Optimal. Leaf size=333 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}+\frac{5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
[Out]
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Rubi [A] time = 0.662723, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}+\frac{5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
Antiderivative was successfully verified.
[In] Int[x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 67.1689, size = 354, normalized size = 1.06 \[ \frac{B x^{2} \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (16 B a c + \frac{9 b \left (18 A c - 11 B b\right )}{4} - \frac{7 c x \left (18 A c - 11 B b\right )}{2}\right )}{504 c^{3}} - \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right )}{768 c^{4}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right )}{12288 c^{5}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right )}{32768 c^{6}} + \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{65536 c^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 1.09158, size = 481, normalized size = 1.44 \[ \frac{315 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^4 c^2 \left (10143 a^2 B-3 a c x (791 A+372 B x)+2 c^2 x^3 (81 A+44 B x)\right )-32 b^3 c^3 \left (3 a^2 (2359 A+879 B x)-4 a c x^2 (213 A+107 B x)+8 c^2 x^4 (9 A+5 B x)\right )-192 b^2 c^3 \left (1221 a^3 B-a^2 c x (597 A+266 B x)+4 a c^2 x^3 (27 A+14 B x)+8 c^3 x^5 (243 A+206 B x)\right )-128 b c^4 \left (-13 a^3 (153 A+53 B x)+6 a^2 c x^2 (87 A+41 B x)+24 a c^2 x^4 (307 A+251 B x)+16 c^3 x^6 (297 A+259 B x)\right )-256 c^4 \left (-256 a^4 B+a^3 c x (315 A+128 B x)+6 a^2 c^2 x^3 (413 A+320 B x)+8 a c^3 x^5 (357 A+304 B x)+112 c^4 x^7 (9 A+8 B x)\right )+84 b^6 c (c x (45 A+22 B x)-485 a B)+72 b^5 c^2 \left (7 a (125 A+49 B x)-2 c x^2 (21 A+11 B x)\right )-210 b^7 c (27 A+11 B x)+3465 b^8 B\right )}{4128768 c^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.017, size = 1277, normalized size = 3.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x+A)*(c*x^2+b*x+a)^(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((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.516283, 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)^(5/2)*(B*x + A)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.291733, size = 868, normalized size = 2.61 \[ \frac{1}{2064384} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, B c^{2} x + \frac{37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac{309 \, B b^{2} c^{8} + 608 \, B a c^{9} + 594 \, A b c^{9}}{c^{8}}\right )} x + \frac{5 \, B b^{3} c^{7} + 3012 \, B a b c^{8} + 1458 \, A b^{2} c^{8} + 2856 \, A a c^{9}}{c^{8}}\right )} x - \frac{11 \, B b^{4} c^{6} - 84 \, B a b^{2} c^{7} - 18 \, A b^{3} c^{7} - 3840 \, B a^{2} c^{8} - 7368 \, A a b c^{8}}{c^{8}}\right )} x + \frac{99 \, B b^{5} c^{5} - 856 \, B a b^{3} c^{6} - 162 \, A b^{4} c^{6} + 1968 \, B a^{2} b c^{7} + 1296 \, A a b^{2} c^{7} + 39648 \, A a^{2} c^{8}}{c^{8}}\right )} x - \frac{231 \, B b^{6} c^{4} - 2232 \, B a b^{4} c^{5} - 378 \, A b^{5} c^{5} + 6384 \, B a^{2} b^{2} c^{6} + 3408 \, A a b^{3} c^{6} - 4096 \, B a^{3} c^{7} - 8352 \, A a^{2} b c^{7}}{c^{8}}\right )} x + \frac{1155 \, B b^{7} c^{3} - 12348 \, B a b^{5} c^{4} - 1890 \, A b^{6} c^{4} + 42192 \, B a^{2} b^{3} c^{5} + 18984 \, A a b^{4} c^{5} - 44096 \, B a^{3} b c^{6} - 57312 \, A a^{2} b^{2} c^{6} + 40320 \, A a^{3} c^{7}}{c^{8}}\right )} x - \frac{3465 \, B b^{8} c^{2} - 40740 \, B a b^{6} c^{3} - 5670 \, A b^{7} c^{3} + 162288 \, B a^{2} b^{4} c^{4} + 63000 \, A a b^{5} c^{4} - 234432 \, B a^{3} b^{2} c^{5} - 226464 \, A a^{2} b^{3} c^{5} + 65536 \, B a^{4} c^{6} + 254592 \, A a^{3} b c^{6}}{c^{8}}\right )} - \frac{5 \,{\left (11 \, B b^{9} - 144 \, B a b^{7} c - 18 \, A b^{8} c + 672 \, B a^{2} b^{5} c^{2} + 224 \, A a b^{6} c^{2} - 1280 \, B a^{3} b^{3} c^{3} - 960 \, A a^{2} b^{4} c^{3} + 768 \, B a^{4} b c^{4} + 1536 \, A a^{3} b^{2} c^{4} - 512 \, A a^{4} c^{5}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x^2,x, algorithm="giac")
[Out]