Optimal. Leaf size=346 \[ \frac{(11 b c-3 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}+\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac{x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{7/2}}{7 d^2} \]
[Out]
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Rubi [A] time = 0.697139, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{(11 b c-3 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}+\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac{x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{7/2}}{7 d^2} \]
Antiderivative was successfully verified.
[In] Int[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 111.288, size = 321, normalized size = 0.93 \[ \frac{2 b^{2} x^{\frac{7}{2}}}{7 d^{2}} + \frac{x^{\frac{7}{2}} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} - \frac{x^{\frac{3}{2}} \left (a d - b c\right ) \left (3 a d - 11 b c\right )}{6 c d^{3}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 \sqrt [4]{c} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 \sqrt [4]{c} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 \sqrt [4]{c} d^{\frac{15}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 \sqrt [4]{c} d^{\frac{15}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.311673, size = 337, normalized size = 0.97 \[ \frac{\frac{21 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{c}}-\frac{21 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{c}}-\frac{42 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c}}+\frac{42 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{c}}-448 b d^{3/4} x^{3/2} (b c-a d)-\frac{168 d^{3/4} x^{3/2} (b c-a d)^2}{c+d x^2}+96 b^2 d^{7/4} x^{7/2}}{336 d^{15/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]
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Maple [A] time = 0.025, size = 523, normalized size = 1.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278114, size = 2029, normalized size = 5.86 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.246771, size = 558, normalized size = 1.61 \[ -\frac{b^{2} c^{2} x^{\frac{3}{2}} - 2 \, a b c d x^{\frac{3}{2}} + a^{2} d^{2} x^{\frac{3}{2}}}{2 \,{\left (d x^{2} + c\right )} d^{3}} + \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c d^{6}} + \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c d^{6}} - \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c d^{6}} + \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c d^{6}} + \frac{2 \,{\left (3 \, b^{2} d^{12} x^{\frac{7}{2}} - 14 \, b^{2} c d^{11} x^{\frac{3}{2}} + 14 \, a b d^{12} x^{\frac{3}{2}}\right )}}{21 \, d^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^2,x, algorithm="giac")
[Out]