3.1192 \(\int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=207 \[ \frac{3 d^4 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{5/2}}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}+\frac{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c} \]

[Out]

(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^2) + ((b^2 - 4
*a*c)^2*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(512*c^2) - ((b^2 - 4*a*c)*d^4*
(b + 2*c*x)^5*Sqrt[a + b*x + c*x^2])/(128*c^2) + (d^4*(b + 2*c*x)^5*(a + b*x + c
*x^2)^(3/2))/(16*c) + (3*(b^2 - 4*a*c)^4*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(2048*c^(5/2))

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Rubi [A]  time = 0.368385, 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 \[ \frac{3 d^4 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{5/2}}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}+\frac{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2),x]

[Out]

(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^2) + ((b^2 - 4
*a*c)^2*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(512*c^2) - ((b^2 - 4*a*c)*d^4*
(b + 2*c*x)^5*Sqrt[a + b*x + c*x^2])/(128*c^2) + (d^4*(b + 2*c*x)^5*(a + b*x + c
*x^2)^(3/2))/(16*c) + (3*(b^2 - 4*a*c)^4*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(2048*c^(5/2))

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Rubi in Sympy [A]  time = 69.6274, size = 199, normalized size = 0.96 \[ \frac{d^{4} \left (b + 2 c x\right )^{5} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{16 c} - \frac{d^{4} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{128 c^{2}} + \frac{d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{512 c^{2}} + \frac{3 d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{3} \sqrt{a + b x + c x^{2}}}{1024 c^{2}} + \frac{3 d^{4} \left (- 4 a c + b^{2}\right )^{4} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**(3/2),x)

[Out]

d**4*(b + 2*c*x)**5*(a + b*x + c*x**2)**(3/2)/(16*c) - d**4*(b + 2*c*x)**5*(-4*a
*c + b**2)*sqrt(a + b*x + c*x**2)/(128*c**2) + d**4*(b + 2*c*x)**3*(-4*a*c + b**
2)**2*sqrt(a + b*x + c*x**2)/(512*c**2) + 3*d**4*(b + 2*c*x)*(-4*a*c + b**2)**3*
sqrt(a + b*x + c*x**2)/(1024*c**2) + 3*d**4*(-4*a*c + b**2)**4*atanh((b + 2*c*x)
/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2048*c**(5/2))

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Mathematica [A]  time = 0.350634, size = 218, normalized size = 1.05 \[ \frac{d^4 \left (3 \left (b^2-4 a c\right )^4 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (-16 b^2 c^2 \left (11 a^2+140 a c x^2+216 c^2 x^4\right )-128 b c^3 x \left (a^2+24 a c x^2+24 c^2 x^4\right )-64 c^3 \left (-3 a^3+2 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right )-4 b^4 c \left (11 a+98 c x^2\right )-64 b^3 c^2 x \left (11 a+28 c x^2\right )+3 b^6-8 b^5 c x\right )\right )}{2048 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2),x]

[Out]

(d^4*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(3*b^6 - 8*b^5*c*x - 64*b^3*c
^2*x*(11*a + 28*c*x^2) - 4*b^4*c*(11*a + 98*c*x^2) - 128*b*c^3*x*(a^2 + 24*a*c*x
^2 + 24*c^2*x^4) - 16*b^2*c^2*(11*a^2 + 140*a*c*x^2 + 216*c^2*x^4) - 64*c^3*(-3*
a^3 + 2*a^2*c*x^2 + 24*a*c^2*x^4 + 16*c^3*x^6)) + 3*(b^2 - 4*a*c)^4*Log[b + 2*c*
x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(2048*c^(5/2))

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Maple [B]  time = 0.026, size = 641, 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)^4*(c*x^2+b*x+a)^(3/2),x)

[Out]

-1/8*d^4*c*b^2*a*(c*x^2+b*x+a)^(3/2)*x-9/32*d^4*c*b^2*a^2*(c*x^2+b*x+a)^(1/2)*x+
3/8*d^4*b^3*(c*x^2+b*x+a)^(5/2)+1/4*d^4*c^2*a^2*(c*x^2+b*x+a)^(3/2)*x+2*d^4*c^3*
x^3*(c*x^2+b*x+a)^(5/2)+3/8*d^4*c^(3/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))+1/64*d^4*b^4*(c*x^2+b*x+a)^(3/2)*x+1/128*d^4*b^5/c*(c*x^2+b*x+a)^(3/2)-3
/1024*d^4*b^7/c^2*(c*x^2+b*x+a)^(1/2)+3/2048*d^4*b^8/c^(5/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))-1/16*d^4*b^3*a*(c*x^2+b*x+a)^(3/2)-9/64*d^4*b^3*a^2*(c*
x^2+b*x+a)^(1/2)+9/256*d^4*b^5/c*(c*x^2+b*x+a)^(1/2)*a+9/64*d^4*b^4/c^(1/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-3/8*d^4*c^(1/2)*b^2*a^3*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/128*d^4*b^6/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*a+3*d^4*c^2*b*x^2*(c*x^2+b*x+a)^(5/2)+1/8*d^4*c*a^2*(c*x^2+b*x
+a)^(3/2)*b+3/8*d^4*c^2*a^3*(c*x^2+b*x+a)^(1/2)*x+3/16*d^4*c*a^3*(c*x^2+b*x+a)^(
1/2)*b-1/2*d^4*c*b*a*(c*x^2+b*x+a)^(5/2)+7/4*d^4*c*b^2*x*(c*x^2+b*x+a)^(5/2)-d^4
*c^2*a*x*(c*x^2+b*x+a)^(5/2)+9/128*d^4*b^4*(c*x^2+b*x+a)^(1/2)*x*a-3/512*d^4*b^6
/c*(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)^4*(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276042, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{4} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \,{\left (2048 \, c^{7} d^{4} x^{7} + 7168 \, b c^{6} d^{4} x^{6} + 768 \,{\left (13 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{4} x^{5} + 640 \,{\left (11 \, b^{3} c^{4} + 12 \, a b c^{5}\right )} d^{4} x^{4} + 16 \,{\left (161 \, b^{4} c^{3} + 472 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 24 \,{\left (17 \, b^{5} c^{2} + 152 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 2 \,{\left (b^{6} c + 396 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} - 192 \, a^{3} c^{4}\right )} d^{4} x -{\left (3 \, b^{7} - 44 \, a b^{5} c - 176 \, a^{2} b^{3} c^{2} + 192 \, a^{3} b c^{3}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{4096 \, c^{\frac{5}{2}}}, \frac{3 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{4} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \,{\left (2048 \, c^{7} d^{4} x^{7} + 7168 \, b c^{6} d^{4} x^{6} + 768 \,{\left (13 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{4} x^{5} + 640 \,{\left (11 \, b^{3} c^{4} + 12 \, a b c^{5}\right )} d^{4} x^{4} + 16 \,{\left (161 \, b^{4} c^{3} + 472 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 24 \,{\left (17 \, b^{5} c^{2} + 152 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 2 \,{\left (b^{6} c + 396 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} - 192 \, a^{3} c^{4}\right )} d^{4} x -{\left (3 \, b^{7} - 44 \, a b^{5} c - 176 \, a^{2} b^{3} c^{2} + 192 \, a^{3} b c^{3}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{2048 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

[1/4096*(3*(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
^4*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*(2048*c^7*d^4*x^7 + 7168*b*c^6*d^4*x^6 + 768*(13*b^2*c^5 + 4*
a*c^6)*d^4*x^5 + 640*(11*b^3*c^4 + 12*a*b*c^5)*d^4*x^4 + 16*(161*b^4*c^3 + 472*a
*b^2*c^4 + 16*a^2*c^5)*d^4*x^3 + 24*(17*b^5*c^2 + 152*a*b^3*c^3 + 16*a^2*b*c^4)*
d^4*x^2 + 2*(b^6*c + 396*a*b^4*c^2 + 240*a^2*b^2*c^3 - 192*a^3*c^4)*d^4*x - (3*b
^7 - 44*a*b^5*c - 176*a^2*b^3*c^2 + 192*a^3*b*c^3)*d^4)*sqrt(c*x^2 + b*x + a)*sq
rt(c))/c^(5/2), 1/2048*(3*(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^4*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) + 2
*(2048*c^7*d^4*x^7 + 7168*b*c^6*d^4*x^6 + 768*(13*b^2*c^5 + 4*a*c^6)*d^4*x^5 + 6
40*(11*b^3*c^4 + 12*a*b*c^5)*d^4*x^4 + 16*(161*b^4*c^3 + 472*a*b^2*c^4 + 16*a^2*
c^5)*d^4*x^3 + 24*(17*b^5*c^2 + 152*a*b^3*c^3 + 16*a^2*b*c^4)*d^4*x^2 + 2*(b^6*c
+ 396*a*b^4*c^2 + 240*a^2*b^2*c^3 - 192*a^3*c^4)*d^4*x - (3*b^7 - 44*a*b^5*c -
176*a^2*b^3*c^2 + 192*a^3*b*c^3)*d^4)*sqrt(c*x^2 + b*x + a)*sqrt(-c))/(sqrt(-c)*
c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{4} \left (\int a b^{4} \sqrt{a + b x + c x^{2}}\, dx + \int b^{5} x \sqrt{a + b x + c x^{2}}\, dx + \int 16 c^{5} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a c^{4} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 48 b c^{4} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 56 b^{2} c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 32 b^{3} c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 9 b^{4} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 32 a b c^{3} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 24 a b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 a b^{3} c x \sqrt{a + b x + c x^{2}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**(3/2),x)

[Out]

d**4*(Integral(a*b**4*sqrt(a + b*x + c*x**2), x) + Integral(b**5*x*sqrt(a + b*x
+ c*x**2), x) + Integral(16*c**5*x**6*sqrt(a + b*x + c*x**2), x) + Integral(16*a
*c**4*x**4*sqrt(a + b*x + c*x**2), x) + Integral(48*b*c**4*x**5*sqrt(a + b*x + c
*x**2), x) + Integral(56*b**2*c**3*x**4*sqrt(a + b*x + c*x**2), x) + Integral(32
*b**3*c**2*x**3*sqrt(a + b*x + c*x**2), x) + Integral(9*b**4*c*x**2*sqrt(a + b*x
+ c*x**2), x) + Integral(32*a*b*c**3*x**3*sqrt(a + b*x + c*x**2), x) + Integral
(24*a*b**2*c**2*x**2*sqrt(a + b*x + c*x**2), x) + Integral(8*a*b**3*c*x*sqrt(a +
b*x + c*x**2), x))

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GIAC/XCAS [A]  time = 0.233477, size = 528, normalized size = 2.55 \[ \frac{1}{1024} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (2 \, c^{5} d^{4} x + 7 \, b c^{4} d^{4}\right )} x + \frac{3 \,{\left (13 \, b^{2} c^{10} d^{4} + 4 \, a c^{11} d^{4}\right )}}{c^{7}}\right )} x + \frac{5 \,{\left (11 \, b^{3} c^{9} d^{4} + 12 \, a b c^{10} d^{4}\right )}}{c^{7}}\right )} x + \frac{161 \, b^{4} c^{8} d^{4} + 472 \, a b^{2} c^{9} d^{4} + 16 \, a^{2} c^{10} d^{4}}{c^{7}}\right )} x + \frac{3 \,{\left (17 \, b^{5} c^{7} d^{4} + 152 \, a b^{3} c^{8} d^{4} + 16 \, a^{2} b c^{9} d^{4}\right )}}{c^{7}}\right )} x + \frac{b^{6} c^{6} d^{4} + 396 \, a b^{4} c^{7} d^{4} + 240 \, a^{2} b^{2} c^{8} d^{4} - 192 \, a^{3} c^{9} d^{4}}{c^{7}}\right )} x - \frac{3 \, b^{7} c^{5} d^{4} - 44 \, a b^{5} c^{6} d^{4} - 176 \, a^{2} b^{3} c^{7} d^{4} + 192 \, a^{3} b c^{8} d^{4}}{c^{7}}\right )} - \frac{3 \,{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

1/1024*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(2*c^5*d^4*x + 7*b*c^4*d^4)*x + 3
*(13*b^2*c^10*d^4 + 4*a*c^11*d^4)/c^7)*x + 5*(11*b^3*c^9*d^4 + 12*a*b*c^10*d^4)/
c^7)*x + (161*b^4*c^8*d^4 + 472*a*b^2*c^9*d^4 + 16*a^2*c^10*d^4)/c^7)*x + 3*(17*
b^5*c^7*d^4 + 152*a*b^3*c^8*d^4 + 16*a^2*b*c^9*d^4)/c^7)*x + (b^6*c^6*d^4 + 396*
a*b^4*c^7*d^4 + 240*a^2*b^2*c^8*d^4 - 192*a^3*c^9*d^4)/c^7)*x - (3*b^7*c^5*d^4 -
44*a*b^5*c^6*d^4 - 176*a^2*b^3*c^7*d^4 + 192*a^3*b*c^8*d^4)/c^7) - 3/2048*(b^8*
d^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^2*d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^4*d^
4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)