Optimal. Leaf size=122 \[ \frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac{32 c}{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{2}{3 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.17699, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac{32 c}{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{2}{3 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 43.416, size = 114, normalized size = 0.93 \[ \frac{256 c^{2} \sqrt{a + b x + c x^{2}}}{3 d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{3}} + \frac{32 c}{3 d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} - \frac{2}{3 d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.181524, size = 108, normalized size = 0.89 \[ \frac{32 c^2 \left (3 a^2+12 a c x^2+8 c^2 x^4\right )+48 b^2 c \left (a+6 c x^2\right )+128 b c^2 x \left (3 a+4 c x^2\right )-2 b^4+32 b^3 c x}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.014, size = 133, normalized size = 1.1 \[ -{\frac{256\,{c}^{4}{x}^{4}+512\,b{c}^{3}{x}^{3}+384\,a{c}^{3}{x}^{2}+288\,{b}^{2}{c}^{2}{x}^{2}+384\,ab{c}^{2}x+32\,{b}^{3}cx+96\,{a}^{2}{c}^{2}+48\,ac{b}^{2}-2\,{b}^{4}}{ \left ( 6\,cx+3\,b \right ){d}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.462014, size = 505, normalized size = 4.14 \[ \frac{2 \,{\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} - b^{4} + 24 \, a b^{2} c + 48 \, a^{2} c^{2} + 48 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (2 \,{\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{2} x^{5} + 5 \,{\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{2} x^{4} + 4 \,{\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} d^{2} x^{3} +{\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} d^{2} x^{2} + 2 \,{\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} d^{2} x +{\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{a^{2} b^{2} \sqrt{a + b x + c x^{2}} + 4 a^{2} b c x \sqrt{a + b x + c x^{2}} + 4 a^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 2 a b^{3} x \sqrt{a + b x + c x^{2}} + 10 a b^{2} c x^{2} \sqrt{a + b x + c x^{2}} + 16 a b c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 8 a c^{3} x^{4} \sqrt{a + b x + c x^{2}} + b^{4} x^{2} \sqrt{a + b x + c x^{2}} + 6 b^{3} c x^{3} \sqrt{a + b x + c x^{2}} + 13 b^{2} c^{2} x^{4} \sqrt{a + b x + c x^{2}} + 12 b c^{3} x^{5} \sqrt{a + b x + c x^{2}} + 4 c^{4} x^{6} \sqrt{a + b x + c x^{2}}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{2}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(5/2)),x, algorithm="giac")
[Out]