Optimal. Leaf size=618 \[ -\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{13/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2} \]
[Out]
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Rubi [A]
time = 0.65, antiderivative size = 618, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 483,
597, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{13/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}+\frac {-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt {x} (b c-a d)}+\frac {d^{9/4} (13 b c-9 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac {d}{2 c x^{5/2} \left (c+d x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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[Out]
Rule 210
Rule 303
Rule 477
Rule 483
Rule 597
Rule 598
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {4 b c-9 a d-9 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {5 \left (4 b^2 c^2+4 a b c d-9 a^2 d^2\right )+5 b d (4 b c-9 a d) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{10 a c^2 (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (5 \left (4 b^3 c^3+4 a b^2 c^2 d+4 a^2 b c d^2-9 a^3 d^3\right )+5 b d \left (4 b^2 c^2+4 a b c d-9 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{10 a^2 c^3 (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {20 b^4 c^3 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {5 a^2 d^3 (-13 b c+9 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{10 a^2 c^3 (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^2}-\frac {\left (d^3 (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^3 (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^2}+\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^2}+\frac {\left (d^{5/2} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 (b c-a d)^2}-\frac {\left (d^{5/2} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {b^3 \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 )}{2 a^2 (b c-a d)^2}+\frac {b^3 \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 )}{2 a^2 (b c-a d)^2}+\frac {b^{13/4} \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 )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \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 )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {\left (d^2 (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^3 (b c-a d)^2}-\frac {\left (d^2 (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^3 (b c-a d)^2}-\frac {\left (d^{9/4} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {\left (d^{9/4} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}-\frac {\left (d^{9/4} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {\left (d^{9/4} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{13/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 1.28, size = 378, normalized size = 0.61 \begin {gather*} \frac {1}{40} \left (-\frac {4 \left (20 b^2 c^2 x^2 \left (c+d x^2\right )+a^2 d \left (4 c^2-36 c d x^2-45 d^2 x^4\right )-4 a b c \left (c^2-4 c d x^2-5 d^2 x^4\right )\right )}{a^2 c^3 (-b c+a d) x^{5/2} \left (c+d x^2\right )}-\frac {20 \sqrt {2} b^{13/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} (b c-a d)^2}+\frac {5 \sqrt {2} d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{13/4} (b c-a d)^2}-\frac {20 \sqrt {2} b^{13/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} (b c-a d)^2}+\frac {5 \sqrt {2} d^{9/4} (13 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{13/4} (b c-a d)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 306, normalized size = 0.50
method | result | size |
derivativedivides | \(\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3} \left (a d -b c \right )^{2}}-\frac {2}{5 c^{2} a \,x^{\frac {5}{2}}}-\frac {2 \left (-2 a d -b c \right )}{a^{2} c^{3} \sqrt {x}}\) | \(306\) |
default | \(\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3} \left (a d -b c \right )^{2}}-\frac {2}{5 c^{2} a \,x^{\frac {5}{2}}}-\frac {2 \left (-2 a d -b c \right )}{a^{2} c^{3} \sqrt {x}}\) | \(306\) |
risch | \(-\frac {2 \left (-10 a d \,x^{2}-5 c \,x^{2} b +a c \right )}{5 a^{2} c^{3} x^{\frac {5}{2}}}+\frac {b^{3} \sqrt {2}\, \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 )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {b^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {b^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {a \,d^{4} x^{\frac {3}{2}}}{2 c^{3} \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}-\frac {d^{3} x^{\frac {3}{2}} b}{2 c^{2} \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {9 a \,d^{3} \sqrt {2}\, \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )}{16 c^{3} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 a \,d^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 c^{3} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 a \,d^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 c^{3} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {13 d^{2} \sqrt {2}\, b \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )}{16 c^{2} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {13 d^{2} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 c^{2} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {13 d^{2} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 c^{2} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(606\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 551, normalized size = 0.89 \begin {gather*} \frac {b^{4} {\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 )}}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac {{\left (13 \, b c d^{3} - 9 \, a d^{4}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {4 \, a b c^{3} - 4 \, a^{2} c^{2} d - 5 \, {\left (4 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 9 \, a^{2} d^{3}\right )} x^{4} - 4 \, {\left (5 \, b^{2} c^{3} + 4 \, a b c^{2} d - 9 \, a^{2} c d^{2}\right )} x^{2}}{10 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{\frac {9}{2}} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{\frac {5}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3728 vs.
\(2 (467) = 934\).
time = 100.56, size = 3728, normalized size = 6.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.81, size = 715, normalized size = 1.16 \begin {gather*} -\frac {d^{3} x^{\frac {3}{2}}}{2 \, {\left (b c^{4} - a c^{3} d\right )} {\left (d x^{2} + c\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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 )}{\sqrt {2} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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 )}{\sqrt {2} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} + \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} + \frac {2 \, {\left (5 \, b c x^{2} + 10 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.00, size = 2500, normalized size = 4.05 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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