Optimal. Leaf size=121 \[ \frac{3 \left (b^2-4 a c\right )}{64 c^4 d^5 \sqrt{b d+2 c d x}}-\frac{3 \left (b^2-4 a c\right )^2}{320 c^4 d^3 (b d+2 c d x)^{5/2}}+\frac{\left (b^2-4 a c\right )^3}{576 c^4 d (b d+2 c d x)^{9/2}}+\frac{(b d+2 c d x)^{3/2}}{192 c^4 d^7} \]
[Out]
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Rubi [A] time = 0.144496, antiderivative size = 121, 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 \[ \frac{3 \left (b^2-4 a c\right )}{64 c^4 d^5 \sqrt{b d+2 c d x}}-\frac{3 \left (b^2-4 a c\right )^2}{320 c^4 d^3 (b d+2 c d x)^{5/2}}+\frac{\left (b^2-4 a c\right )^3}{576 c^4 d (b d+2 c d x)^{9/2}}+\frac{(b d+2 c d x)^{3/2}}{192 c^4 d^7} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 36.1051, size = 117, normalized size = 0.97 \[ \frac{\left (- 4 a c + b^{2}\right )^{3}}{576 c^{4} d \left (b d + 2 c d x\right )^{\frac{9}{2}}} - \frac{3 \left (- 4 a c + b^{2}\right )^{2}}{320 c^{4} d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{3 \left (- 4 a c + b^{2}\right )}{64 c^{4} d^{5} \sqrt{b d + 2 c d x}} + \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{192 c^{4} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(11/2),x)
[Out]
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Mathematica [A] time = 0.271499, size = 93, normalized size = 0.77 \[ \frac{(b+2 c x)^6 \left (\frac{5 \left (b^2-4 a c\right )^3}{(b+2 c x)^5}-\frac{27 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{135 \left (b^2-4 a c\right )}{b+2 c x}+15 b+30 c x\right )}{2880 c^4 (d (b+2 c x))^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(11/2),x]
[Out]
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Maple [A] time = 0.01, size = 174, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -15\,{c}^{6}{x}^{6}-45\,b{c}^{5}{x}^{5}+135\,a{c}^{5}{x}^{4}-90\,{b}^{2}{c}^{4}{x}^{4}+270\,ab{c}^{4}{x}^{3}-105\,{b}^{3}{c}^{3}{x}^{3}+27\,{a}^{2}{c}^{4}{x}^{2}+189\,a{b}^{2}{c}^{3}{x}^{2}-63\,{b}^{4}{c}^{2}{x}^{2}+27\,{a}^{2}b{c}^{3}x+54\,a{b}^{3}{c}^{2}x-18\,{b}^{5}cx+5\,{a}^{3}{c}^{3}+3\,{a}^{2}{b}^{2}{c}^{2}+6\,a{b}^{4}c-2\,{b}^{6} \right ) }{45\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{11}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(11/2),x)
[Out]
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Maxima [A] time = 0.692706, size = 186, normalized size = 1.54 \[ \frac{\frac{15 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{c^{3} d^{6}} + \frac{135 \,{\left (2 \, c d x + b d\right )}^{4}{\left (b^{2} - 4 \, a c\right )} - 27 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{2} d^{2} + 5 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{4}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{3} d^{4}}}{2880 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209387, size = 302, normalized size = 2.5 \[ \frac{15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} + 2 \, b^{6} - 6 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 5 \, a^{3} c^{3} + 45 \,{\left (2 \, b^{2} c^{4} - 3 \, a c^{5}\right )} x^{4} + 15 \,{\left (7 \, b^{3} c^{3} - 18 \, a b c^{4}\right )} x^{3} + 9 \,{\left (7 \, b^{4} c^{2} - 21 \, a b^{2} c^{3} - 3 \, a^{2} c^{4}\right )} x^{2} + 9 \,{\left (2 \, b^{5} c - 6 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x}{45 \,{\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )} \sqrt{2 \, c d x + b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(11/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 51.6244, size = 1731, normalized size = 14.31 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.241902, size = 238, normalized size = 1.97 \[ \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{192 \, c^{4} d^{7}} + \frac{5 \, b^{6} d^{4} - 60 \, a b^{4} c d^{4} + 240 \, a^{2} b^{2} c^{2} d^{4} - 320 \, a^{3} c^{3} d^{4} - 27 \,{\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 216 \,{\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 432 \,{\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 135 \,{\left (2 \, c d x + b d\right )}^{4} b^{2} - 540 \,{\left (2 \, c d x + b d\right )}^{4} a c}{2880 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{4} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(11/2),x, algorithm="giac")
[Out]