3.1269 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{5/2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7} \]

[Out]

(b^2 - 4*a*c)^3/(192*c^4*d*(b*d + 2*c*d*x)^(3/2)) + (3*(b^2 - 4*a*c)^2*Sqrt[b*d
+ 2*c*d*x])/(64*c^4*d^3) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/2))/(320*c^4*d^5)
+ (b*d + 2*c*d*x)^(9/2)/(576*c^4*d^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.14426, 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 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(5/2),x]

[Out]

(b^2 - 4*a*c)^3/(192*c^4*d*(b*d + 2*c*d*x)^(3/2)) + (3*(b^2 - 4*a*c)^2*Sqrt[b*d
+ 2*c*d*x])/(64*c^4*d^3) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/2))/(320*c^4*d^5)
+ (b*d + 2*c*d*x)^(9/2)/(576*c^4*d^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.9823, size = 117, normalized size = 0.97 \[ \frac{\left (- 4 a c + b^{2}\right )^{3}}{192 c^{4} d \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x}}{64 c^{4} d^{3}} - \frac{3 \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}}{320 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}}}{576 c^{4} d^{7}} \]

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)**(5/2),x)

[Out]

(-4*a*c + b**2)**3/(192*c**4*d*(b*d + 2*c*d*x)**(3/2)) + 3*(-4*a*c + b**2)**2*sq
rt(b*d + 2*c*d*x)/(64*c**4*d**3) - 3*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)/(320
*c**4*d**5) + (b*d + 2*c*d*x)**(9/2)/(576*c**4*d**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.321842, size = 120, normalized size = 0.99 \[ \frac{(b+2 c x)^3 \left (\frac{2160 a^2 c^2-972 a b^2 c+113 b^4}{c^4}+\frac{432 a b c x-68 b^3 x}{c^3}+\frac{15 \left (b^2-4 a c\right )^3}{c^4 (b+2 c x)^2}+\frac{12 x^2 \left (36 a c+b^2\right )}{c^2}+\frac{160 b x^3}{c}+80 x^4\right )}{2880 (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(5/2),x]

[Out]

((b + 2*c*x)^3*((113*b^4 - 972*a*b^2*c + 2160*a^2*c^2)/c^4 + (12*(b^2 + 36*a*c)*
x^2)/c^2 + (160*b*x^3)/c + 80*x^4 + (15*(b^2 - 4*a*c)^3)/(c^4*(b + 2*c*x)^2) + (
-68*b^3*x + 432*a*b*c*x)/c^3))/(2880*(d*(b + 2*c*x))^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 173, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -5\,{c}^{6}{x}^{6}-15\,b{c}^{5}{x}^{5}-27\,a{c}^{5}{x}^{4}-12\,{b}^{2}{c}^{4}{x}^{4}-54\,ab{c}^{4}{x}^{3}+{b}^{3}{c}^{3}{x}^{3}-135\,{a}^{2}{c}^{4}{x}^{2}+27\,a{b}^{2}{c}^{3}{x}^{2}-3\,{b}^{4}{c}^{2}{x}^{2}-135\,{a}^{2}b{c}^{3}x+54\,a{b}^{3}{c}^{2}x-6\,{b}^{5}cx+15\,{a}^{3}{c}^{3}-45\,{a}^{2}{b}^{2}{c}^{2}+18\,a{b}^{4}c-2\,{b}^{6} \right ) }{45\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

-1/45*(2*c*x+b)*(-5*c^6*x^6-15*b*c^5*x^5-27*a*c^5*x^4-12*b^2*c^4*x^4-54*a*b*c^4*
x^3+b^3*c^3*x^3-135*a^2*c^4*x^2+27*a*b^2*c^3*x^2-3*b^4*c^2*x^2-135*a^2*b*c^3*x+5
4*a*b^3*c^2*x-6*b^5*c*x+15*a^3*c^3-45*a^2*b^2*c^2+18*a*b^4*c-2*b^6)/c^4/(2*c*d*x
+b*d)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.68368, size = 184, normalized size = 1.52 \[ \frac{\frac{15 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{3}} - \frac{27 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 135 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{2 \, c d x + b d} d^{4} - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{c^{3} d^{6}}}{2880 \, c d} \]

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)^(5/2),x, algorithm="maxima")

[Out]

1/2880*(15*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/((2*c*d*x + b*d)^(3/
2)*c^3) - (27*(2*c*d*x + b*d)^(5/2)*(b^2 - 4*a*c)*d^2 - 135*(b^4 - 8*a*b^2*c + 1
6*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^4 - 5*(2*c*d*x + b*d)^(9/2))/(c^3*d^6))/(c*d)

_______________________________________________________________________________________

Fricas [A]  time = 0.211238, size = 243, normalized size = 2.01 \[ \frac{5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} + 2 \, b^{6} - 18 \, a b^{4} c + 45 \, a^{2} b^{2} c^{2} - 15 \, a^{3} c^{3} + 3 \,{\left (4 \, b^{2} c^{4} + 9 \, a c^{5}\right )} x^{4} -{\left (b^{3} c^{3} - 54 \, a b c^{4}\right )} x^{3} + 3 \,{\left (b^{4} c^{2} - 9 \, a b^{2} c^{3} + 45 \, a^{2} c^{4}\right )} x^{2} + 3 \,{\left (2 \, b^{5} c - 18 \, a b^{3} c^{2} + 45 \, a^{2} b c^{3}\right )} x}{45 \,{\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )} \sqrt{2 \, c d x + b d}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

1/45*(5*c^6*x^6 + 15*b*c^5*x^5 + 2*b^6 - 18*a*b^4*c + 45*a^2*b^2*c^2 - 15*a^3*c^
3 + 3*(4*b^2*c^4 + 9*a*c^5)*x^4 - (b^3*c^3 - 54*a*b*c^4)*x^3 + 3*(b^4*c^2 - 9*a*
b^2*c^3 + 45*a^2*c^4)*x^2 + 3*(2*b^5*c - 18*a*b^3*c^2 + 45*a^2*b*c^3)*x)/((2*c^5
*d^2*x + b*c^4*d^2)*sqrt(2*c*d*x + b*d))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{3}}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral((a + b*x + c*x**2)**3/(d*(b + 2*c*x))**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.246727, size = 252, normalized size = 2.08 \[ \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{192 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{4} d} + \frac{135 \, \sqrt{2 \, c d x + b d} b^{4} c^{32} d^{60} - 1080 \, \sqrt{2 \, c d x + b d} a b^{2} c^{33} d^{60} + 2160 \, \sqrt{2 \, c d x + b d} a^{2} c^{34} d^{60} - 27 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} c^{32} d^{58} + 108 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} a c^{33} d^{58} + 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{32} d^{56}}{2880 \, c^{36} d^{63}} \]

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)^(5/2),x, algorithm="giac")

[Out]

1/192*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/((2*c*d*x + b*d)^(3/2)*c^
4*d) + 1/2880*(135*sqrt(2*c*d*x + b*d)*b^4*c^32*d^60 - 1080*sqrt(2*c*d*x + b*d)*
a*b^2*c^33*d^60 + 2160*sqrt(2*c*d*x + b*d)*a^2*c^34*d^60 - 27*(2*c*d*x + b*d)^(5
/2)*b^2*c^32*d^58 + 108*(2*c*d*x + b*d)^(5/2)*a*c^33*d^58 + 5*(2*c*d*x + b*d)^(9
/2)*c^32*d^56)/(c^36*d^63)