Integrand size = 20, antiderivative size = 315 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}-\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{7/4} c^{7/4}} \]
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Time = 0.20 (sec) , antiderivative size = 315, 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, 837, 841, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{7/4} c^{7/4}}-\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \]
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Rule 210
Rule 631
Rule 642
Rule 833
Rule 837
Rule 841
Rule 1176
Rule 1179
Rule 1182
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\int \frac {\frac {a A}{2}+\frac {3 a B x}{2}}{\sqrt {x} \left (a+c x^2\right )^2} \, dx}{4 a c} \\ & = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}-\frac {\int \frac {-\frac {3}{4} a^2 A c-\frac {3}{4} a^2 B c x}{\sqrt {x} \left (a+c x^2\right )} \, dx}{8 a^3 c^2} \\ & = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} a^2 A c-\frac {3}{4} a^2 B c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a^3 c^2} \\ & = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}-\frac {\left (3 \left (\sqrt {a} B-A \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} c^2}+\frac {\left (3 \left (\sqrt {a} B+A \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} c^2} \\ & = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}+\frac {\left (3 \left (\sqrt {a} B+A \sqrt {c}\right )\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^{3/2} c^2}+\frac {\left (3 \left (\sqrt {a} B+A \sqrt {c}\right )\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^{3/2} c^2}+\frac {\left (3 \left (\sqrt {a} B-A \sqrt {c}\right )\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^{7/4} c^{7/4}}+\frac {\left (3 \left (\sqrt {a} B-A \sqrt {c}\right )\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^{7/4} c^{7/4}} \\ & = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}+\frac {3 \left (\sqrt {a} B-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^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-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^{7/4} c^{7/4}}+\frac {\left (3 \left (\sqrt {a} B+A \sqrt {c}\right )\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^{7/4} c^{7/4}}-\frac {\left (3 \left (\sqrt {a} B+A \sqrt {c}\right )\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^{7/4} c^{7/4}} \\ & = -\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )}-\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B-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^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-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^{7/4} c^{7/4}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.57 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {4 a^{3/4} c^{3/4} \sqrt {x} \left (-a (3 A+B x)+c x^2 (A+3 B x)\right )}{\left (a+c x^2\right )^2}-3 \sqrt {2} \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )-3 \sqrt {2} \left (\sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{64 a^{7/4} c^{7/4}} \]
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Time = 0.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {3 B \,x^{\frac {7}{2}}}{16 a}+\frac {A \,x^{\frac {5}{2}}}{16 a}-\frac {B \,x^{\frac {3}{2}}}{16 c}-\frac {3 A \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}+\frac {3 B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a c}\) | \(271\) |
default | \(\frac {\frac {3 B \,x^{\frac {7}{2}}}{16 a}+\frac {A \,x^{\frac {5}{2}}}{16 a}-\frac {B \,x^{\frac {3}{2}}}{16 c}-\frac {3 A \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}+\frac {3 B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a c}\) | \(271\) |
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Leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (219) = 438\).
Time = 0.34 (sec) , antiderivative size = 982, normalized size of antiderivative = 3.12 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - A B^{2} a^{3} c^{2} + A^{3} a^{2} c^{3}\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}}\right ) - 3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - A B^{2} a^{3} c^{2} + A^{3} a^{2} c^{3}\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}}\right ) - 3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + A B^{2} a^{3} c^{2} - A^{3} a^{2} c^{3}\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}}\right ) + 3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + A B^{2} a^{3} c^{2} - A^{3} a^{2} c^{3}\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}}\right ) + 4 \, {\left (3 \, B c x^{3} + A c x^{2} - B a x - 3 \, A a\right )} \sqrt {x}}{64 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} \]
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Timed out. \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, B c x^{\frac {7}{2}} + A c x^{\frac {5}{2}} - B a x^{\frac {3}{2}} - 3 \, A a \sqrt {x}}{16 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{128 \, a c} \]
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Time = 0.28 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, B c x^{\frac {7}{2}} + A c x^{\frac {5}{2}} - B a x^{\frac {3}{2}} - 3 \, A a \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a c} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{2} c^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{2} c^{4}} \]
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Time = 10.15 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.21 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {A\,x^{5/2}}{16\,a}+\frac {3\,B\,x^{7/2}}{16\,a}-\frac {3\,A\,\sqrt {x}}{16\,c}-\frac {B\,x^{3/2}}{16\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}-\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}}}{32\,\left (\frac {27\,B^3}{2048\,a\,c^2}-\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^6\,c^4}-\frac {27\,A^2\,B}{2048\,a^2\,c}+\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^5}\right )}-\frac {9\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}-\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}}}{32\,\left (\frac {27\,B^3}{2048\,c^2}-\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^4}-\frac {27\,A^2\,B}{2048\,a\,c}+\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^4\,c^5}\right )}\right )\,\sqrt {-\frac {9\,\left (A^2\,c\,\sqrt {-a^7\,c^7}-B^2\,a\,\sqrt {-a^7\,c^7}+2\,A\,B\,a^4\,c^4\right )}{4096\,a^7\,c^7}}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}-\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}}}{32\,\left (\frac {27\,B^3}{2048\,a\,c^2}+\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^6\,c^4}-\frac {27\,A^2\,B}{2048\,a^2\,c}-\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^5}\right )}-\frac {9\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}-\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}}}{32\,\left (\frac {27\,B^3}{2048\,c^2}+\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^4}-\frac {27\,A^2\,B}{2048\,a\,c}-\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^4\,c^5}\right )}\right )\,\sqrt {-\frac {9\,\left (B^2\,a\,\sqrt {-a^7\,c^7}-A^2\,c\,\sqrt {-a^7\,c^7}+2\,A\,B\,a^4\,c^4\right )}{4096\,a^7\,c^7}} \]
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