3.1196 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx\)

Optimal. Leaf size=115 \[ \frac{\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16 c^{5/2} d}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]

[Out]

-((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])/(8*c^2*d) + (a + b*x + c*x^2)^(3/2)/(6*c*
d) + ((b^2 - 4*a*c)^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*
a*c]])/(16*c^(5/2)*d)

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Rubi [A]  time = 0.226103, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16 c^{5/2} d}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])/(8*c^2*d) + (a + b*x + c*x^2)^(3/2)/(6*c*
d) + ((b^2 - 4*a*c)^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*
a*c]])/(16*c^(5/2)*d)

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Rubi in Sympy [A]  time = 51.2958, size = 100, normalized size = 0.87 \[ \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{6 c d} - \frac{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{8 c^{2} d} + \frac{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{16 c^{\frac{5}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x + c*x**2)**(3/2)/(6*c*d) - (-4*a*c + b**2)*sqrt(a + b*x + c*x**2)/(8*c*
*2*d) + (-4*a*c + b**2)**(3/2)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c
+ b**2))/(16*c**(5/2)*d)

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Mathematica [A]  time = 0.413829, size = 165, normalized size = 1.43 \[ \frac{-3 \left (b^2-4 a c\right )^2 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)} \left (4 c \left (4 a+c x^2\right )-3 b^2+4 b c x\right )+3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{48 c^{5/2} d \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 4*b*c*x + 4*c*(4*a
+ c*x^2)) + 3*(b^2 - 4*a*c)^2*Log[b + 2*c*x] - 3*(b^2 - 4*a*c)^2*Log[-(b^2*Sqrt
[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])/(48*c^(5/2)*
Sqrt[-b^2 + 4*a*c]*d)

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Maple [B]  time = 0.012, size = 430, normalized size = 3.7 \[{\frac{1}{6\,cd} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4\,cd}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{b}^{2}}{16\,{c}^{2}d}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{a}^{2}}{cd}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{a{b}^{2}}{2\,{c}^{2}d}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{{b}^{4}}{16\,d{c}^{3}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6/d/c*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+1/4/d/c*(4*(x+1/2*b/c)^2*c+(4*
a*c-b^2)/c)^(1/2)*a-1/16/d/c^2*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^2-1/d/c
/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2
*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a^2+1/2/d/c^2/((4*a*c-b^2)/c)^(1/2)
*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/
c)^(1/2))/(x+1/2*b/c))*a*b^2-1/16/d/c^3/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2
)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/
c))*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)/(2*c*d*x + b*d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27119, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}}{96 \, c^{2} d}, \frac{3 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) + 2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}}{48 \, c^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/96*(3*(b^2 - 4*a*c)*sqrt(-(b^2 - 4*a*c)/c)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 +
8*a*c - 4*sqrt(c*x^2 + b*x + a)*c*sqrt(-(b^2 - 4*a*c)/c))/(4*c^2*x^2 + 4*b*c*x
+ b^2)) - 4*(4*c^2*x^2 + 4*b*c*x - 3*b^2 + 16*a*c)*sqrt(c*x^2 + b*x + a))/(c^2*d
), 1/48*(3*(b^2 - 4*a*c)*sqrt((b^2 - 4*a*c)/c)*arctan(-1/2*(b^2 - 4*a*c)/(sqrt(c
*x^2 + b*x + a)*c*sqrt((b^2 - 4*a*c)/c))) + 2*(4*c^2*x^2 + 4*b*c*x - 3*b^2 + 16*
a*c)*sqrt(c*x^2 + b*x + a))/(c^2*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224536, size = 201, normalized size = 1.75 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (4 \, x{\left (\frac{x}{d} + \frac{b}{c d}\right )} - \frac{3 \, b^{2} c^{3} d^{3} - 16 \, a c^{4} d^{3}}{c^{5} d^{4}}\right )} + \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{8 \, \sqrt{b^{2} c - 4 \, a c^{2}} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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