3.1237 \(\int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{12 c \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac{2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}} \]

[Out]

-2/((b^2 - 4*a*c)*d^3*(b + 2*c*x)^2*Sqrt[a + b*x + c*x^2]) - (12*c*Sqrt[a + b*x
+ c*x^2])/((b^2 - 4*a*c)^2*d^3*(b + 2*c*x)^2) - (6*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*d^3)

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Rubi [A]  time = 0.220401, antiderivative size = 132, 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 \[ -\frac{12 c \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac{2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d^3*(b + 2*c*x)^2*Sqrt[a + b*x + c*x^2]) - (12*c*Sqrt[a + b*x
+ c*x^2])/((b^2 - 4*a*c)^2*d^3*(b + 2*c*x)^2) - (6*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*d^3)

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Rubi in Sympy [A]  time = 54.5126, size = 131, normalized size = 0.99 \[ - \frac{6 \sqrt{c} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{12 c \sqrt{a + b x + c x^{2}}}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{2}} - \frac{2}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(3/2),x)

[Out]

-6*sqrt(c)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(d**3*(-4*
a*c + b**2)**(5/2)) - 12*c*sqrt(a + b*x + c*x**2)/(d**3*(b + 2*c*x)**2*(-4*a*c +
b**2)**2) - 2/(d**3*(b + 2*c*x)**2*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.698234, size = 157, normalized size = 1.19 \[ \frac{\frac{6 \sqrt{c} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{5/2}}-\frac{2 \left (2 c \left (a+3 c x^2\right )+b^2+6 b c x\right )}{\left (b^2-4 a c\right )^2 (b+2 c x)^2 \sqrt{a+x (b+c x)}}-\frac{6 \sqrt{c} \log (b+2 c x)}{\left (4 a c-b^2\right )^{5/2}}}{d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((-2*(b^2 + 6*b*c*x + 2*c*(a + 3*c*x^2)))/((b^2 - 4*a*c)^2*(b + 2*c*x)^2*Sqrt[a
+ x*(b + c*x)]) - (6*Sqrt[c]*Log[b + 2*c*x])/(-b^2 + 4*a*c)^(5/2) + (6*Sqrt[c]*L
og[-(b^2*Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])
/(-b^2 + 4*a*c)^(5/2))/d^3

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Maple [A]  time = 0.018, size = 218, normalized size = 1.7 \[ -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}{\frac{1}{\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}}}-3\,{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}+6\,{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(3/2),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)^(1/2)
-3/d^3/(4*a*c-b^2)^2/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)+6/d^3/(4*a*c-b^2)
^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))

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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(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.389565, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 2 \,{\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt{c x^{2} + b x + a}}{4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} +{\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} +{\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x +{\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}, \frac{2 \,{\left (3 \,{\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (\frac{1}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}\right ) -{\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt{c x^{2} + b x + a}\right )}}{4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} +{\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} +{\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x +{\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(3/2)),x, algorithm="fricas")

[Out]

[(3*(4*c^3*x^4 + 8*b*c^2*x^3 + a*b^2 + (5*b^2*c + 4*a*c^2)*x^2 + (b^3 + 4*a*b*c)
*x)*sqrt(-c/(b^2 - 4*a*c))*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(c*x^
2 + b*x + a)*(b^2 - 4*a*c)*sqrt(-c/(b^2 - 4*a*c)))/(4*c^2*x^2 + 4*b*c*x + b^2))
- 2*(6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)*sqrt(c*x^2 + b*x + a))/(4*(b^4*c^3 - 8*a
*b^2*c^4 + 16*a^2*c^5)*d^3*x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x^
3 + (5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*x^2 + (b^7 - 4*a*
b^5*c - 16*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^3*x + (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2
*c^2)*d^3), 2*(3*(4*c^3*x^4 + 8*b*c^2*x^3 + a*b^2 + (5*b^2*c + 4*a*c^2)*x^2 + (b
^3 + 4*a*b*c)*x)*sqrt(c/(b^2 - 4*a*c))*arctan(1/2/(sqrt(c*x^2 + b*x + a)*sqrt(c/
(b^2 - 4*a*c)))) - (6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)*sqrt(c*x^2 + b*x + a))/(4
*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^
2*b*c^4)*d^3*x^3 + (5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*x^
2 + (b^7 - 4*a*b^5*c - 16*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^3*x + (a*b^6 - 8*a^2*b^4
*c + 16*a^3*b^2*c^2)*d^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{a b^{3} \sqrt{a + b x + c x^{2}} + 6 a b^{2} c x \sqrt{a + b x + c x^{2}} + 12 a b c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 8 a c^{3} x^{3} \sqrt{a + b x + c x^{2}} + b^{4} x \sqrt{a + b x + c x^{2}} + 7 b^{3} c x^{2} \sqrt{a + b x + c x^{2}} + 18 b^{2} c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 20 b c^{3} x^{4} \sqrt{a + b x + c x^{2}} + 8 c^{4} x^{5} \sqrt{a + b x + c x^{2}}}\, dx}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/(a*b**3*sqrt(a + b*x + c*x**2) + 6*a*b**2*c*x*sqrt(a + b*x + c*x**2)
+ 12*a*b*c**2*x**2*sqrt(a + b*x + c*x**2) + 8*a*c**3*x**3*sqrt(a + b*x + c*x**2)
+ b**4*x*sqrt(a + b*x + c*x**2) + 7*b**3*c*x**2*sqrt(a + b*x + c*x**2) + 18*b**
2*c**2*x**3*sqrt(a + b*x + c*x**2) + 20*b*c**3*x**4*sqrt(a + b*x + c*x**2) + 8*c
**4*x**5*sqrt(a + b*x + c*x**2)), x)/d**3

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(3/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError