Optimal. Leaf size=627 \[ \frac{\left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{\sqrt{a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{256 c^2 e^6}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{96 c e^4}-\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^7}-\frac{\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2} \]
[Out]
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Rubi [A] time = 2.51007, antiderivative size = 627, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{\sqrt{a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{256 c^2 e^6}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{96 c e^4}-\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^7}-\frac{\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d),x)
[Out]
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Mathematica [A] time = 2.00752, size = 637, normalized size = 1.02 \[ \frac{\frac{15 \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)+320 c^3 e^3 (b d-a e)^2 (a e-4 b d)+b^6 e^6+1024 c^6 d^6\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{5/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left (-16 c^3 e^2 \left (2 a^2 e^2 (368 d-165 e x)-2 a b e \left (975 d^2-446 d e x+283 e^2 x^2\right )+b^2 \left (1200 d^3-555 d^2 e x+356 d e^2 x^2-261 e^3 x^3\right )\right )+8 b c^2 e^3 \left (1066 a^2 e^2+2 a b e (341 e x-830 d)+b^2 \left (735 d^2-310 d e x+191 e^2 x^2\right )\right )+10 b^3 c e^4 (28 a e-12 b d+b e x)+64 c^4 e \left (a e \left (-280 d^3+135 d^2 e x-88 d e^2 x^2+65 e^3 x^3\right )+b \left (330 d^4-160 d^3 e x+105 d^2 e^2 x^2-78 d e^3 x^3+62 e^4 x^4\right )\right )-15 b^5 e^5-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )}{c^2}-7680 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^{5/2}+7680 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^{5/2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{7680 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),x]
[Out]
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Maple [B] time = 0.018, size = 6077, normalized size = 9.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),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((c*x^2 + b*x + a)^(5/2)*(2*c*x + b)/(e*x + d),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((c*x^2 + b*x + a)^(5/2)*(2*c*x + b)/(e*x + d),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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(2*c*x + b)/(e*x + d),x, algorithm="giac")
[Out]