Integrand size = 22, antiderivative size = 293 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {3 (A b+7 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (A b+7 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}} \]
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Time = 0.16 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {468, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {3 (7 a B+A b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (7 a B+A b) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (7 a B+A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (7 a B+A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {x^{3/2} (7 a B+A b)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
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Rule 210
Rule 294
Rule 303
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {A b}{2}+\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b+7 a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a b^2} \\ & = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a b^2} \\ & = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a b^{5/2}}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a b^{5/2}} \\ & = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a b^3}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a b^3}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{5/4} b^{11/4}} \\ & = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {3 (A b+7 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (A b+7 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}+\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}-\frac {(3 (A b+7 a B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}} \\ & = \frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {3 (A b+7 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (A b+7 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.58 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (7 a^2 B-3 A b^2 x^2+a b \left (A+11 B x^2\right )\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} (A b+7 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} (A b+7 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{5/4} b^{11/4}} \]
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Time = 2.74 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57
method | result | size |
derivativedivides | \(\frac {\frac {\left (3 A b -11 B a \right ) x^{\frac {7}{2}}}{16 a b}-\frac {\left (A b +7 B a \right ) x^{\frac {3}{2}}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (A b +7 B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{3} a \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(166\) |
default | \(\frac {\frac {\left (3 A b -11 B a \right ) x^{\frac {7}{2}}}{16 a b}-\frac {\left (A b +7 B a \right ) x^{\frac {3}{2}}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (A b +7 B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{3} a \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(166\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.97 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \log \left (27 \, a^{4} b^{8} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} a^{3} + 147 \, A B^{2} a^{2} b + 21 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) - 3 \, {\left (i \, a b^{4} x^{4} + 2 i \, a^{2} b^{3} x^{2} + i \, a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \log \left (27 i \, a^{4} b^{8} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} a^{3} + 147 \, A B^{2} a^{2} b + 21 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) - 3 \, {\left (-i \, a b^{4} x^{4} - 2 i \, a^{2} b^{3} x^{2} - i \, a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \log \left (-27 i \, a^{4} b^{8} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} a^{3} + 147 \, A B^{2} a^{2} b + 21 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) - 3 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \log \left (-27 \, a^{4} b^{8} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} a^{3} + 147 \, A B^{2} a^{2} b + 21 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (11 \, B a b - 3 \, A b^{2}\right )} x^{3} + {\left (7 \, B a^{2} + A a b\right )} x\right )} \sqrt {x}}{64 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
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Timed out. \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
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none
Time = 0.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.86 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (11 \, B a b - 3 \, A b^{2}\right )} x^{\frac {7}{2}} + {\left (7 \, B a^{2} + A a b\right )} x^{\frac {3}{2}}}{16 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {3 \, {\left (7 \, B a + A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {11 \, B a b x^{\frac {7}{2}} - 3 \, A b^{2} x^{\frac {7}{2}} + 7 \, B a^{2} x^{\frac {3}{2}} + A a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{5}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.42 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+7\,B\,a\right )}{32\,{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+7\,B\,a\right )}{32\,{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {\frac {x^{3/2}\,\left (A\,b+7\,B\,a\right )}{16\,b^2}-\frac {x^{7/2}\,\left (3\,A\,b-11\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \]
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