Optimal. Leaf size=326 \[ \frac{2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}+\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{17/4}}+\frac{2 \sqrt{x} (b c-a d)^3}{b^4}+\frac{2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac{2 d^3 x^{13/2}}{13 b} \]
[Out]
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Rubi [A] time = 0.575506, antiderivative size = 326, 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 \[ \frac{2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}+\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{17/4}}+\frac{2 \sqrt{x} (b c-a d)^3}{b^4}+\frac{2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac{2 d^3 x^{13/2}}{13 b} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
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Rubi in Sympy [A] time = 105.181, size = 309, normalized size = 0.95 \[ - \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{17}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{17}{4}}} + \frac{2 d^{3} x^{\frac{13}{2}}}{13 b} - \frac{2 d^{2} x^{\frac{9}{2}} \left (a d - 3 b c\right )}{9 b^{2}} + \frac{2 d x^{\frac{5}{2}} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{5 b^{3}} - \frac{2 \sqrt{x} \left (a d - b c\right )^{3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.219743, size = 314, normalized size = 0.96 \[ \frac{936 b^{5/4} d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+520 b^{9/4} d^2 x^{9/2} (3 b c-a d)+4680 \sqrt [4]{b} \sqrt{x} (b c-a d)^3-585 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+585 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1170 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+1170 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+360 b^{13/4} d^3 x^{13/2}}{2340 b^{17/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
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Maple [B] time = 0.015, size = 712, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(d*x^2+c)^3/(b*x^2+a),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((d*x^2 + c)^3*x^(3/2)/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264048, size = 2106, normalized size = 6.46 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^(3/2)/(b*x^2 + a),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)*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.317018, size = 717, normalized size = 2.2 \[ -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \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 )}{2 \, b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \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 )}{2 \, b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{5}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{5}} + \frac{2 \,{\left (45 \, b^{12} d^{3} x^{\frac{13}{2}} + 195 \, b^{12} c d^{2} x^{\frac{9}{2}} - 65 \, a b^{11} d^{3} x^{\frac{9}{2}} + 351 \, b^{12} c^{2} d x^{\frac{5}{2}} - 351 \, a b^{11} c d^{2} x^{\frac{5}{2}} + 117 \, a^{2} b^{10} d^{3} x^{\frac{5}{2}} + 585 \, b^{12} c^{3} \sqrt{x} - 1755 \, a b^{11} c^{2} d \sqrt{x} + 1755 \, a^{2} b^{10} c d^{2} \sqrt{x} - 585 \, a^{3} b^{9} d^{3} \sqrt{x}\right )}}{585 \, b^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^(3/2)/(b*x^2 + a),x, algorithm="giac")
[Out]