Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac{(b d+2 c d x)^{13/2}}{208 c^3 d^5} \]
[Out]
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Rubi [A] time = 0.147152, 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 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac{(b d+2 c d x)^{13/2}}{208 c^3 d^5} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 28.8273, size = 82, normalized size = 0.93 \[ \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{80 c^{3} d} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{9}{2}}}{72 c^{3} d^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{13}{2}}}{208 c^{3} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(3/2)*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.109338, size = 92, normalized size = 1.05 \[ \frac{\left (c^2 \left (117 a^2+130 a c x^2+45 c^2 x^4\right )+b^2 c \left (35 c x^2-26 a\right )+10 b c^2 x \left (13 a+9 c x^2\right )+2 b^4-10 b^3 c x\right ) (d (b+2 c x))^{5/2}}{585 c^3 d} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 96, normalized size = 1.1 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 45\,{c}^{4}{x}^{4}+90\,b{x}^{3}{c}^{3}+130\,a{c}^{3}{x}^{2}+35\,{b}^{2}{c}^{2}{x}^{2}+130\,ab{c}^{2}x-10\,{b}^{3}cx+117\,{a}^{2}{c}^{2}-26\,ac{b}^{2}+2\,{b}^{4} \right ) }{585\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.686126, size = 109, normalized size = 1.24 \[ -\frac{130 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 117 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} d^{4} - 45 \,{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}{9360 \, c^{3} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208711, size = 221, normalized size = 2.51 \[ \frac{{\left (180 \, c^{6} d x^{6} + 540 \, b c^{5} d x^{5} + 5 \,{\left (109 \, b^{2} c^{4} + 104 \, a c^{5}\right )} d x^{4} + 10 \,{\left (19 \, b^{3} c^{3} + 104 \, a b c^{4}\right )} d x^{3} + 3 \,{\left (b^{4} c^{2} + 182 \, a b^{2} c^{3} + 156 \, a^{2} c^{4}\right )} d x^{2} - 2 \,{\left (b^{5} c - 13 \, a b^{3} c^{2} - 234 \, a^{2} b c^{3}\right )} d x +{\left (2 \, b^{6} - 26 \, a b^{4} c + 117 \, a^{2} b^{2} c^{2}\right )} d\right )} \sqrt{2 \, c d x + b d}}{585 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.79901, size = 695, normalized size = 7.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**(3/2)*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.240189, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^2,x, algorithm="giac")
[Out]