3.1209 \(\int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=207 \[ -\frac{5 d^2 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}-\frac{5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c} \]

[Out]

(-5*(b^2 - 4*a*c)^3*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4096*c^3) + (5*(b^2
- 4*a*c)^2*d^2*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(2048*c^3) - (5*(b^2 - 4*a*c
)*d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2))/(384*c^2) + (d^2*(b + 2*c*x)^3*(a +
b*x + c*x^2)^(5/2))/(16*c) - (5*(b^2 - 4*a*c)^4*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(8192*c^(7/2))

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Rubi [A]  time = 0.337236, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{5 d^2 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}-\frac{5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b^2 - 4*a*c)^3*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4096*c^3) + (5*(b^2
- 4*a*c)^2*d^2*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(2048*c^3) - (5*(b^2 - 4*a*c
)*d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2))/(384*c^2) + (d^2*(b + 2*c*x)^3*(a +
b*x + c*x^2)^(5/2))/(16*c) - (5*(b^2 - 4*a*c)^4*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(8192*c^(7/2))

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Rubi in Sympy [A]  time = 69.0222, size = 202, normalized size = 0.98 \[ \frac{d^{2} \left (b + 2 c x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{16 c} - \frac{5 d^{2} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{2}} + \frac{5 d^{2} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{2048 c^{3}} - \frac{5 d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{3} \sqrt{a + b x + c x^{2}}}{4096 c^{3}} - \frac{5 d^{2} \left (- 4 a c + b^{2}\right )^{4} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8192 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(b + 2*c*x)**3*(a + b*x + c*x**2)**(5/2)/(16*c) - 5*d**2*(b + 2*c*x)**3*(-4
*a*c + b**2)*(a + b*x + c*x**2)**(3/2)/(384*c**2) + 5*d**2*(b + 2*c*x)**3*(-4*a*
c + b**2)**2*sqrt(a + b*x + c*x**2)/(2048*c**3) - 5*d**2*(b + 2*c*x)*(-4*a*c + b
**2)**3*sqrt(a + b*x + c*x**2)/(4096*c**3) - 5*d**2*(-4*a*c + b**2)**4*atanh((b
+ 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8192*c**(7/2))

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Mathematica [A]  time = 0.340082, size = 220, normalized size = 1.06 \[ \frac{d^2 \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (16 b^2 c^2 \left (73 a^2+580 a c x^2+584 c^2 x^4\right )+128 b c^3 x \left (59 a^2+136 a c x^2+72 c^2 x^4\right )+64 c^3 \left (15 a^3+118 a^2 c x^2+136 a c^2 x^4+48 c^3 x^6\right )+44 b^4 c \left (2 c x^2-5 a\right )+64 b^3 c^2 x \left (9 a+52 c x^2\right )+15 b^6-40 b^5 c x\right )-15 \left (b^2-4 a c\right )^4 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{24576 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^6 - 40*b^5*c*x + 44*b^4*
c*(-5*a + 2*c*x^2) + 64*b^3*c^2*x*(9*a + 52*c*x^2) + 128*b*c^3*x*(59*a^2 + 136*a
*c*x^2 + 72*c^2*x^4) + 16*b^2*c^2*(73*a^2 + 580*a*c*x^2 + 584*c^2*x^4) + 64*c^3*
(15*a^3 + 118*a^2*c*x^2 + 136*a*c^2*x^4 + 48*c^3*x^6)) - 15*(b^2 - 4*a*c)^4*Log[
b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(24576*c^(7/2))

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Maple [B]  time = 0.015, size = 634, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/512*d^2*b^4/c*(c*x^2+b*x+a)^(1/2)*x*a+1/4*d^2*b*(c*x^2+b*x+a)^(7/2)-5/96*d^2
*a^2*(c*x^2+b*x+a)^(3/2)*b-5/64*d^2*a^3*(c*x^2+b*x+a)^(1/2)*b-5/32*d^2*c^(1/2)*a
^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/1536*d^2*b^5/c^2*(c*x^2+b*x+a)^
(3/2)+5/4096*d^2*b^7/c^3*(c*x^2+b*x+a)^(1/2)-5/8192*d^2*b^8/c^(7/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/48*d^2*b^2*(c*x^2+b*x+a)^(5/2)*x+1/96*d^2*b^3/
c*(c*x^2+b*x+a)^(5/2)+1/2*d^2*c*x*(c*x^2+b*x+a)^(7/2)-1/24*d^2*a*(c*x^2+b*x+a)^(
5/2)*b-5/48*d^2*c*a^2*(c*x^2+b*x+a)^(3/2)*x+5/96*d^2*b^2*(c*x^2+b*x+a)^(3/2)*x*a
-5/768*d^2*b^4/c*(c*x^2+b*x+a)^(3/2)*x+5/192*d^2*b^3/c*(c*x^2+b*x+a)^(3/2)*a+15/
128*d^2*b^2*(c*x^2+b*x+a)^(1/2)*x*a^2+5/2048*d^2*b^6/c^2*(c*x^2+b*x+a)^(1/2)*x+1
5/256*d^2*b^3/c*(c*x^2+b*x+a)^(1/2)*a^2-1/12*d^2*c*a*(c*x^2+b*x+a)^(5/2)*x-15/10
24*d^2*b^5/c^2*(c*x^2+b*x+a)^(1/2)*a+5/32*d^2*b^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*a^3-15/256*d^2*b^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a^2+5/512*d^2*b^6/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*a-5/32*d^2*c*a^3*(c*x^2+b*x+a)^(1/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279802, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{2} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \,{\left (6144 \, c^{7} d^{2} x^{7} + 21504 \, b c^{6} d^{2} x^{6} + 256 \,{\left (109 \, b^{2} c^{5} + 68 \, a c^{6}\right )} d^{2} x^{5} + 640 \,{\left (25 \, b^{3} c^{4} + 68 \, a b c^{5}\right )} d^{2} x^{4} + 16 \,{\left (219 \, b^{4} c^{3} + 2248 \, a b^{2} c^{4} + 944 \, a^{2} c^{5}\right )} d^{2} x^{3} + 8 \,{\left (b^{5} c^{2} + 1304 \, a b^{3} c^{3} + 2832 \, a^{2} b c^{4}\right )} d^{2} x^{2} - 2 \,{\left (5 \, b^{6} c - 68 \, a b^{4} c^{2} - 4944 \, a^{2} b^{2} c^{3} - 960 \, a^{3} c^{4}\right )} d^{2} x +{\left (15 \, b^{7} - 220 \, a b^{5} c + 1168 \, a^{2} b^{3} c^{2} + 960 \, a^{3} b c^{3}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{49152 \, c^{\frac{7}{2}}}, -\frac{15 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{2} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) - 2 \,{\left (6144 \, c^{7} d^{2} x^{7} + 21504 \, b c^{6} d^{2} x^{6} + 256 \,{\left (109 \, b^{2} c^{5} + 68 \, a c^{6}\right )} d^{2} x^{5} + 640 \,{\left (25 \, b^{3} c^{4} + 68 \, a b c^{5}\right )} d^{2} x^{4} + 16 \,{\left (219 \, b^{4} c^{3} + 2248 \, a b^{2} c^{4} + 944 \, a^{2} c^{5}\right )} d^{2} x^{3} + 8 \,{\left (b^{5} c^{2} + 1304 \, a b^{3} c^{3} + 2832 \, a^{2} b c^{4}\right )} d^{2} x^{2} - 2 \,{\left (5 \, b^{6} c - 68 \, a b^{4} c^{2} - 4944 \, a^{2} b^{2} c^{3} - 960 \, a^{3} c^{4}\right )} d^{2} x +{\left (15 \, b^{7} - 220 \, a b^{5} c + 1168 \, a^{2} b^{3} c^{2} + 960 \, a^{3} b c^{3}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{24576 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/49152*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)
*d^2*log(4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 +
4*a*c)*sqrt(c)) + 4*(6144*c^7*d^2*x^7 + 21504*b*c^6*d^2*x^6 + 256*(109*b^2*c^5 +
68*a*c^6)*d^2*x^5 + 640*(25*b^3*c^4 + 68*a*b*c^5)*d^2*x^4 + 16*(219*b^4*c^3 + 2
248*a*b^2*c^4 + 944*a^2*c^5)*d^2*x^3 + 8*(b^5*c^2 + 1304*a*b^3*c^3 + 2832*a^2*b*
c^4)*d^2*x^2 - 2*(5*b^6*c - 68*a*b^4*c^2 - 4944*a^2*b^2*c^3 - 960*a^3*c^4)*d^2*x
+ (15*b^7 - 220*a*b^5*c + 1168*a^2*b^3*c^2 + 960*a^3*b*c^3)*d^2)*sqrt(c*x^2 + b
*x + a)*sqrt(c))/c^(7/2), -1/24576*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*
a^3*b^2*c^3 + 256*a^4*c^4)*d^2*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x
+ a)*c)) - 2*(6144*c^7*d^2*x^7 + 21504*b*c^6*d^2*x^6 + 256*(109*b^2*c^5 + 68*a*
c^6)*d^2*x^5 + 640*(25*b^3*c^4 + 68*a*b*c^5)*d^2*x^4 + 16*(219*b^4*c^3 + 2248*a*
b^2*c^4 + 944*a^2*c^5)*d^2*x^3 + 8*(b^5*c^2 + 1304*a*b^3*c^3 + 2832*a^2*b*c^4)*d
^2*x^2 - 2*(5*b^6*c - 68*a*b^4*c^2 - 4944*a^2*b^2*c^3 - 960*a^3*c^4)*d^2*x + (15
*b^7 - 220*a*b^5*c + 1168*a^2*b^3*c^2 + 960*a^3*b*c^3)*d^2)*sqrt(c*x^2 + b*x + a
)*sqrt(-c))/(sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \left (\int a^{2} b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int b^{4} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{4} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 2 a b^{3} x \sqrt{a + b x + c x^{2}}\, dx + \int 8 a c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 12 b c^{3} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 13 b^{2} c^{2} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 6 b^{3} c x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a b c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 10 a b^{2} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a^{2} b c x \sqrt{a + b x + c x^{2}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(Integral(a**2*b**2*sqrt(a + b*x + c*x**2), x) + Integral(b**4*x**2*sqrt(a
+ b*x + c*x**2), x) + Integral(4*c**4*x**6*sqrt(a + b*x + c*x**2), x) + Integral
(2*a*b**3*x*sqrt(a + b*x + c*x**2), x) + Integral(8*a*c**3*x**4*sqrt(a + b*x + c
*x**2), x) + Integral(4*a**2*c**2*x**2*sqrt(a + b*x + c*x**2), x) + Integral(12*
b*c**3*x**5*sqrt(a + b*x + c*x**2), x) + Integral(13*b**2*c**2*x**4*sqrt(a + b*x
+ c*x**2), x) + Integral(6*b**3*c*x**3*sqrt(a + b*x + c*x**2), x) + Integral(16
*a*b*c**2*x**3*sqrt(a + b*x + c*x**2), x) + Integral(10*a*b**2*c*x**2*sqrt(a + b
*x + c*x**2), x) + Integral(4*a**2*b*c*x*sqrt(a + b*x + c*x**2), x))

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GIAC/XCAS [A]  time = 0.233517, size = 525, normalized size = 2.54 \[ \frac{1}{12288} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (2 \, c^{4} d^{2} x + 7 \, b c^{3} d^{2}\right )} x + \frac{109 \, b^{2} c^{9} d^{2} + 68 \, a c^{10} d^{2}}{c^{7}}\right )} x + \frac{5 \,{\left (25 \, b^{3} c^{8} d^{2} + 68 \, a b c^{9} d^{2}\right )}}{c^{7}}\right )} x + \frac{219 \, b^{4} c^{7} d^{2} + 2248 \, a b^{2} c^{8} d^{2} + 944 \, a^{2} c^{9} d^{2}}{c^{7}}\right )} x + \frac{b^{5} c^{6} d^{2} + 1304 \, a b^{3} c^{7} d^{2} + 2832 \, a^{2} b c^{8} d^{2}}{c^{7}}\right )} x - \frac{5 \, b^{6} c^{5} d^{2} - 68 \, a b^{4} c^{6} d^{2} - 4944 \, a^{2} b^{2} c^{7} d^{2} - 960 \, a^{3} c^{8} d^{2}}{c^{7}}\right )} x + \frac{15 \, b^{7} c^{4} d^{2} - 220 \, a b^{5} c^{5} d^{2} + 1168 \, a^{2} b^{3} c^{6} d^{2} + 960 \, a^{3} b c^{7} d^{2}}{c^{7}}\right )} + \frac{5 \,{\left (b^{8} d^{2} - 16 \, a b^{6} c d^{2} + 96 \, a^{2} b^{4} c^{2} d^{2} - 256 \, a^{3} b^{2} c^{3} d^{2} + 256 \, a^{4} c^{4} d^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8192 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12288*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(2*c^4*d^2*x + 7*b*c^3*d^2)*x +
(109*b^2*c^9*d^2 + 68*a*c^10*d^2)/c^7)*x + 5*(25*b^3*c^8*d^2 + 68*a*b*c^9*d^2)/
c^7)*x + (219*b^4*c^7*d^2 + 2248*a*b^2*c^8*d^2 + 944*a^2*c^9*d^2)/c^7)*x + (b^5*
c^6*d^2 + 1304*a*b^3*c^7*d^2 + 2832*a^2*b*c^8*d^2)/c^7)*x - (5*b^6*c^5*d^2 - 68*
a*b^4*c^6*d^2 - 4944*a^2*b^2*c^7*d^2 - 960*a^3*c^8*d^2)/c^7)*x + (15*b^7*c^4*d^2
- 220*a*b^5*c^5*d^2 + 1168*a^2*b^3*c^6*d^2 + 960*a^3*b*c^7*d^2)/c^7) + 5/8192*(
b^8*d^2 - 16*a*b^6*c*d^2 + 96*a^2*b^4*c^2*d^2 - 256*a^3*b^2*c^3*d^2 + 256*a^4*c^
4*d^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)