Optimal. Leaf size=59 \[ \frac{4}{35} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac{2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2} \]
[Out]
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Rubi [A] time = 0.0793353, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{4}{35} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac{2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 17.4373, size = 56, normalized size = 0.95 \[ \frac{2 d^{3} \left (b + 2 c x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{35} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**3*(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0986639, size = 44, normalized size = 0.75 \[ -\frac{2}{35} d^3 (a+x (b+c x))^{5/2} \left (4 c \left (2 a-5 c x^2\right )-7 b^2-20 b c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 41, normalized size = 0.7 \[ -{\frac{ \left ( -40\,{c}^{2}{x}^{2}-40\,bxc+16\,ac-14\,{b}^{2} \right ){d}^{3}}{35} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^(3/2),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((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238441, size = 201, normalized size = 3.41 \[ \frac{2}{35} \,{\left (20 \, c^{4} d^{3} x^{6} + 60 \, b c^{3} d^{3} x^{5} +{\left (67 \, b^{2} c^{2} + 32 \, a c^{3}\right )} d^{3} x^{4} + 2 \,{\left (17 \, b^{3} c + 32 \, a b c^{2}\right )} d^{3} x^{3} +{\left (7 \, b^{4} + 46 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{3} x^{2} + 2 \,{\left (7 \, a b^{3} + 2 \, a^{2} b c\right )} d^{3} x +{\left (7 \, a^{2} b^{2} - 8 \, a^{3} c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.05103, size = 371, normalized size = 6.29 \[ - \frac{16 a^{3} c d^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{2 a^{2} b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{8 a^{2} b c d^{3} x \sqrt{a + b x + c x^{2}}}{35} + \frac{8 a^{2} c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{35} + \frac{4 a b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{5} + \frac{92 a b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{35} + \frac{128 a b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{64 a c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{35} + \frac{2 b^{4} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{68 b^{3} c d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{134 b^{2} c^{2} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{35} + \frac{24 b c^{3} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{7} + \frac{8 c^{4} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**3*(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222887, size = 269, normalized size = 4.56 \[ \frac{2}{35} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left (20 \,{\left (c^{4} d^{3} x + 3 \, b c^{3} d^{3}\right )} x + \frac{67 \, b^{2} c^{8} d^{3} + 32 \, a c^{9} d^{3}}{c^{6}}\right )} x + \frac{2 \,{\left (17 \, b^{3} c^{7} d^{3} + 32 \, a b c^{8} d^{3}\right )}}{c^{6}}\right )} x + \frac{7 \, b^{4} c^{6} d^{3} + 46 \, a b^{2} c^{7} d^{3} + 4 \, a^{2} c^{8} d^{3}}{c^{6}}\right )} x + \frac{2 \,{\left (7 \, a b^{3} c^{6} d^{3} + 2 \, a^{2} b c^{7} d^{3}\right )}}{c^{6}}\right )} x + \frac{7 \, a^{2} b^{2} c^{6} d^{3} - 8 \, a^{3} c^{7} d^{3}}{c^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]