Optimal. Leaf size=136 \[ 40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.232751, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ 40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 47.2282, size = 134, normalized size = 0.99 \[ 40 c^{\frac{3}{2}} d^{6} \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} + 80 c^{2} d^{6} \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} - \frac{40 c d^{6} \left (b + 2 c x\right )^{3}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d^{6} \left (b + 2 c x\right )^{5}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.277867, size = 141, normalized size = 1.04 \[ d^6 \left (-\frac{2 (b+2 c x) \left (-8 c^2 \left (15 a^2+20 a c x^2+3 c^2 x^4\right )+4 b^2 c \left (5 a+c x^2\right )-16 b c^2 x \left (10 a+3 c x^2\right )+b^4+28 b^3 c x\right )}{3 (a+x (b+c x))^{3/2}}-40 c^{3/2} \left (4 a c-b^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.036, size = 997, normalized size = 7.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),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((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.534342, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (30 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} +{\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \,{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \,{\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \,{\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac{2 \,{\left (60 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} +{\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \,{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt{-c} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) +{\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \,{\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \,{\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^(5/2),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((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.240516, size = 713, normalized size = 5.24 \[ -\frac{40 \,{\left (b^{2} c^{2} d^{6} - 4 \, a c^{3} d^{6}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} + \frac{2 \,{\left (2 \,{\left (2 \,{\left (2 \,{\left (3 \,{\left (\frac{2 \,{\left (b^{4} c^{8} d^{6} - 8 \, a b^{2} c^{9} d^{6} + 16 \, a^{2} c^{10} d^{6}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac{5 \,{\left (b^{5} c^{7} d^{6} - 8 \, a b^{3} c^{8} d^{6} + 16 \, a^{2} b c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac{5 \,{\left (b^{6} c^{6} d^{6} - 48 \, a^{2} b^{2} c^{8} d^{6} + 128 \, a^{3} c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{15 \,{\left (b^{7} c^{5} d^{6} - 16 \, a b^{5} c^{6} d^{6} + 80 \, a^{2} b^{3} c^{7} d^{6} - 128 \, a^{3} b c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{15 \,{\left (b^{8} c^{4} d^{6} - 12 \, a b^{6} c^{5} d^{6} + 40 \, a^{2} b^{4} c^{6} d^{6} - 128 \, a^{4} c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{b^{9} c^{3} d^{6} + 12 \, a b^{7} c^{4} d^{6} - 264 \, a^{2} b^{5} c^{5} d^{6} + 1280 \, a^{3} b^{3} c^{6} d^{6} - 1920 \, a^{4} b c^{7} d^{6}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]