,A^2\,c\,\sqrt {-a^9\,c^5}-9\,B^2\,a\,\sqrt {-a^9\,c^5}+30\,A\,B\,a^5\,c^3}{4096\,a^9\,c^5}} \]
[In]
[Out]