Integrand size = 20, antiderivative size = 320 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}} \]
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Time = 0.21 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {833, 841, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{3/4} c^{11/4}}+\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{3/4} c^{11/4}}-\frac {\sqrt {x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac {x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]
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Rule 210
Rule 631
Rule 642
Rule 833
Rule 841
Rule 1176
Rule 1179
Rule 1182
Rubi steps \begin{align*} \text {integral}& = -\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 {\text {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 ) \text {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 ) \text {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 ) \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 c^3}+\frac {\left (21 B+\frac {5 A \sqrt {c}}{\sqrt {a}}\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 c^3}+\frac {\left (21 \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^{3/4} c^{11/4}}+\frac {\left (21 \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^{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 ) \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^{3/4} c^{11/4}}-\frac {\left (21 \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^{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*}
Time = 0.77 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.58 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {-\frac {4 c^{3/4} \sqrt {x} \left (5 a A+7 a B x+9 A c x^2+11 B c x^3\right )}{\left (a+c x^2\right )^2}-\frac {\sqrt {2} \left (21 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{a^{3/4}}-\frac {\sqrt {2} \left (21 \sqrt {a} B-5 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{a^{3/4}}}{64 c^{11/4}} \]
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Time = 0.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {11 B \,x^{\frac {7}{2}}}{16 c}-\frac {9 A \,x^{\frac {5}{2}}}{16 c}-\frac {7 B a \,x^{\frac {3}{2}}}{16 c^{2}}-\frac {5 a A \sqrt {x}}{16 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {5 A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {21 B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 c^{2}}\) | \(270\) |
default | \(\frac {-\frac {11 B \,x^{\frac {7}{2}}}{16 c}-\frac {9 A \,x^{\frac {5}{2}}}{16 c}-\frac {7 B a \,x^{\frac {3}{2}}}{16 c^{2}}-\frac {5 a A \sqrt {x}}{16 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {5 A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {21 B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 c^{2}}\) | \(270\) |
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Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (224) = 448\).
Time = 0.74 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.08 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\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 )}} \]
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Timed out. \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.91 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {11 \, B c x^{\frac {7}{2}} + 9 \, A c x^{\frac {5}{2}} + 7 \, B a x^{\frac {3}{2}} + 5 \, A a \sqrt {x}}{16 \, {\left (c^{4} x^{4} + 2 \, a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, B \sqrt {a} + 5 \, A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (21 \, B \sqrt {a} + 5 \, A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (21 \, B \sqrt {a} - 5 \, A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (21 \, B \sqrt {a} - 5 \, A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{128 \, c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.92 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {11 \, B c x^{\frac {7}{2}} + 9 \, A c x^{\frac {5}{2}} + 7 \, B a x^{\frac {3}{2}} + 5 \, A a \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} c^{2}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 21 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a c^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 21 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a c^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 21 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a c^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 21 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a c^{5}} \]
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Time = 10.21 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.14 \[ \int \frac {x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-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} \]
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