Optimal. Leaf size=55 \[ \frac{(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d} \]
[Out]
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Rubi [A] time = 0.0779594, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 14.9307, size = 49, normalized size = 0.89 \[ - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}}{7 c^{2} d} + \frac{\left (b d + 2 c d x\right )^{\frac{11}{2}}}{44 c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(5/2)*(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0686104, size = 45, normalized size = 0.82 \[ \frac{\left (c \left (11 a+7 c x^2\right )-b^2+7 b c x\right ) (d (b+2 c x))^{7/2}}{77 c^2 d} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.007, size = 46, normalized size = 0.8 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 7\,{c}^{2}{x}^{2}+7\,bxc+11\,ac-{b}^{2} \right ) }{77\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.684355, size = 62, normalized size = 1.13 \[ -\frac{11 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{308 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206441, size = 165, normalized size = 3. \[ \frac{{\left (56 \, c^{5} d^{2} x^{5} + 140 \, b c^{4} d^{2} x^{4} + 2 \,{\left (59 \, b^{2} c^{3} + 44 \, a c^{4}\right )} d^{2} x^{3} +{\left (37 \, b^{3} c^{2} + 132 \, a b c^{3}\right )} d^{2} x^{2} +{\left (b^{4} c + 66 \, a b^{2} c^{2}\right )} d^{2} x -{\left (b^{5} - 11 \, a b^{3} c\right )} d^{2}\right )} \sqrt{2 \, c d x + b d}}{77 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.4193, size = 289, normalized size = 5.25 \[ \begin{cases} \frac{a b^{3} d^{2} \sqrt{b d + 2 c d x}}{7 c} + \frac{6 a b^{2} d^{2} x \sqrt{b d + 2 c d x}}{7} + \frac{12 a b c d^{2} x^{2} \sqrt{b d + 2 c d x}}{7} + \frac{8 a c^{2} d^{2} x^{3} \sqrt{b d + 2 c d x}}{7} - \frac{b^{5} d^{2} \sqrt{b d + 2 c d x}}{77 c^{2}} + \frac{b^{4} d^{2} x \sqrt{b d + 2 c d x}}{77 c} + \frac{37 b^{3} d^{2} x^{2} \sqrt{b d + 2 c d x}}{77} + \frac{118 b^{2} c d^{2} x^{3} \sqrt{b d + 2 c d x}}{77} + \frac{20 b c^{2} d^{2} x^{4} \sqrt{b d + 2 c d x}}{11} + \frac{8 c^{3} d^{2} x^{5} \sqrt{b d + 2 c d x}}{11} & \text{for}\: c \neq 0 \\\left (b d\right )^{\frac{5}{2}} \left (a x + \frac{b x^{2}}{2}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**(5/2)*(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.22472, size = 599, normalized size = 10.89 \[ \frac{4620 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} d - 1848 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} a b - \frac{462 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} b^{3}}{c} + \frac{165 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{12} d^{14} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b c^{12} d^{13} + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{12} d^{12}\right )} b^{2}}{c^{13} d^{13}} + \frac{132 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{12} d^{14} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b c^{12} d^{13} + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{12} d^{12}\right )} a}{c^{12} d^{13}} - \frac{44 \,{\left (105 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{3} c^{24} d^{27} - 189 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} c^{24} d^{26} + 135 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b c^{24} d^{25} - 35 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{24} d^{24}\right )} b}{c^{25} d^{26}} + \frac{1155 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c^{40} d^{44} - 2772 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{3} c^{40} d^{43} + 2970 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b^{2} c^{40} d^{42} - 1540 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} b c^{40} d^{41} + 315 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}} c^{40} d^{40}}{c^{41} d^{43}}}{13860 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a),x, algorithm="giac")
[Out]