Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{56 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{48 c^3 d}+\frac{(b d+2 c d x)^{11/2}}{176 c^3 d^5} \]
[Out]
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Rubi [A] time = 0.115111, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{56 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{48 c^3 d}+\frac{(b d+2 c d x)^{11/2}}{176 c^3 d^5} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 28.4676, size = 82, normalized size = 0.93 \[ \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{48 c^{3} d} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}}{56 c^{3} d^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{11}{2}}}{176 c^{3} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.103095, size = 92, normalized size = 1.05 \[ \frac{\left (c^2 \left (77 a^2+66 a c x^2+21 c^2 x^4\right )+b^2 c \left (15 c x^2-22 a\right )+6 b c^2 x \left (11 a+7 c x^2\right )+2 b^4-6 b^3 c x\right ) (d (b+2 c x))^{3/2}}{231 c^3 d} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 96, normalized size = 1.1 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 21\,{c}^{4}{x}^{4}+42\,b{x}^{3}{c}^{3}+66\,a{c}^{3}{x}^{2}+15\,{b}^{2}{c}^{2}{x}^{2}+66\,ab{c}^{2}x-6\,{b}^{3}cx+77\,{a}^{2}{c}^{2}-22\,ac{b}^{2}+2\,{b}^{4} \right ) }{231\,{c}^{3}}\sqrt{2\,cdx+bd}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.684367, size = 109, normalized size = 1.24 \[ -\frac{66 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 77 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} d^{4} - 21 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{3696 \, c^{3} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210355, size = 163, normalized size = 1.85 \[ \frac{{\left (42 \, c^{5} x^{5} + 105 \, b c^{4} x^{4} + 2 \, b^{5} - 22 \, a b^{3} c + 77 \, a^{2} b c^{2} + 12 \,{\left (6 \, b^{2} c^{3} + 11 \, a c^{4}\right )} x^{3} + 3 \,{\left (b^{3} c^{2} + 66 \, a b c^{3}\right )} x^{2} - 2 \,{\left (b^{4} c - 11 \, a b^{2} c^{2} - 77 \, a^{2} c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{231 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.29109, size = 94, normalized size = 1.07 \[ \frac{\frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (16 a^{2} c^{2} - 8 a b^{2} c + b^{4}\right )}{48 c^{2}} + \frac{\left (4 a c - b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}}{56 c^{2} d^{2}} + \frac{\left (b d + 2 c d x\right )^{\frac{11}{2}}}{176 c^{2} d^{4}}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.237592, size = 559, normalized size = 6.35 \[ \frac{18480 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a^{2} - \frac{3696 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} a b}{c d} + \frac{132 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{12} d^{14} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b c^{12} d^{13} + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{12} d^{12}\right )} b^{2}}{c^{14} d^{14}} + \frac{264 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{12} d^{14} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b c^{12} d^{13} + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{12} d^{12}\right )} a}{c^{13} d^{14}} - \frac{44 \,{\left (105 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{3} c^{24} d^{27} - 189 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} c^{24} d^{26} + 135 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b c^{24} d^{25} - 35 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{24} d^{24}\right )} b}{c^{26} d^{27}} + \frac{1155 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c^{40} d^{44} - 2772 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{3} c^{40} d^{43} + 2970 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b^{2} c^{40} d^{42} - 1540 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} b c^{40} d^{41} + 315 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}} c^{40} d^{40}}{c^{42} d^{44}}}{55440 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^2,x, algorithm="giac")
[Out]