Optimal. Leaf size=269 \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac{\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]
[Out]
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Rubi [A] time = 0.561258, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac{\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[x^2*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 49.6611, size = 280, normalized size = 1.04 \[ \frac{B x^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (12 B a c + \frac{7 b \left (14 A c - 9 B b\right )}{4} - \frac{5 c x \left (14 A c - 9 B b\right )}{2}\right )}{210 c^{3}} - \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (8 A a c^{2} - 14 A b^{2} c - 12 B a b c + 9 B b^{3}\right )}{384 c^{4}} + \frac{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (8 A a c^{2} - 14 A b^{2} c - 12 B a b c + 9 B b^{3}\right )}{1024 c^{5}} - \frac{\left (- 4 a c + b^{2}\right )^{2} \left (8 A a c^{2} - 14 A b^{2} c - 12 B a b c + 9 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.620383, size = 328, normalized size = 1.22 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (48 b^2 c^2 \left (343 a^2 B-2 a c x (63 A+31 B x)+2 c^2 x^3 (7 A+4 B x)\right )+32 b c^3 \left (-3 a^2 (189 A+73 B x)+6 a c x^2 (21 A+11 B x)+8 c^2 x^4 (91 A+75 B x)\right )+64 c^3 \left (-96 a^3 B+3 a^2 c x (35 A+16 B x)+2 a c^2 x^3 (245 A+192 B x)+40 c^3 x^5 (7 A+6 B x)\right )+28 b^4 c (c x (35 A+18 B x)-270 a B)+16 b^3 c^2 \left (7 a (95 A+39 B x)-c x^2 (49 A+27 B x)\right )-210 b^5 c (7 A+3 B x)+945 b^6 B\right )-105 \left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{215040 c^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.016, size = 838, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x+A)*(c*x^2+b*x+a)^(3/2),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((c*x^2 + b*x + a)^(3/2)*(B*x + A)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.41847, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28466, size = 570, normalized size = 2.12 \[ \frac{1}{107520} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B c x + \frac{15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac{3 \, B b^{2} c^{5} + 192 \, B a c^{6} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac{27 \, B b^{3} c^{4} - 132 \, B a b c^{5} - 42 \, A b^{2} c^{5} - 1960 \, A a c^{6}}{c^{6}}\right )} x + \frac{63 \, B b^{4} c^{3} - 372 \, B a b^{2} c^{4} - 98 \, A b^{3} c^{4} + 384 \, B a^{2} c^{5} + 504 \, A a b c^{5}}{c^{6}}\right )} x - \frac{315 \, B b^{5} c^{2} - 2184 \, B a b^{3} c^{3} - 490 \, A b^{4} c^{3} + 3504 \, B a^{2} b c^{4} + 3024 \, A a b^{2} c^{4} - 3360 \, A a^{2} c^{5}}{c^{6}}\right )} x + \frac{945 \, B b^{6} c - 7560 \, B a b^{4} c^{2} - 1470 \, A b^{5} c^{2} + 16464 \, B a^{2} b^{2} c^{3} + 10640 \, A a b^{3} c^{3} - 6144 \, B a^{3} c^{4} - 18144 \, A a^{2} b c^{4}}{c^{6}}\right )} + \frac{{\left (9 \, B b^{7} - 84 \, B a b^{5} c - 14 \, A b^{6} c + 240 \, B a^{2} b^{3} c^{2} + 120 \, A a b^{4} c^{2} - 192 \, B a^{3} b c^{3} - 288 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*x^2,x, algorithm="giac")
[Out]