Optimal. Leaf size=189 \[ -\frac{2 \left (x \left (24 a^2 B c^2+8 a A b c^2-22 a b^2 B c+3 b^4 B\right )+a \left (16 a A c^2-20 a b B c+3 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]
[Out]
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Rubi [A] time = 0.352753, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 \left (x \left (24 a^2 B c^2+8 a A b c^2-22 a b^2 B c+3 b^4 B\right )+a \left (16 a A c^2-20 a b B c+3 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x))/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 37.55, size = 192, normalized size = 1.02 \[ \frac{B \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} + \frac{2 x^{2} \left (a \left (2 A c - B b\right ) - x \left (- A b c - 2 B a c + B b^{2}\right )\right )}{3 c \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \left (\frac{a \left (16 A a c^{2} - 20 B a b c + 3 B b^{3}\right )}{2} + x \left (4 A a b c^{2} + 12 B a^{2} c^{2} - 11 B a b^{2} c + \frac{3 B b^{4}}{2}\right )\right )}{3 c^{2} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x+A)/(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.640398, size = 234, normalized size = 1.24 \[ \frac{-\frac{2 \left (a^2 c (2 c (A+B x)-3 b B)+a b \left (-b c (A+4 B x)+3 A c^2 x+b^2 B\right )+b^3 x (b B-A c)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}}-\frac{2 \left (8 a^2 c^3 (3 A+4 B x)+b^3 c (10 a B-A c x)-2 a b^2 c^2 (3 A+14 B x)-4 a b c^2 (8 a B-3 A c x)+b^4 c (A+4 B x)+b^5 (-B)\right )}{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)}}+3 B \sqrt{c} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x))/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.014, size = 860, normalized size = 4.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x+A)/(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((B*x + A)*x^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.491863, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^3/(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(x**3*(B*x+A)/(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.289606, size = 424, normalized size = 2.24 \[ -\frac{2 \,{\left ({\left ({\left (\frac{{\left (4 \, B b^{4} c - 28 \, B a b^{2} c^{2} - A b^{3} c^{2} + 32 \, B a^{2} c^{3} + 12 \, A a b c^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (B b^{5} - 6 \, B a b^{3} c + 2 \, A a b^{2} c^{2} + 8 \, A a^{2} c^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (B a b^{4} - 7 \, B a^{2} b^{2} c + 4 \, B a^{3} c^{2} + 4 \, A a^{2} b c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \, B a^{2} b^{3} - 20 \, B a^{3} b c + 16 \, A a^{3} c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} - \frac{B{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]