3.1239 \(\int \frac{(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=136 \[ 40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d^6*(b + 2*c*x)^5)/(3*(a + b*x + c*x^2)^(3/2)) - (40*c*d^6*(b + 2*c*x)^3)/(3
*Sqrt[a + b*x + c*x^2]) + 80*c^2*d^6*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 40*c^(3
/2)*(b^2 - 4*a*c)*d^6*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]

_______________________________________________________________________________________

Rubi [A]  time = 0.232751, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ 40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d^6*(b + 2*c*x)^5)/(3*(a + b*x + c*x^2)^(3/2)) - (40*c*d^6*(b + 2*c*x)^3)/(3
*Sqrt[a + b*x + c*x^2]) + 80*c^2*d^6*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 40*c^(3
/2)*(b^2 - 4*a*c)*d^6*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.2282, size = 134, normalized size = 0.99 \[ 40 c^{\frac{3}{2}} d^{6} \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} + 80 c^{2} d^{6} \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} - \frac{40 c d^{6} \left (b + 2 c x\right )^{3}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d^{6} \left (b + 2 c x\right )^{5}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**(5/2),x)

[Out]

40*c**(3/2)*d**6*(-4*a*c + b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x
**2))) + 80*c**2*d**6*(b + 2*c*x)*sqrt(a + b*x + c*x**2) - 40*c*d**6*(b + 2*c*x)
**3/(3*sqrt(a + b*x + c*x**2)) - 2*d**6*(b + 2*c*x)**5/(3*(a + b*x + c*x**2)**(3
/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.277867, size = 141, normalized size = 1.04 \[ d^6 \left (-\frac{2 (b+2 c x) \left (-8 c^2 \left (15 a^2+20 a c x^2+3 c^2 x^4\right )+4 b^2 c \left (5 a+c x^2\right )-16 b c^2 x \left (10 a+3 c x^2\right )+b^4+28 b^3 c x\right )}{3 (a+x (b+c x))^{3/2}}-40 c^{3/2} \left (4 a c-b^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]

[Out]

d^6*((-2*(b + 2*c*x)*(b^4 + 28*b^3*c*x + 4*b^2*c*(5*a + c*x^2) - 16*b*c^2*x*(10*
a + 3*c*x^2) - 8*c^2*(15*a^2 + 20*a*c*x^2 + 3*c^2*x^4)))/(3*(a + x*(b + c*x))^(3
/2)) - 40*c^(3/2)*(-b^2 + 4*a*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]
])

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 997, normalized size = 7.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x)

[Out]

-120*d^6*c^2*b^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-960*d^6*c^3*b^4*a/(4*a*c-b^
2)^2/(c*x^2+b*x+a)^(1/2)*x+240*d^6*c^3*b^2*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x
+1920*d^6*c^4*b^2*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-160*d^6*c^3*a*b^2/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+60*d^6*c*b^7/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+120
*d^6*c^2*b^3*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+960*d^6*c^3*b^3*a^2/(4*a*c-b^2)
^2/(c*x^2+b*x+a)^(1/2)+15*d^6*c*b^6/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+120*d^6*c^
2*b^6/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-60*d^6*c*b^5*a/(4*a*c-b^2)/(c*x^2+b*x+
a)^(3/2)-480*d^6*c^2*b^5*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-80*d^6*c^2*a*b^3/(4
*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+40*d^6*c^2*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+4
1/6*d^6*b^5/(c*x^2+b*x+a)^(3/2)-310/3*d^6*c*b^3*a/(c*x^2+b*x+a)^(3/2)-40*d^6*c^2
*b^2*x/(c*x^2+b*x+a)^(1/2)+20*d^6*c*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+160/3*d^
6*c^4*a*x^3/(c*x^2+b*x+a)^(3/2)+160*d^6*c^3*a*x/(c*x^2+b*x+a)^(1/2)-80*d^6*c^2*a
*b/(c*x^2+b*x+a)^(1/2)+160*d^6*c^2*b*a^2/(c*x^2+b*x+a)^(3/2)+80*d^6*c^4*b*x^4/(c
*x^2+b*x+a)^(3/2)-40/3*d^6*c^3*b^2*x^3/(c*x^2+b*x+a)^(3/2)-140*d^6*c^2*b^3*x^2/(
c*x^2+b*x+a)^(3/2)-65*d^6*c*b^4*x/(c*x^2+b*x+a)^(3/2)+40*d^6*c^(3/2)*b^2*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-160*d^6*c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))+15/2*d^6*b^7/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+20*d^6*c*b^3/(c*
x^2+b*x+a)^(1/2)+32*d^6*c^5*x^5/(c*x^2+b*x+a)^(3/2)+240*d^6*c^3*b*a*x^2/(c*x^2+b
*x+a)^(3/2)+60*d^6*c^2*b^2*a*x/(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)^6/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.534342, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (30 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} +{\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \,{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \,{\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \,{\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac{2 \,{\left (60 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} +{\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \,{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt{-c} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) +{\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \,{\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \,{\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[-2/3*(30*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*
c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^
2*c - 4*a^3*c^2)*d^6)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*
x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(
b^2*c^3 + 8*a*c^4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*
b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a*b^3*c - 120*a^2*b*c^2)*d^6)*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), 2/3*(60*(
(b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2
*c^2 - 8*a^2*c^3)*d^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3
*c^2)*d^6)*sqrt(-c)*arctan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) + (
48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 + 8*a*c^4)*d^6*x^3 - 60*(b^3*c^
2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*
a*b^3*c - 120*a^2*b*c^2)*d^6)*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240516, size = 713, normalized size = 5.24 \[ -\frac{40 \,{\left (b^{2} c^{2} d^{6} - 4 \, a c^{3} d^{6}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} + \frac{2 \,{\left (2 \,{\left (2 \,{\left (2 \,{\left (3 \,{\left (\frac{2 \,{\left (b^{4} c^{8} d^{6} - 8 \, a b^{2} c^{9} d^{6} + 16 \, a^{2} c^{10} d^{6}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac{5 \,{\left (b^{5} c^{7} d^{6} - 8 \, a b^{3} c^{8} d^{6} + 16 \, a^{2} b c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac{5 \,{\left (b^{6} c^{6} d^{6} - 48 \, a^{2} b^{2} c^{8} d^{6} + 128 \, a^{3} c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{15 \,{\left (b^{7} c^{5} d^{6} - 16 \, a b^{5} c^{6} d^{6} + 80 \, a^{2} b^{3} c^{7} d^{6} - 128 \, a^{3} b c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{15 \,{\left (b^{8} c^{4} d^{6} - 12 \, a b^{6} c^{5} d^{6} + 40 \, a^{2} b^{4} c^{6} d^{6} - 128 \, a^{4} c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{b^{9} c^{3} d^{6} + 12 \, a b^{7} c^{4} d^{6} - 264 \, a^{2} b^{5} c^{5} d^{6} + 1280 \, a^{3} b^{3} c^{6} d^{6} - 1920 \, a^{4} b c^{7} d^{6}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

-40*(b^2*c^2*d^6 - 4*a*c^3*d^6)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sq
rt(c) - b))/sqrt(c) + 2/3*(2*(2*(2*(3*(2*(b^4*c^8*d^6 - 8*a*b^2*c^9*d^6 + 16*a^2
*c^10*d^6)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) + 5*(b^5*c^7*d^6 - 8*a*b^3*c^8
*d^6 + 16*a^2*b*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 5*(b^6*c^6*d^
6 - 48*a^2*b^2*c^8*d^6 + 128*a^3*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*
x - 15*(b^7*c^5*d^6 - 16*a*b^5*c^6*d^6 + 80*a^2*b^3*c^7*d^6 - 128*a^3*b*c^8*d^6)
/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^8*c^4*d^6 - 12*a*b^6*c^5*d^6 +
40*a^2*b^4*c^6*d^6 - 128*a^4*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x -
(b^9*c^3*d^6 + 12*a*b^7*c^4*d^6 - 264*a^2*b^5*c^5*d^6 + 1280*a^3*b^3*c^6*d^6 - 1
920*a^4*b*c^7*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))/(c*x^2 + b*x + a)^(3/2)