3.1240 \(\int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d^5*(b + 2*c*x)^4)/(3*(a + b*x + c*x^2)^(3/2)) - (32*c*d^5*(b + 2*c*x)^2)/(3
*Sqrt[a + b*x + c*x^2]) + (256*c^2*d^5*Sqrt[a + b*x + c*x^2])/3

_______________________________________________________________________________________

Rubi [A]  time = 0.13166, antiderivative size = 84, 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 \[ \frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d^5*(b + 2*c*x)^4)/(3*(a + b*x + c*x^2)^(3/2)) - (32*c*d^5*(b + 2*c*x)^2)/(3
*Sqrt[a + b*x + c*x^2]) + (256*c^2*d^5*Sqrt[a + b*x + c*x^2])/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.1435, size = 82, normalized size = 0.98 \[ \frac{256 c^{2} d^{5} \sqrt{a + b x + c x^{2}}}{3} - \frac{32 c d^{5} \left (b + 2 c x\right )^{2}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d^{5} \left (b + 2 c x\right )^{4}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

256*c**2*d**5*sqrt(a + b*x + c*x**2)/3 - 32*c*d**5*(b + 2*c*x)**2/(3*sqrt(a + b*
x + c*x**2)) - 2*d**5*(b + 2*c*x)**4/(3*(a + b*x + c*x**2)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.232779, size = 72, normalized size = 0.86 \[ d^5 \sqrt{a+x (b+c x)} \left (\frac{16 c \left (4 a c-b^2\right )}{a+x (b+c x)}-\frac{2 \left (b^2-4 a c\right )^2}{3 (a+x (b+c x))^2}+32 c^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^5*Sqrt[a + x*(b + c*x)]*(32*c^2 - (2*(b^2 - 4*a*c)^2)/(3*(a + x*(b + c*x))^2)
+ (16*c*(-b^2 + 4*a*c))/(a + x*(b + c*x)))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 91, normalized size = 1.1 \[{\frac{2\,{d}^{5} \left ( 48\,{c}^{4}{x}^{4}+96\,b{c}^{3}{x}^{3}+192\,a{c}^{3}{x}^{2}+24\,{b}^{2}{c}^{2}{x}^{2}+192\,ab{c}^{2}x-24\,{b}^{3}cx+128\,{a}^{2}{c}^{2}-16\,ac{b}^{2}-{b}^{4} \right ) }{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*d^5*(48*c^4*x^4+96*b*c^3*x^3+192*a*c^3*x^2+24*b^2*c^2*x^2+192*a*b*c^2*x-24*b
^3*c*x+128*a^2*c^2-16*a*b^2*c-b^4)/(c*x^2+b*x+a)^(3/2)

_______________________________________________________________________________________

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)^5/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.401209, size = 189, normalized size = 2.25 \[ \frac{2 \,{\left (48 \, c^{4} d^{5} x^{4} + 96 \, b c^{3} d^{5} x^{3} + 24 \,{\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{5} x^{2} - 24 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} d^{5} x -{\left (b^{4} + 16 \, a b^{2} c - 128 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(48*c^4*d^5*x^4 + 96*b*c^3*d^5*x^3 + 24*(b^2*c^2 + 8*a*c^3)*d^5*x^2 - 24*(b^
3*c - 8*a*b*c^2)*d^5*x - (b^4 + 16*a*b^2*c - 128*a^2*c^2)*d^5)*sqrt(c*x^2 + b*x
+ a)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

_______________________________________________________________________________________

Sympy [A]  time = 4.84015, size = 615, normalized size = 7.32 \[ \frac{256 a^{2} c^{2} d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{32 a b^{2} c d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{384 a b c^{2} d^{5} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{384 a c^{3} d^{5} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 b^{4} d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{48 b^{3} c d^{5} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{48 b^{2} c^{2} d^{5} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{192 b c^{3} d^{5} x^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{96 c^{4} d^{5} x^{4}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

256*a**2*c**2*d**5/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) +
3*c*x**2*sqrt(a + b*x + c*x**2)) - 32*a*b**2*c*d**5/(3*a*sqrt(a + b*x + c*x**2)
+ 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) + 384*a*b*c**2
*d**5*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sq
rt(a + b*x + c*x**2)) + 384*a*c**3*d**5*x**2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x
*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 2*b**4*d**5/(3*a*sq
rt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*
x**2)) - 48*b**3*c*d**5*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x
**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) + 48*b**2*c**2*d**5*x**2/(3*a*sqrt(a + b
*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) +
192*b*c**3*d**5*x**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2)
+ 3*c*x**2*sqrt(a + b*x + c*x**2)) + 96*c**4*d**5*x**4/(3*a*sqrt(a + b*x + c*x*
*2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.233801, size = 521, normalized size = 6.2 \[ \frac{2 \,{\left (24 \,{\left ({\left (2 \,{\left (\frac{{\left (b^{4} c^{6} d^{5} - 8 \, a b^{2} c^{7} d^{5} + 16 \, a^{2} c^{8} d^{5}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{2 \,{\left (b^{5} c^{5} d^{5} - 8 \, a b^{3} c^{6} d^{5} + 16 \, a^{2} b c^{7} d^{5}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{6} c^{4} d^{5} - 48 \, a^{2} b^{2} c^{6} d^{5} + 128 \, a^{3} c^{7} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{7} c^{3} d^{5} - 16 \, a b^{5} c^{4} d^{5} + 80 \, a^{2} b^{3} c^{5} d^{5} - 128 \, a^{3} b c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{8} c^{2} d^{5} + 8 \, a b^{6} c^{3} d^{5} - 240 \, a^{2} b^{4} c^{4} d^{5} + 1280 \, a^{3} b^{2} c^{5} d^{5} - 2048 \, a^{4} c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

2/3*(24*((2*((b^4*c^6*d^5 - 8*a*b^2*c^7*d^5 + 16*a^2*c^8*d^5)*x/(b^4*c^2 - 8*a*b
^2*c^3 + 16*a^2*c^4) + 2*(b^5*c^5*d^5 - 8*a*b^3*c^6*d^5 + 16*a^2*b*c^7*d^5)/(b^4
*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^6*c^4*d^5 - 48*a^2*b^2*c^6*d^5 + 128*a^
3*c^7*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (b^7*c^3*d^5 - 16*a*b^5*c^4
*d^5 + 80*a^2*b^3*c^5*d^5 - 128*a^3*b*c^6*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c
^4))*x - (b^8*c^2*d^5 + 8*a*b^6*c^3*d^5 - 240*a^2*b^4*c^4*d^5 + 1280*a^3*b^2*c^5
*d^5 - 2048*a^4*c^6*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)
^(3/2)