Integrand size = 20, antiderivative size = 325 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}-\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} c^{9/4}}+\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} c^{9/4}}-\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{5/4} c^{9/4}}+\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{5/4} c^{9/4}} \]
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Time = 0.19 (sec) , antiderivative size = 325, 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 = {833, 835, 841, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} c^{9/4}}+\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{5/4} c^{9/4}}-\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{5/4} c^{9/4}}+\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{5/4} c^{9/4}}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}-\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]
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Rule 210
Rule 631
Rule 642
Rule 833
Rule 835
Rule 841
Rule 1176
Rule 1179
Rule 1182
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\int \frac {\sqrt {x} \left (\frac {3 a A}{2}+\frac {5 a B x}{2}\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c} \\ & = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}+\frac {\int \frac {\frac {5 a^2 B}{4}+\frac {3}{4} a A c x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2} \\ & = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {5 a^2 B}{4}+\frac {3}{4} a A c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 c^2} \\ & = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}-\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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 c^2}+\frac {\left (3 A+\frac {5 \sqrt {a} B}{\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 c^2} \\ & = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}+\frac {\left (3 A+\frac {5 \sqrt {a} B}{\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 c^2}+\frac {\left (3 A+\frac {5 \sqrt {a} B}{\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 c^2}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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^{5/4} c^{7/4}}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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^{5/4} c^{7/4}} \\ & = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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^{5/4} c^{7/4}}-\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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^{5/4} c^{7/4}}+\frac {\left (5 \sqrt {a} B+3 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^{5/4} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 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^{5/4} c^{9/4}} \\ & = -\frac {x^{3/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 a B-3 A c x)}{16 a c^2 \left (a+c x^2\right )}-\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 )}{32 \sqrt {2} a^{5/4} 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 )}{32 \sqrt {2} a^{5/4} c^{9/4}}+\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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^{5/4} c^{7/4}}-\frac {\left (3 A-\frac {5 \sqrt {a} B}{\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^{5/4} c^{7/4}} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.58 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {x} \left (5 a^2 B-3 A c^2 x^3+a c x (A+9 B x)\right )}{\left (a+c x^2\right )^2}-\sqrt {2} \left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+\sqrt {2} \left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{64 a^{5/4} c^{9/4}} \]
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Time = 0.15 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {3 A \,x^{\frac {7}{2}}}{16 a}-\frac {9 B \,x^{\frac {5}{2}}}{16 c}-\frac {A \,x^{\frac {3}{2}}}{16 c}-\frac {5 a B \sqrt {x}}{16 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {5 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {3 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 a \,c^{2}}\) | \(266\) |
default | \(\frac {\frac {3 A \,x^{\frac {7}{2}}}{16 a}-\frac {9 B \,x^{\frac {5}{2}}}{16 c}-\frac {A \,x^{\frac {3}{2}}}{16 c}-\frac {5 a B \sqrt {x}}{16 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {5 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {3 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 a \,c^{2}}\) | \(266\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (229) = 458\).
Time = 0.37 (sec) , antiderivative size = 1031, normalized size of antiderivative = 3.17 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {{\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} \sqrt {-\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} + 30 \, A B}{a^{2} c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, A a^{4} c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} + 125 \, B^{3} a^{3} c^{2} - 45 \, A^{2} B a^{2} c^{3}\right )} \sqrt {-\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} + 30 \, A B}{a^{2} c^{4}}}\right ) - {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} \sqrt {-\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} + 30 \, A B}{a^{2} c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, A a^{4} c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} + 125 \, B^{3} a^{3} c^{2} - 45 \, A^{2} B a^{2} c^{3}\right )} \sqrt {-\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} + 30 \, A B}{a^{2} c^{4}}}\right ) - {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} \sqrt {\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} - 30 \, A B}{a^{2} c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, A a^{4} c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} - 125 \, B^{3} a^{3} c^{2} + 45 \, A^{2} B a^{2} c^{3}\right )} \sqrt {\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} - 30 \, A B}{a^{2} c^{4}}}\right ) + {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} \sqrt {\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} - 30 \, A B}{a^{2} c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, A a^{4} c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} - 125 \, B^{3} a^{3} c^{2} + 45 \, A^{2} B a^{2} c^{3}\right )} \sqrt {\frac {a^{2} c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a^{5} c^{9}}} - 30 \, A B}{a^{2} c^{4}}}\right ) - 4 \, {\left (3 \, A c^{2} x^{3} - 9 \, B a c x^{2} - A a c x - 5 \, B a^{2}\right )} \sqrt {x}}{64 \, {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}} \]
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Timed out. \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.96 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, A c^{2} x^{\frac {7}{2}} - 9 \, B a c x^{\frac {5}{2}} - A a c x^{\frac {3}{2}} - 5 \, B a^{2} \sqrt {x}}{16 \, {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a \sqrt {c} + 3 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (5 \, B a \sqrt {c} + 3 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (5 \, B a \sqrt {c} - 3 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (5 \, B a \sqrt {c} - 3 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{128 \, a c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.92 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{2} c^{4}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{2} c^{4}} + \frac {3 \, A c^{2} x^{\frac {7}{2}} - 9 \, B a c x^{\frac {5}{2}} - A a c x^{\frac {3}{2}} - 5 \, B a^{2} \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a c^{2}} \]
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Time = 10.22 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.14 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=2\,\mathrm {atanh}\left (\frac {25\,B^2\,\sqrt {x}\,\sqrt {\frac {25\,B^2\,\sqrt {-a^5\,c^9}}{4096\,a^4\,c^9}-\frac {9\,A^2\,\sqrt {-a^5\,c^9}}{4096\,a^5\,c^8}-\frac {15\,A\,B}{2048\,a^2\,c^4}}}{32\,\left (\frac {27\,A^3}{2048\,a^2\,c}-\frac {75\,A\,B^2}{2048\,a\,c^2}+\frac {125\,B^3\,\sqrt {-a^5\,c^9}}{2048\,a^3\,c^7}-\frac {45\,A^2\,B\,\sqrt {-a^5\,c^9}}{2048\,a^4\,c^6}\right )}+\frac {9\,A^2\,\sqrt {x}\,\sqrt {\frac {25\,B^2\,\sqrt {-a^5\,c^9}}{4096\,a^4\,c^9}-\frac {9\,A^2\,\sqrt {-a^5\,c^9}}{4096\,a^5\,c^8}-\frac {15\,A\,B}{2048\,a^2\,c^4}}}{32\,\left (\frac {75\,A\,B^2}{2048\,c^3}-\frac {27\,A^3}{2048\,a\,c^2}-\frac {125\,B^3\,\sqrt {-a^5\,c^9}}{2048\,a^2\,c^8}+\frac {45\,A^2\,B\,\sqrt {-a^5\,c^9}}{2048\,a^3\,c^7}\right )}\right )\,\sqrt {-\frac {9\,A^2\,c\,\sqrt {-a^5\,c^9}-25\,B^2\,a\,\sqrt {-a^5\,c^9}+30\,A\,B\,a^3\,c^5}{4096\,a^5\,c^9}}+2\,\mathrm {atanh}\left (\frac {25\,B^2\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^5\,c^9}}{4096\,a^5\,c^8}-\frac {15\,A\,B}{2048\,a^2\,c^4}-\frac {25\,B^2\,\sqrt {-a^5\,c^9}}{4096\,a^4\,c^9}}}{32\,\left (\frac {27\,A^3}{2048\,a^2\,c}-\frac {75\,A\,B^2}{2048\,a\,c^2}-\frac {125\,B^3\,\sqrt {-a^5\,c^9}}{2048\,a^3\,c^7}+\frac {45\,A^2\,B\,\sqrt {-a^5\,c^9}}{2048\,a^4\,c^6}\right )}+\frac {9\,A^2\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^5\,c^9}}{4096\,a^5\,c^8}-\frac {15\,A\,B}{2048\,a^2\,c^4}-\frac {25\,B^2\,\sqrt {-a^5\,c^9}}{4096\,a^4\,c^9}}}{32\,\left (\frac {75\,A\,B^2}{2048\,c^3}-\frac {27\,A^3}{2048\,a\,c^2}+\frac {125\,B^3\,\sqrt {-a^5\,c^9}}{2048\,a^2\,c^8}-\frac {45\,A^2\,B\,\sqrt {-a^5\,c^9}}{2048\,a^3\,c^7}\right )}\right )\,\sqrt {-\frac {25\,B^2\,a\,\sqrt {-a^5\,c^9}-9\,A^2\,c\,\sqrt {-a^5\,c^9}+30\,A\,B\,a^3\,c^5}{4096\,a^5\,c^9}}-\frac {\frac {A\,x^{3/2}}{16\,c}-\frac {3\,A\,x^{7/2}}{16\,a}+\frac {9\,B\,x^{5/2}}{16\,c}+\frac {5\,B\,a\,\sqrt {x}}{16\,c^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \]
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