3.2335 \(\int \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=112 \[ \frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^2) + ((b + 2*c*x)*(a
+ b*x + c*x^2)^(3/2))/(8*c) + (3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*
Sqrt[a + b*x + c*x^2])])/(128*c^(5/2))

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Rubi [A]  time = 0.0935915, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2),x]

[Out]

(-3*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^2) + ((b + 2*c*x)*(a
+ b*x + c*x^2)^(3/2))/(8*c) + (3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*
Sqrt[a + b*x + c*x^2])])/(128*c^(5/2))

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Rubi in Sympy [A]  time = 8.8518, size = 105, normalized size = 0.94 \[ \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{8 c} - \frac{3 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{2}} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b + 2*c*x)*(a + b*x + c*x**2)**(3/2)/(8*c) - 3*(b + 2*c*x)*(-4*a*c + b**2)*sqrt
(a + b*x + c*x**2)/(64*c**2) + 3*(-4*a*c + b**2)**2*atanh((b + 2*c*x)/(2*sqrt(c)
*sqrt(a + b*x + c*x**2)))/(128*c**(5/2))

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Mathematica [A]  time = 0.112391, size = 99, normalized size = 0.88 \[ \frac{2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{128 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*
x^2)) + 3*(b^2 - 4*a*c)^2*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(128
*c^(5/2))

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Maple [B]  time = 0.005, size = 201, normalized size = 1.8 \[{\frac{2\,cx+b}{8\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ax}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,ab}{16\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,a{b}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2),x)

[Out]

1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/8*(c*x^2+b*x+a)^(1/2)*x*a-3/32/c*(c*x^2+b*
x+a)^(1/2)*x*b^2+3/16/c*(c*x^2+b*x+a)^(1/2)*b*a-3/64/c^2*(c*x^2+b*x+a)^(1/2)*b^3
+3/8/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-3/16/c^(3/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*b^2+3/128/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*b^4

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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((c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245165, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} - 3 \, b^{3} + 20 \, a b c + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{256 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} - 3 \, b^{3} + 20 \, a b c + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{128 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(4*(16*c^3*x^3 + 24*b*c^2*x^2 - 3*b^3 + 20*a*b*c + 2*(b^2*c + 20*a*c^2)*x
)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^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)))/
c^(5/2), 1/128*(2*(16*c^3*x^3 + 24*b*c^2*x^2 - 3*b^3 + 20*a*b*c + 2*(b^2*c + 20*
a*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*arct
an(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.216883, size = 166, normalized size = 1.48 \[ \frac{1}{64} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c x + 3 \, b\right )} x + \frac{b^{2} c^{2} + 20 \, a c^{3}}{c^{3}}\right )} x - \frac{3 \, b^{3} c - 20 \, a b c^{2}}{c^{3}}\right )} - \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*sqrt(c*x^2 + b*x + a)*(2*(4*(2*c*x + 3*b)*x + (b^2*c^2 + 20*a*c^3)/c^3)*x -
(3*b^3*c - 20*a*b*c^2)/c^3) - 3/128*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*ln(abs(-2*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)