3.1194 \(\int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=165 \[ \frac{d^2 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^2}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{64 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \]

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^2) - ((b^2 - 4*a*
c)*d^2*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(64*c^2) + (d^2*(b + 2*c*x)^3*(a + b
*x + c*x^2)^(3/2))/(12*c) + ((b^2 - 4*a*c)^3*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*
Sqrt[a + b*x + c*x^2])])/(256*c^(5/2))

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Rubi [A]  time = 0.248576, antiderivative size = 165, 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 \[ \frac{d^2 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^2}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{64 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^2) - ((b^2 - 4*a*
c)*d^2*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2])/(64*c^2) + (d^2*(b + 2*c*x)^3*(a + b
*x + c*x^2)^(3/2))/(12*c) + ((b^2 - 4*a*c)^3*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*
Sqrt[a + b*x + c*x^2])])/(256*c^(5/2))

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Rubi in Sympy [A]  time = 49.9997, size = 155, normalized size = 0.94 \[ \frac{d^{2} \left (b + 2 c x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{12 c} - \frac{d^{2} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{2}} + \frac{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{128 c^{2}} + \frac{d^{2} \left (- 4 a c + b^{2}\right )^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

d**2*(b + 2*c*x)**3*(a + b*x + c*x**2)**(3/2)/(12*c) - d**2*(b + 2*c*x)**3*(-4*a
*c + b**2)*sqrt(a + b*x + c*x**2)/(64*c**2) + d**2*(b + 2*c*x)*(-4*a*c + b**2)**
2*sqrt(a + b*x + c*x**2)/(128*c**2) + d**2*(-4*a*c + b**2)**3*atanh((b + 2*c*x)/
(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(256*c**(5/2))

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Mathematica [A]  time = 0.20763, size = 150, normalized size = 0.91 \[ \frac{d^2 \left (3 \left (b^2-4 a c\right )^3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (-16 c^2 \left (3 a^2+14 a c x^2+8 c^2 x^4\right )-8 b^2 c \left (4 a+17 c x^2\right )-32 b c^2 x \left (7 a+8 c x^2\right )+3 b^4-8 b^3 c x\right )\right )}{768 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(3*b^4 - 8*b^3*c*x - 32*b*c^2
*x*(7*a + 8*c*x^2) - 8*b^2*c*(4*a + 17*c*x^2) - 16*c^2*(3*a^2 + 14*a*c*x^2 + 8*c
^2*x^4)) + 3*(b^2 - 4*a*c)^3*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/
(768*c^(5/2))

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Maple [B]  time = 0.013, size = 406, normalized size = 2.5 \[{\frac{{d}^{2}b}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,c{d}^{2}x}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a{d}^{2}b}{12} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{2}b}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{{d}^{2}{b}^{4}x}{64\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{3}{d}^{2}a}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}{d}^{2}{a}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,{d}^{2}{b}^{4}a}{64}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{d}^{2}x}{24} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}{d}^{2}}{48\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{2}{b}^{5}}{128\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{d}^{2}{b}^{6}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{d}^{2}ax}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{c{d}^{2}ax}{6} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}c{d}^{2}x}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{3}{d}^{2}}{4}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]

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)^(3/2),x)

[Out]

1/3*d^2*b*(c*x^2+b*x+a)^(5/2)+2/3*d^2*c*x*(c*x^2+b*x+a)^(5/2)-1/12*d^2*a*(c*x^2+
b*x+a)^(3/2)*b-1/8*d^2*a^2*(c*x^2+b*x+a)^(1/2)*b-1/64*d^2*b^4/c*(c*x^2+b*x+a)^(1
/2)*x+1/16*d^2*b^3/c*(c*x^2+b*x+a)^(1/2)*a+3/16*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^2-3/64*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+1/24*d^2*b^2*(c*x^2+b*x+a)^(3/2)*x+1/48*d^2*b^3/c*(c*x^2+b*x+
a)^(3/2)-1/128*d^2*b^5/c^2*(c*x^2+b*x+a)^(1/2)+1/256*d^2*b^6/c^(5/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/8*d^2*b^2*(c*x^2+b*x+a)^(1/2)*x*a-1/6*d^2*c*a
*(c*x^2+b*x+a)^(3/2)*x-1/4*d^2*c*a^2*(c*x^2+b*x+a)^(1/2)*x-1/4*d^2*c^(1/2)*a^3*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258352, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) - 4 \,{\left (256 \, c^{5} d^{2} x^{5} + 640 \, b c^{4} d^{2} x^{4} + 16 \,{\left (33 \, b^{2} c^{3} + 28 \, a c^{4}\right )} d^{2} x^{3} + 8 \,{\left (19 \, b^{3} c^{2} + 84 \, a b c^{3}\right )} d^{2} x^{2} + 2 \,{\left (b^{4} c + 144 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} d^{2} x -{\left (3 \, b^{5} - 32 \, a b^{3} c - 48 \, a^{2} b c^{2}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{1536 \, c^{\frac{5}{2}}}, \frac{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \,{\left (256 \, c^{5} d^{2} x^{5} + 640 \, b c^{4} d^{2} x^{4} + 16 \,{\left (33 \, b^{2} c^{3} + 28 \, a c^{4}\right )} d^{2} x^{3} + 8 \,{\left (19 \, b^{3} c^{2} + 84 \, a b c^{3}\right )} d^{2} x^{2} + 2 \,{\left (b^{4} c + 144 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} d^{2} x -{\left (3 \, b^{5} - 32 \, a b^{3} c - 48 \, a^{2} b c^{2}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{768 \, \sqrt{-c} c^{2}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[-1/1536*(3*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*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*
(256*c^5*d^2*x^5 + 640*b*c^4*d^2*x^4 + 16*(33*b^2*c^3 + 28*a*c^4)*d^2*x^3 + 8*(1
9*b^3*c^2 + 84*a*b*c^3)*d^2*x^2 + 2*(b^4*c + 144*a*b^2*c^2 + 48*a^2*c^3)*d^2*x -
(3*b^5 - 32*a*b^3*c - 48*a^2*b*c^2)*d^2)*sqrt(c*x^2 + b*x + a)*sqrt(c))/c^(5/2)
, 1/768*(3*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^2*arctan(1/2*(2*c*
x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) + 2*(256*c^5*d^2*x^5 + 640*b*c^4*d^2*
x^4 + 16*(33*b^2*c^3 + 28*a*c^4)*d^2*x^3 + 8*(19*b^3*c^2 + 84*a*b*c^3)*d^2*x^2 +
2*(b^4*c + 144*a*b^2*c^2 + 48*a^2*c^3)*d^2*x - (3*b^5 - 32*a*b^3*c - 48*a^2*b*c
^2)*d^2)*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. \[ d^{2} \left (\int a b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int b^{3} x \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 b c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 5 b^{2} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a b c x \sqrt{a + b x + c x^{2}}\, dx\right ) \]

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)**(3/2),x)

[Out]

d**2*(Integral(a*b**2*sqrt(a + b*x + c*x**2), x) + Integral(b**3*x*sqrt(a + b*x
+ c*x**2), x) + Integral(4*c**3*x**4*sqrt(a + b*x + c*x**2), x) + Integral(4*a*c
**2*x**2*sqrt(a + b*x + c*x**2), x) + Integral(8*b*c**2*x**3*sqrt(a + b*x + c*x*
*2), x) + Integral(5*b**2*c*x**2*sqrt(a + b*x + c*x**2), x) + Integral(4*a*b*c*x
*sqrt(a + b*x + c*x**2), x))

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GIAC/XCAS [A]  time = 0.229692, size = 350, normalized size = 2.12 \[ \frac{1}{384} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, c^{3} d^{2} x + 5 \, b c^{2} d^{2}\right )} x + \frac{33 \, b^{2} c^{6} d^{2} + 28 \, a c^{7} d^{2}}{c^{5}}\right )} x + \frac{19 \, b^{3} c^{5} d^{2} + 84 \, a b c^{6} d^{2}}{c^{5}}\right )} x + \frac{b^{4} c^{4} d^{2} + 144 \, a b^{2} c^{5} d^{2} + 48 \, a^{2} c^{6} d^{2}}{c^{5}}\right )} x - \frac{3 \, b^{5} c^{3} d^{2} - 32 \, a b^{3} c^{4} d^{2} - 48 \, a^{2} b c^{5} d^{2}}{c^{5}}\right )} - \frac{{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{5}{2}}} \]

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)^(3/2),x, algorithm="giac")

[Out]

1/384*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*c^3*d^2*x + 5*b*c^2*d^2)*x + (33*b^2*
c^6*d^2 + 28*a*c^7*d^2)/c^5)*x + (19*b^3*c^5*d^2 + 84*a*b*c^6*d^2)/c^5)*x + (b^4
*c^4*d^2 + 144*a*b^2*c^5*d^2 + 48*a^2*c^6*d^2)/c^5)*x - (3*b^5*c^3*d^2 - 32*a*b^
3*c^4*d^2 - 48*a^2*b*c^5*d^2)/c^5) - 1/256*(b^6*d^2 - 12*a*b^4*c*d^2 + 48*a^2*b^
2*c^2*d^2 - 64*a^3*c^3*d^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c
) - b))/c^(5/2)