3.1182 \(\int (b d+2 c d x)^2 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=123 \[ -\frac{d^2 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{3/2}}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c} \]

[Out]

-((b^2 - 4*a*c)*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16*c) + (d^2*(b + 2*c*x)
^3*Sqrt[a + b*x + c*x^2])/(8*c) - ((b^2 - 4*a*c)^2*d^2*ArcTanh[(b + 2*c*x)/(2*Sq
rt[c]*Sqrt[a + b*x + c*x^2])])/(32*c^(3/2))

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Rubi [A]  time = 0.169088, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{d^2 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{3/2}}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16*c) + (d^2*(b + 2*c*x)
^3*Sqrt[a + b*x + c*x^2])/(8*c) - ((b^2 - 4*a*c)^2*d^2*ArcTanh[(b + 2*c*x)/(2*Sq
rt[c]*Sqrt[a + b*x + c*x^2])])/(32*c^(3/2))

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Rubi in Sympy [A]  time = 32.3924, size = 112, normalized size = 0.91 \[ \frac{d^{2} \left (b + 2 c x\right )^{3} \sqrt{a + b x + c x^{2}}}{8 c} - \frac{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{16 c} - \frac{d^{2} \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{32 c^{\frac{3}{2}}} \]

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)**(1/2),x)

[Out]

d**2*(b + 2*c*x)**3*sqrt(a + b*x + c*x**2)/(8*c) - d**2*(b + 2*c*x)*(-4*a*c + b*
*2)*sqrt(a + b*x + c*x**2)/(16*c) - d**2*(-4*a*c + b**2)**2*atanh((b + 2*c*x)/(2
*sqrt(c)*sqrt(a + b*x + c*x**2)))/(32*c**(3/2))

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Mathematica [A]  time = 0.171472, size = 98, normalized size = 0.8 \[ \frac{d^2 \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )-\left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{32 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(b^2 + 8*b*c*x + 4*c*(a + 2*c*
x^2)) - (b^2 - 4*a*c)^2*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(32*c
^(3/2))

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Maple [B]  time = 0.013, size = 230, normalized size = 1.9 \[{\frac{{b}^{2}{d}^{2}x}{8}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{3}{d}^{2}}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}{d}^{2}a}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{d}^{2}{b}^{4}}{32}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{d}^{2}cx \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}+{\frac{{d}^{2}b}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{c{d}^{2}ax}{2}\sqrt{c{x}^{2}+bx+a}}-{\frac{a{d}^{2}b}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}{d}^{2}}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]

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)^(1/2),x)

[Out]

1/8*d^2*b^2*(c*x^2+b*x+a)^(1/2)*x+1/16*d^2*b^3/c*(c*x^2+b*x+a)^(1/2)+1/4*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-1/32*d^2*b^4/c^(3/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+d^2*c*x*(c*x^2+b*x+a)^(3/2)+1/2*d^2*b*(c
*x^2+b*x+a)^(3/2)-1/2*d^2*c*a*(c*x^2+b*x+a)^(1/2)*x-1/4*d^2*a*(c*x^2+b*x+a)^(1/2
)*b-1/2*d^2*c^(1/2)*a^2*ln((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*sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239968, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \,{\left (16 \, c^{3} d^{2} x^{3} + 24 \, b c^{2} d^{2} x^{2} + 2 \,{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} + 4 \, a b c\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{64 \, c^{\frac{3}{2}}}, -\frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) - 2 \,{\left (16 \, c^{3} d^{2} x^{3} + 24 \, b c^{2} d^{2} x^{2} + 2 \,{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} + 4 \, a b c\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{32 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/64*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*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*(16*c^3*d^2*x^3 + 24*b*
c^2*d^2*x^2 + 2*(5*b^2*c + 4*a*c^2)*d^2*x + (b^3 + 4*a*b*c)*d^2)*sqrt(c*x^2 + b*
x + a)*sqrt(c))/c^(3/2), -1/32*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2*arctan(1/2*(2
*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) - 2*(16*c^3*d^2*x^3 + 24*b*c^2*d^2
*x^2 + 2*(5*b^2*c + 4*a*c^2)*d^2*x + (b^3 + 4*a*b*c)*d^2)*sqrt(c*x^2 + b*x + a)*
sqrt(-c))/(sqrt(-c)*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \left (\int b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 b c x \sqrt{a + b x + c x^{2}}\, dx\right ) \]

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)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229922, size = 209, normalized size = 1.7 \[ \frac{1}{16} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c^{2} d^{2} x + 3 \, b c d^{2}\right )} x + \frac{5 \, b^{2} c^{3} d^{2} + 4 \, a c^{4} d^{2}}{c^{3}}\right )} x + \frac{b^{3} c^{2} d^{2} + 4 \, a b c^{3} d^{2}}{c^{3}}\right )} + \frac{{\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{32 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(c*x^2 + b*x + a)*(2*(4*(2*c^2*d^2*x + 3*b*c*d^2)*x + (5*b^2*c^3*d^2 +
4*a*c^4*d^2)/c^3)*x + (b^3*c^2*d^2 + 4*a*b*c^3*d^2)/c^3) + 1/32*(b^4*d^2 - 8*a*b
^2*c*d^2 + 16*a^2*c^2*d^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)
- b))/c^(3/2)