3.1245 \(\int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{16 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac{8 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.214038, antiderivative size = 118, 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 \[ \frac{16 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac{8 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 54.8232, size = 109, normalized size = 0.92 \[ \frac{16 c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{d \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{8 c}{d \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} - \frac{2}{3 d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.813468, size = 154, normalized size = 1.31 \[ \frac{2 \left (\frac{24 c^{3/2} \log (b+2 c x)}{\left (4 a c-b^2\right )^{5/2}}-\frac{24 c^{3/2} \log \left (\sqrt{c} \left (-2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}-4 a c+b^2\right )\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{4 c \left (4 a+3 c x^2\right )-b^2+12 b c x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((-b^2 + 12*b*c*x + 4*c*(4*a + 3*c*x^2))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^(
3/2)) + (24*c^(3/2)*Log[b + 2*c*x])/(-b^2 + 4*a*c)^(5/2) - (24*c^(3/2)*Log[Sqrt[
c]*(b^2 - 4*a*c - 2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)])])/(-b^2 +
4*a*c)^(5/2)))/(3*d)

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Maple [B]  time = 0.016, size = 207, normalized size = 1.8 \[{\frac{2}{3\,d \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{-{\frac{3}{2}}}}+8\,{\frac{c}{d \left ( 4\,ac-{b}^{2} \right ) ^{2}}{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}-16\,{\frac{c}{d \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(5/2),x)

[Out]

2/3/d/(4*a*c-b^2)/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+8/d*c/(4*a*c-b^2)^2/
((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)-16/d*c/(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)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.361972, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (12 \,{\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c +{\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) +{\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x +{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}, -\frac{2 \,{\left (24 \,{\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c +{\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (\frac{1}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}\right ) -{\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x +{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="fricas")

[Out]

[2/3*(12*(c^3*x^4 + 2*b*c^2*x^3 + 2*a*b*c*x + a^2*c + (b^2*c + 2*a*c^2)*x^2)*sqr
t(-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)) + (12*c
^2*x^2 + 12*b*c*x - b^2 + 16*a*c)*sqrt(c*x^2 + b*x + a))/((b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a
*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^2*b
^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d), -2/3*(24*(c^3*x^4 + 2*b*c^2*x^3 + 2*a*b*c*x +
a^2*c + (b^2*c + 2*a*c^2)*x^2)*sqrt(c/(b^2 - 4*a*c))*arctan(1/2/(sqrt(c*x^2 + b
*x + a)*sqrt(c/(b^2 - 4*a*c)))) - (12*c^2*x^2 + 12*b*c*x - b^2 + 16*a*c)*sqrt(c*
x^2 + b*x + a))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3
*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 - 8
*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/(a**2*b*sqrt(a + b*x + c*x**2) + 2*a**2*c*x*sqrt(a + b*x + c*x**2) +
2*a*b**2*x*sqrt(a + b*x + c*x**2) + 6*a*b*c*x**2*sqrt(a + b*x + c*x**2) + 4*a*c*
*2*x**3*sqrt(a + b*x + c*x**2) + b**3*x**2*sqrt(a + b*x + c*x**2) + 4*b**2*c*x**
3*sqrt(a + b*x + c*x**2) + 5*b*c**2*x**4*sqrt(a + b*x + c*x**2) + 2*c**3*x**5*sq
rt(a + b*x + c*x**2)), x)/d

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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)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError