Optimal. Leaf size=288 \[ \frac{\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}+\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{13/4}}+\frac{2 \sqrt{x} (b c-a d)^2}{d^3}-\frac{2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac{2 b^2 x^{9/2}}{9 d} \]
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Rubi [A] time = 0.506879, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}+\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{13/4}}+\frac{2 \sqrt{x} (b c-a d)^2}{d^3}-\frac{2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac{2 b^2 x^{9/2}}{9 d} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 97.4904, size = 270, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{9}{2}}}{9 d} + \frac{2 b x^{\frac{5}{2}} \left (2 a d - b c\right )}{5 d^{2}} + \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{13}{4}}} + \frac{2 \sqrt{x} \left (a d - b c\right )^{2}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.163061, size = 276, normalized size = 0.96 \[ \frac{-72 b d^{5/4} x^{5/2} (b c-2 a d)+360 \sqrt [4]{d} \sqrt{x} (b c-a d)^2+45 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-45 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+90 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )-90 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )+40 b^2 d^{9/4} x^{9/2}}{180 d^{13/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2),x]
[Out]
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Maple [B] time = 0.016, size = 495, normalized size = 1.7 \[{\frac{2\,{b}^{2}}{9\,d}{x}^{{\frac{9}{2}}}}+{\frac{4\,ab}{5\,d}{x}^{{\frac{5}{2}}}}-{\frac{2\,{b}^{2}c}{5\,{d}^{2}}{x}^{{\frac{5}{2}}}}+2\,{\frac{{a}^{2}\sqrt{x}}{d}}-4\,{\frac{abc\sqrt{x}}{{d}^{2}}}+2\,{\frac{{b}^{2}{c}^{2}\sqrt{x}}{{d}^{3}}}-{\frac{\sqrt{2}{a}^{2}}{2\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}abc}{{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}{b}^{2}{c}^{2}}{2\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}{a}^{2}}{4\,d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}abc}{2\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}{b}^{2}{c}^{2}}{4\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}{a}^{2}}{2\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}abc}{{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}{b}^{2}{c}^{2}}{2\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(b*x^2+a)^2/(d*x^2+c),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^(3/2)/(d*x^2 + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.259936, size = 1401, normalized size = 4.86 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(3/2)/(d*x^2 + c),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**(3/2)*(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.251715, size = 520, normalized size = 1.81 \[ -\frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{4}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{4}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{4}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{4}} + \frac{2 \,{\left (5 \, b^{2} d^{8} x^{\frac{9}{2}} - 9 \, b^{2} c d^{7} x^{\frac{5}{2}} + 18 \, a b d^{8} x^{\frac{5}{2}} + 45 \, b^{2} c^{2} d^{6} \sqrt{x} - 90 \, a b c d^{7} \sqrt{x} + 45 \, a^{2} d^{8} \sqrt{x}\right )}}{45 \, d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^(3/2)/(d*x^2 + c),x, algorithm="giac")
[Out]