3.1198 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx\)

Optimal. Leaf size=115 \[ -\frac{3 \sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{32 c^{5/2} d^3}+\frac{3 \sqrt{a+b x+c x^2}}{16 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2} \]

[Out]

(3*Sqrt[a + b*x + c*x^2])/(16*c^2*d^3) - (a + b*x + c*x^2)^(3/2)/(4*c*d^3*(b + 2
*c*x)^2) - (3*Sqrt[b^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^
2 - 4*a*c]])/(32*c^(5/2)*d^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.206146, antiderivative size = 115, 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 \[ -\frac{3 \sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{32 c^{5/2} d^3}+\frac{3 \sqrt{a+b x+c x^2}}{16 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[a + b*x + c*x^2])/(16*c^2*d^3) - (a + b*x + c*x^2)^(3/2)/(4*c*d^3*(b + 2
*c*x)^2) - (3*Sqrt[b^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^
2 - 4*a*c]])/(32*c^(5/2)*d^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.1406, size = 109, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{4 c d^{3} \left (b + 2 c x\right )^{2}} + \frac{3 \sqrt{a + b x + c x^{2}}}{16 c^{2} d^{3}} - \frac{3 \sqrt{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{32 c^{\frac{5}{2}} d^{3}} \]

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)**3,x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(4*c*d**3*(b + 2*c*x)**2) + 3*sqrt(a + b*x + c*x**2)/
(16*c**2*d**3) - 3*sqrt(-4*a*c + b**2)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqr
t(-4*a*c + b**2))/(32*c**(5/2)*d**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.448311, size = 141, normalized size = 1.23 \[ \frac{-3 \sqrt{4 a c-b^2} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (\frac{b^2-4 a c}{(b+2 c x)^2}+2\right )+3 \sqrt{4 a c-b^2} \log (b+2 c x)}{32 c^{5/2} d^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(2 + (b^2 - 4*a*c)/(b + 2*c*x)^2) + 3*Sqrt[-b^2
+ 4*a*c]*Log[b + 2*c*x] - 3*Sqrt[-b^2 + 4*a*c]*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2)
+ 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])/(32*c^(5/2)*d^3)

_______________________________________________________________________________________

Maple [B]  time = 0.015, size = 562, normalized size = 4.9 \[ -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{4\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a}{8\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{b}^{2}}{32\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{a}^{2}}{2\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{3\,a{b}^{2}}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{3\,{b}^{4}}{32\,{c}^{3}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

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)^3,x)

[Out]

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

_______________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.315619, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \,{\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt{c x^{2} + b x + a}}{64 \,{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}}, -\frac{3 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) - 2 \,{\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt{c x^{2} + b x + a}}{32 \,{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \]

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)**3,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8*c**3*
x**3), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c*x + 12*b*c**2*x
**2 + 8*c**3*x**3), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c
*x + 12*b*c**2*x**2 + 8*c**3*x**3), x))/d**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.645718, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

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)^3,x, algorithm="giac")

[Out]

sage0*x