Optimal. Leaf size=242 \[ -\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^6 d^3 (p+4)}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^6 d^3 (p+2)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^6 d^3 (p+6)} \]
[Out]
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Rubi [A] time = 0.410663, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^6 d^3 (p+4)}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^6 d^3 (p+2)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^6 d^3 (p+6)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*Sqrt[c + d*x])^p,x]
[Out]
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Rubi in Sympy [A] time = 33.7381, size = 221, normalized size = 0.91 \[ - \frac{2 a \left (a + b \sqrt{c + d x}\right )^{p + 1} \left (a^{2} - b^{2} c\right )^{2}}{b^{6} d^{3} \left (p + 1\right )} - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{p + 3} \left (5 a^{2} - 3 b^{2} c\right )}{b^{6} d^{3} \left (p + 3\right )} - \frac{10 a \left (a + b \sqrt{c + d x}\right )^{p + 5}}{b^{6} d^{3} \left (p + 5\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 2} \left (5 a^{4} - 6 a^{2} b^{2} c + b^{4} c^{2}\right )}{b^{6} d^{3} \left (p + 2\right )} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{p + 4} \left (5 a^{2} - b^{2} c\right )}{b^{6} d^{3} \left (p + 4\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 6}}{b^{6} d^{3} \left (p + 6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(a+b*(d*x+c)**(1/2))**p,x)
[Out]
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Mathematica [A] time = 0.57644, size = 285, normalized size = 1.18 \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (-120 a^5+120 a^4 b (p+1) \sqrt{c+d x}+12 a^3 b^2 \left (-4 c \left (p^2+p-5\right )-5 d \left (p^2+3 p+2\right ) x\right )-4 a^2 b^3 (p+1) \sqrt{c+d x} \left (c \left (-2 p^2+8 p+60\right )-5 d \left (p^2+5 p+6\right ) x\right )-a b^4 \left (-8 c^2 \left (2 p^3+12 p^2+10 p-15\right )+4 c d \left (p^4+4 p^3-10 p^2-43 p-30\right ) x+5 d^2 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2\right )+b^5 \left (p^3+9 p^2+23 p+15\right ) \sqrt{c+d x} \left (8 c^2-4 c d (p+2) x+d^2 \left (p^2+6 p+8\right ) x^2\right )\right )}{b^6 d^3 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*Sqrt[c + d*x])^p,x]
[Out]
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Maple [F] time = 0.006, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(a+b*(d*x+c)^(1/2))^p,x)
[Out]
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Maxima [A] time = 0.729036, size = 543, normalized size = 2.24 \[ \frac{2 \,{\left (\frac{{\left ({\left (d x + c\right )} b^{2}{\left (p + 1\right )} + \sqrt{d x + c} a b p - a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c^{2}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac{2 \,{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )}{\left (d x + c\right )}^{2} b^{4} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} - 3 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt{d x + c} a^{3} b p - 6 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} + \frac{{\left ({\left (p^{5} + 15 \, p^{4} + 85 \, p^{3} + 225 \, p^{2} + 274 \, p + 120\right )}{\left (d x + c\right )}^{3} b^{6} +{\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )}{\left (d x + c\right )}^{\frac{5}{2}} a b^{5} - 5 \,{\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )}{\left (d x + c\right )}^{2} a^{2} b^{4} + 20 \,{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{3} - 60 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{4} b^{2} + 120 \, \sqrt{d x + c} a^{5} b p - 120 \, a^{6}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{{\left (p^{6} + 21 \, p^{5} + 175 \, p^{4} + 735 \, p^{3} + 1624 \, p^{2} + 1764 \, p + 720\right )} b^{6}}\right )}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.359207, size = 961, normalized size = 3.97 \[ \frac{2 \,{\left (120 \, b^{6} c^{3} - 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c - 120 \, a^{6} + 8 \,{\left (b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2}\right )} p^{3} +{\left (b^{6} d^{3} p^{5} + 15 \, b^{6} d^{3} p^{4} + 85 \, b^{6} d^{3} p^{3} + 225 \, b^{6} d^{3} p^{2} + 274 \, b^{6} d^{3} p + 120 \, b^{6} d^{3}\right )} x^{3} + 24 \,{\left (3 \, b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2} - 2 \, a^{4} b^{2} c\right )} p^{2} +{\left (b^{6} c d^{2} p^{5} +{\left (11 \, b^{6} c - 5 \, a^{2} b^{4}\right )} d^{2} p^{4} +{\left (41 \, b^{6} c - 30 \, a^{2} b^{4}\right )} d^{2} p^{3} +{\left (61 \, b^{6} c - 55 \, a^{2} b^{4}\right )} d^{2} p^{2} + 30 \,{\left (b^{6} c - a^{2} b^{4}\right )} d^{2} p\right )} x^{2} + 8 \,{\left (23 \, b^{6} c^{3} - 24 \, a^{2} b^{4} c^{2} + 9 \, a^{4} b^{2} c\right )} p - 4 \,{\left ({\left (b^{6} c^{2} + a^{2} b^{4} c\right )} d p^{4} + 3 \,{\left (3 \, b^{6} c^{2} - a^{2} b^{4} c\right )} d p^{3} +{\left (23 \, b^{6} c^{2} - 34 \, a^{2} b^{4} c + 15 \, a^{4} b^{2}\right )} d p^{2} + 15 \,{\left (b^{6} c^{2} - 2 \, a^{2} b^{4} c + a^{4} b^{2}\right )} d p\right )} x +{\left (8 \,{\left (3 \, a b^{5} c^{2} + a^{3} b^{3} c\right )} p^{3} + 24 \,{\left (7 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c\right )} p^{2} +{\left (a b^{5} d^{2} p^{5} + 10 \, a b^{5} d^{2} p^{4} + 35 \, a b^{5} d^{2} p^{3} + 50 \, a b^{5} d^{2} p^{2} + 24 \, a b^{5} d^{2} p\right )} x^{2} + 8 \,{\left (33 \, a b^{5} c^{2} - 40 \, a^{3} b^{3} c + 15 \, a^{5} b\right )} p - 4 \,{\left (2 \, a b^{5} c d p^{4} + 5 \,{\left (3 \, a b^{5} c - a^{3} b^{3}\right )} d p^{3} +{\left (31 \, a b^{5} c - 15 \, a^{3} b^{3}\right )} d p^{2} + 2 \,{\left (9 \, a b^{5} c - 5 \, a^{3} b^{3}\right )} d p\right )} x\right )} \sqrt{d x + c}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{b^{6} d^{3} p^{6} + 21 \, b^{6} d^{3} p^{5} + 175 \, b^{6} d^{3} p^{4} + 735 \, b^{6} d^{3} p^{3} + 1624 \, b^{6} d^{3} p^{2} + 1764 \, b^{6} d^{3} p + 720 \, b^{6} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p*x^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**2*(a+b*(d*x+c)**(1/2))**p,x)
[Out]
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GIAC/XCAS [A] time = 2.27939, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p*x^2,x, algorithm="giac")
[Out]