Optimal. Leaf size=881 \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]
[Out]
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Rubi [A] time = 3.69103, antiderivative size = 881, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 4.51477, size = 729, normalized size = 0.83 \[ \frac{1}{640} \left (\frac{120 \sqrt{2} b^{17/4} (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{120 \sqrt{2} b^{17/4} (7 a d-3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (7 a d-3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{13/4} (b c-a d)^4}-\frac{320 b^5 x^{3/2}}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{1280 (3 a d+2 b c)}{a^3 c^4 \sqrt{x}}+\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{17/4} (b c-a d)^4}+\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{17/4} (b c-a d)^4}-\frac{256}{a^2 c^3 x^{5/2}}+\frac{40 d^4 x^{3/2} (37 b c-21 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{160 d^4 x^{3/2}}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Maple [A] time = 0.046, size = 1170, normalized size = 1.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(7/2)/(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(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(7/2)),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(1/x**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.5731, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(7/2)),x, algorithm="giac")
[Out]