Integrand size = 24, antiderivative size = 409 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}} \]
[Out]
Time = 0.35 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 478, 584, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{10 b^4}+\frac {\sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (5 b c-17 a d) (b c-a d)^2}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (5 b c-17 a d) (b c-a d)^2}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}+\frac {\sqrt {x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac {d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {17 d^3 x^{13/2}}{26 b^2} \]
[In]
[Out]
Rule 210
Rule 217
Rule 477
Rule 478
Rule 584
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^8 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^4 \left (c+d x^4\right )^2 \left (5 c+17 d x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b} \\ & = -\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {(5 b c-17 a d) (b c-a d)^2}{b^4}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^4}{b^3}+\frac {d^2 (39 b c-17 a d) x^8}{b^2}+\frac {17 d^3 x^{12}}{b}+\frac {-5 a b^3 c^3+27 a^2 b^2 c^2 d-39 a^3 b c d^2+17 a^4 d^3}{b^4 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 b} \\ & = \frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (a (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b^5} \\ & = \frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^5}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^5} \\ & = \frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \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 )}{8 b^{11/2}}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \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 )}{8 b^{11/2}}+\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \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 )}{8 \sqrt {2} b^{21/4}}+\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \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 )}{8 \sqrt {2} b^{21/4}} \\ & = \frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}} \\ & = \frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.74 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-9945 a^4 d^3+117 a^3 b d^2 \left (195 c-68 d x^2\right )+13 a^2 b^2 d \left (-1215 c^2+1404 c d x^2+68 d^2 x^4\right )+a b^3 \left (2925 c^3-12636 c^2 d x^2-2028 c d^2 x^4-340 d^3 x^6\right )+12 b^4 x^2 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )\right )}{a+b x^2}-585 \sqrt {2} \sqrt [4]{a} (b c-a d)^2 (-5 b c+17 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+585 \sqrt {2} \sqrt [4]{a} (b c-a d)^2 (-5 b c+17 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4680 b^{21/4}} \]
[In]
[Out]
Time = 2.88 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {2 \left (-45 b^{3} d^{3} x^{6}+130 a \,b^{2} d^{3} x^{4}-195 b^{3} c \,d^{2} x^{4}-351 x^{2} a^{2} b \,d^{3}+702 x^{2} a \,b^{2} c \,d^{2}-351 x^{2} b^{3} c^{2} d +2340 a^{3} d^{3}-5265 a^{2} b c \,d^{2}+3510 a \,b^{2} c^{2} d -585 b^{3} c^{3}\right ) \sqrt {x}}{585 b^{5}}+\frac {a \left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (17 a d -5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{5}}\) | \(284\) |
derivativedivides | \(-\frac {2 \left (-\frac {d^{3} x^{\frac {13}{2}} b^{3}}{13}+\frac {2 a \,b^{2} d^{3} x^{\frac {9}{2}}}{9}-\frac {b^{3} c \,d^{2} x^{\frac {9}{2}}}{3}-\frac {3 a^{2} b \,d^{3} x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} c \,d^{2} x^{\frac {5}{2}}}{5}-\frac {3 b^{3} c^{2} d \,x^{\frac {5}{2}}}{5}+4 a^{3} d^{3} \sqrt {x}-9 a^{2} b c \,d^{2} \sqrt {x}+6 a \,b^{2} c^{2} d \sqrt {x}-b^{3} c^{3} \sqrt {x}\right )}{b^{5}}+\frac {2 a \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (17 a^{3} d^{3}-39 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{5}}\) | \(327\) |
default | \(-\frac {2 \left (-\frac {d^{3} x^{\frac {13}{2}} b^{3}}{13}+\frac {2 a \,b^{2} d^{3} x^{\frac {9}{2}}}{9}-\frac {b^{3} c \,d^{2} x^{\frac {9}{2}}}{3}-\frac {3 a^{2} b \,d^{3} x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} c \,d^{2} x^{\frac {5}{2}}}{5}-\frac {3 b^{3} c^{2} d \,x^{\frac {5}{2}}}{5}+4 a^{3} d^{3} \sqrt {x}-9 a^{2} b c \,d^{2} \sqrt {x}+6 a \,b^{2} c^{2} d \sqrt {x}-b^{3} c^{3} \sqrt {x}\right )}{b^{5}}+\frac {2 a \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (17 a^{3} d^{3}-39 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{5}}\) | \(327\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1822, normalized size of antiderivative = 4.45 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.22 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {x}}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{16 \, b^{5}} + \frac {2 \, {\left (45 \, b^{3} d^{3} x^{\frac {13}{2}} + 65 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{\frac {9}{2}} + 351 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {5}{2}} + 585 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \sqrt {x}\right )}}{585 \, b^{5}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.47 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\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 )}{8 \, b^{6}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\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 )}{8 \, b^{6}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{6}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{6}} + \frac {a b^{3} c^{3} \sqrt {x} - 3 \, a^{2} b^{2} c^{2} d \sqrt {x} + 3 \, a^{3} b c d^{2} \sqrt {x} - a^{4} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {2 \, {\left (45 \, b^{24} d^{3} x^{\frac {13}{2}} + 195 \, b^{24} c d^{2} x^{\frac {9}{2}} - 130 \, a b^{23} d^{3} x^{\frac {9}{2}} + 351 \, b^{24} c^{2} d x^{\frac {5}{2}} - 702 \, a b^{23} c d^{2} x^{\frac {5}{2}} + 351 \, a^{2} b^{22} d^{3} x^{\frac {5}{2}} + 585 \, b^{24} c^{3} \sqrt {x} - 3510 \, a b^{23} c^{2} d \sqrt {x} + 5265 \, a^{2} b^{22} c d^{2} \sqrt {x} - 2340 \, a^{3} b^{21} d^{3} \sqrt {x}\right )}}{585 \, b^{26}} \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 1850, normalized size of antiderivative = 4.52 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]