Optimal. Leaf size=59 \[ \frac{4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0792496, 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 \[ \frac{4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.3399, size = 56, normalized size = 0.95 \[ \frac{2 d^{3} \left (b + 2 c x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{63} \]
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)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.143072, size = 44, normalized size = 0.75 \[ -\frac{2}{63} d^3 (a+x (b+c x))^{7/2} \left (4 c \left (2 a-7 c x^2\right )-9 b^2-28 b c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 41, normalized size = 0.7 \[ -{\frac{ \left ( -56\,{c}^{2}{x}^{2}-56\,bxc+16\,ac-18\,{b}^{2} \right ){d}^{3}}{63} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \]
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)^(5/2),x)
[Out]
_______________________________________________________________________________________
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)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.269067, size = 290, normalized size = 4.92 \[ \frac{2}{63} \,{\left (28 \, c^{5} d^{3} x^{8} + 112 \, b c^{4} d^{3} x^{7} +{\left (177 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{3} x^{6} +{\left (139 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d^{3} x^{5} + 5 \,{\left (11 \, b^{4} c + 51 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{3} x^{4} +{\left (9 \, b^{5} + 130 \, a b^{3} c + 120 \, a^{2} b c^{2}\right )} d^{3} x^{3} +{\left (27 \, a b^{4} + 87 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{3} x^{2} +{\left (27 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{3} x +{\left (9 \, a^{3} b^{2} - 8 \, a^{4} c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \]
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)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 15.8075, size = 559, normalized size = 9.47 \[ - \frac{16 a^{4} c d^{3} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 a^{3} b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{8 a^{3} b c d^{3} x \sqrt{a + b x + c x^{2}}}{63} + \frac{8 a^{3} c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{63} + \frac{6 a^{2} b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{7} + \frac{58 a^{2} b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{21} + \frac{80 a^{2} b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{21} + \frac{40 a^{2} c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{21} + \frac{6 a b^{4} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{260 a b^{3} c d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{63} + \frac{170 a b^{2} c^{2} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{21} + \frac{152 a b c^{3} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{21} + \frac{152 a c^{4} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 b^{5} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{110 b^{4} c d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{63} + \frac{278 b^{3} c^{2} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{63} + \frac{118 b^{2} c^{3} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{21} + \frac{32 b c^{4} d^{3} x^{7} \sqrt{a + b x + c x^{2}}}{9} + \frac{8 c^{5} d^{3} x^{8} \sqrt{a + b x + c x^{2}}}{9} \]
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)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.22639, size = 389, normalized size = 6.59 \[ \frac{2}{63} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left ({\left ({\left (28 \,{\left (c^{5} d^{3} x + 4 \, b c^{4} d^{3}\right )} x + \frac{177 \, b^{2} c^{11} d^{3} + 76 \, a c^{12} d^{3}}{c^{8}}\right )} x + \frac{139 \, b^{3} c^{10} d^{3} + 228 \, a b c^{11} d^{3}}{c^{8}}\right )} x + \frac{5 \,{\left (11 \, b^{4} c^{9} d^{3} + 51 \, a b^{2} c^{10} d^{3} + 12 \, a^{2} c^{11} d^{3}\right )}}{c^{8}}\right )} x + \frac{9 \, b^{5} c^{8} d^{3} + 130 \, a b^{3} c^{9} d^{3} + 120 \, a^{2} b c^{10} d^{3}}{c^{8}}\right )} x + \frac{27 \, a b^{4} c^{8} d^{3} + 87 \, a^{2} b^{2} c^{9} d^{3} + 4 \, a^{3} c^{10} d^{3}}{c^{8}}\right )} x + \frac{27 \, a^{2} b^{3} c^{8} d^{3} + 4 \, a^{3} b c^{9} d^{3}}{c^{8}}\right )} x + \frac{9 \, a^{3} b^{2} c^{8} d^{3} - 8 \, a^{4} c^{9} d^{3}}{c^{8}}\right )} \]
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)^(5/2),x, algorithm="giac")
[Out]