3.1188 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^5} \, dx\)

Optimal. Leaf size=133 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{32 c^{3/2} d^5 \left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{16 c d^5 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{8 c d^5 (b+2 c x)^4} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(8*c*d^5*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(16*c*(b^
2 - 4*a*c)*d^5*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^
2 - 4*a*c]]/(32*c^(3/2)*(b^2 - 4*a*c)^(3/2)*d^5)

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Rubi [A]  time = 0.224488, antiderivative size = 133, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{32 c^{3/2} d^5 \left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{16 c d^5 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{8 c d^5 (b+2 c x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x + c*x^2]/(8*c*d^5*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(16*c*(b^
2 - 4*a*c)*d^5*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^
2 - 4*a*c]]/(32*c^(3/2)*(b^2 - 4*a*c)^(3/2)*d^5)

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Rubi in Sympy [A]  time = 55.0965, size = 121, normalized size = 0.91 \[ \frac{\sqrt{a + b x + c x^{2}}}{16 c d^{5} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x + c x^{2}}}{8 c d^{5} \left (b + 2 c x\right )^{4}} + \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{32 c^{\frac{3}{2}} d^{5} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b*x + c*x**2)/(16*c*d**5*(b + 2*c*x)**2*(-4*a*c + b**2)) - sqrt(a + b*x
+ c*x**2)/(8*c*d**5*(b + 2*c*x)**4) + atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqr
t(-4*a*c + b**2))/(32*c**(3/2)*d**5*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.842007, size = 162, normalized size = 1.22 \[ \frac{\frac{\log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 c \left (2 a+c x^2\right )-b^2+4 b c x\right )}{\left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\log (b+2 c x)}{\left (4 a c-b^2\right )^{3/2}}}{32 c^{3/2} d^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 400, normalized size = 3. \[ -{\frac{1}{32\,{c}^{4}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-4}}+{\frac{1}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}-{\frac{1}{32\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}+{\frac{a}{8\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{{b}^{2}}{32\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.434168, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} -{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} - 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{64 \,{\left (16 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{5} x^{4} + 32 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{5} x^{3} + 24 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{5} x^{2} + 8 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{5} x +{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{5}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}}}, \frac{2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} -{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{32 \,{\left (16 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{5} x^{4} + 32 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{5} x^{3} + 24 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{5} x^{2} + 8 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{5} x +{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{5}\right )} \sqrt{b^{2} c - 4 \, a c^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/64*(4*(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c)*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 +
b*x + a) - (16*c^4*x^4 + 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4)*log(-
((4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c)*sqrt(-b^2*c + 4*a*c^2) - 4*(b^2*c - 4*a*c^2
)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*b*c*x + b^2)))/((16*(b^2*c^5 - 4*a*c^6)*
d^5*x^4 + 32*(b^3*c^4 - 4*a*b*c^5)*d^5*x^3 + 24*(b^4*c^3 - 4*a*b^2*c^4)*d^5*x^2
+ 8*(b^5*c^2 - 4*a*b^3*c^3)*d^5*x + (b^6*c - 4*a*b^4*c^2)*d^5)*sqrt(-b^2*c + 4*a
*c^2)), 1/32*(2*(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c)*sqrt(b^2*c - 4*a*c^2)*sqrt(c
*x^2 + b*x + a) - (16*c^4*x^4 + 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4)
*arctan(1/2*sqrt(b^2*c - 4*a*c^2)/(sqrt(c*x^2 + b*x + a)*c)))/((16*(b^2*c^5 - 4*
a*c^6)*d^5*x^4 + 32*(b^3*c^4 - 4*a*b*c^5)*d^5*x^3 + 24*(b^4*c^3 - 4*a*b^2*c^4)*d
^5*x^2 + 8*(b^5*c^2 - 4*a*b^3*c^3)*d^5*x + (b^6*c - 4*a*b^4*c^2)*d^5)*sqrt(b^2*c
- 4*a*c^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**
2*c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x)/d**5

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GIAC/XCAS [A]  time = 0.361057, size = 860, normalized size = 6.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6144*(sqrt(-b^2*c + 4*a*c^2)*ln(sqrt(c))*sign(1/(2*c*d*x + b*d))*sign(c)*sign
(d)/(b^8*c^5*d^13 - 16*a*b^6*c^6*d^13 + 96*a^2*b^4*c^7*d^13 - 256*a^3*b^2*c^8*d^
13 + 256*a^4*c^9*d^13) - sqrt(-b^2*c*d^2/(2*c*d*x + b*d)^2 + 4*a*c^2*d^2/(2*c*d*
x + b*d)^2 + c)*((b^2*c^7*d^5*abs(c)*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d) - 4
*a*c^8*d^5*abs(c)*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d))/(b^8*c^12*d^16 - 16*a
*b^6*c^13*d^16 + 96*a^2*b^4*c^14*d^16 - 256*a^3*b^2*c^15*d^16 + 256*a^4*c^16*d^1
6) - 2*(b^4*c^9*d^9*abs(c)*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d) - 8*a*b^2*c^1
0*d^9*abs(c)*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d) + 16*a^2*c^11*d^9*abs(c)*si
gn(1/(2*c*d*x + b*d))*sign(c)*sign(d))/((b^8*c^12*d^16 - 16*a*b^6*c^13*d^16 + 96
*a^2*b^4*c^14*d^16 - 256*a^3*b^2*c^15*d^16 + 256*a^4*c^16*d^16)*(2*c*d*x + b*d)^
2*c^2*d^2))/((2*c*d*x + b*d)*c*d) - sqrt(-b^2*c + 4*a*c^2)*ln(abs(sqrt(-b^2*c*d^
2/(2*c*d*x + b*d)^2 + 4*a*c^2*d^2/(2*c*d*x + b*d)^2 + c) + sqrt(-b^2*c^3*d^4 + 4
*a*c^4*d^4)/((2*c*d*x + b*d)*c*d)))*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d)/((b^
8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^13))*d^
2*abs(c)