Optimal. Leaf size=401 \[ -\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left (\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right )}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.758133, antiderivative size = 401, 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 \[ -\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left (\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right )}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 115.602, size = 388, normalized size = 0.97 \[ \frac{x^{\frac{7}{2}} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} + \frac{x^{\frac{7}{2}} \left (a d - b c\right ) \left (a d + 15 b c\right )}{16 c^{2} d^{2} \left (c + d x^{2}\right )} - \frac{x^{\frac{3}{2}} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right )}{48 c^{2} d^{3}} + \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{5}{4}} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{5}{4}} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{5}{4}} d^{\frac{15}{4}}} + \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{5}{4}} d^{\frac{15}{4}}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.401856, size = 363, normalized size = 0.91 \[ \frac{\frac{24 d^{3/4} x^{3/2} \left (3 a^2 d^2-22 a b c d+19 b^2 c^2\right )}{c \left (c+d x^2\right )}-\frac{3 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{6 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{96 d^{3/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+256 b^2 d^{3/4} x^{3/2}}{384 d^{15/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]
[Out]
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Maple [A] time = 0.029, size = 562, normalized size = 1.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274627, size = 2140, normalized size = 5.34 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^3,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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.262009, size = 576, normalized size = 1.44 \[ \frac{2 \, b^{2} x^{\frac{3}{2}}}{3 \, d^{3}} + \frac{19 \, b^{2} c^{2} d x^{\frac{7}{2}} - 22 \, a b c d^{2} x^{\frac{7}{2}} + 3 \, a^{2} d^{3} x^{\frac{7}{2}} + 15 \, b^{2} c^{3} x^{\frac{3}{2}} - 14 \, a b c^{2} d x^{\frac{3}{2}} - a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c d^{3}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{6}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{6}} + \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{6}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^3,x, algorithm="giac")
[Out]