3.1215 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx\)

Optimal. Leaf size=147 \[ -\frac{15 \sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{512 c^{7/2} d^5}+\frac{15 \sqrt{a+b x+c x^2}}{256 c^3 d^5}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4} \]

[Out]

(15*Sqrt[a + b*x + c*x^2])/(256*c^3*d^5) - (5*(a + b*x + c*x^2)^(3/2))/(64*c^2*d
^5*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(5/2)/(8*c*d^5*(b + 2*c*x)^4) - (15*Sqrt[b
^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(512*c^
(7/2)*d^5)

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Rubi [A]  time = 0.286636, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{15 \sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{512 c^{7/2} d^5}+\frac{15 \sqrt{a+b x+c x^2}}{256 c^3 d^5}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[a + b*x + c*x^2])/(256*c^3*d^5) - (5*(a + b*x + c*x^2)^(3/2))/(64*c^2*d
^5*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(5/2)/(8*c*d^5*(b + 2*c*x)^4) - (15*Sqrt[b
^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(512*c^
(7/2)*d^5)

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Rubi in Sympy [A]  time = 69.0653, size = 141, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{8 c d^{5} \left (b + 2 c x\right )^{4}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{64 c^{2} d^{5} \left (b + 2 c x\right )^{2}} + \frac{15 \sqrt{a + b x + c x^{2}}}{256 c^{3} d^{5}} - \frac{15 \sqrt{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{512 c^{\frac{7}{2}} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x + c*x**2)**(5/2)/(8*c*d**5*(b + 2*c*x)**4) - 5*(a + b*x + c*x**2)**(3/
2)/(64*c**2*d**5*(b + 2*c*x)**2) + 15*sqrt(a + b*x + c*x**2)/(256*c**3*d**5) - 1
5*sqrt(-4*a*c + b**2)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))
/(512*c**(7/2)*d**5)

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Mathematica [A]  time = 0.47388, size = 171, normalized size = 1.16 \[ \frac{-15 \sqrt{4 a c-b^2} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )-\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-9 \left (b^2-4 a c\right ) (b+2 c x)^2+2 \left (b^2-4 a c\right )^2-8 (b+2 c x)^4\right )}{(b+2 c x)^4}+15 \sqrt{4 a c-b^2} \log (b+2 c x)}{512 c^{7/2} d^5} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(2*(b^2 - 4*a*c)^2 - 9*(b^2 - 4*a*c)*(b + 2*c
*x)^2 - 8*(b + 2*c*x)^4))/(b + 2*c*x)^4 + 15*Sqrt[-b^2 + 4*a*c]*Log[b + 2*c*x] -
15*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)]])/(512*c^(7/2)*d^5)

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Maple [B]  time = 0.02, size = 900, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(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)^(7/2
)-3/16/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)^(
7/2)+3/16/d^5/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+5/16/d^5
/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*a-5/64/d^5/c^2/(4*a*c
-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^2+15/32/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)*a^2-15/64/d^5/c^2/(4*a*c-b^2)^2*(4*(x+1/
2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a*b^2+15/512/d^5/c^3/(4*a*c-b^2)^2*(4*(x+1/2*b/c
)^2*c+(4*a*c-b^2)/c)^(1/2)*b^4-15/8/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^3+45/32/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))*a^2*b^2-45/128/d^5/c^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))*a*b^4+15/512/d^5/c^4/(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^6

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.659963, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \,{\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 15 \, b^{4} - 20 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \,{\left (19 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2} + 4 \,{\left (25 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{1024 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}}, -\frac{15 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) - 2 \,{\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 15 \, b^{4} - 20 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \,{\left (19 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2} + 4 \,{\left (25 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{512 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1024*(15*(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)*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*(128*c^4*x^4 + 25
6*b*c^3*x^3 + 15*b^4 - 20*a*b^2*c - 32*a^2*c^2 + 12*(19*b^2*c^2 - 12*a*c^3)*x^2
+ 4*(25*b^3*c - 36*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a))/(16*c^7*d^5*x^4 + 32*b*c^6
*d^5*x^3 + 24*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*c^3*d^5), -1/512*(15*(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)*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*(
128*c^4*x^4 + 256*b*c^3*x^3 + 15*b^4 - 20*a*b^2*c - 32*a^2*c^2 + 12*(19*b^2*c^2
- 12*a*c^3)*x^2 + 4*(25*b^3*c - 36*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a))/(16*c^7*d^
5*x^4 + 32*b*c^6*d^5*x^3 + 24*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*c^3*d^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(Integral(a**2*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) + Integral(b**2*x**2*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) + Integral(c**2*x**4*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 + 3
2*c**5*x**5), x) + Integral(2*a*b*x*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) + In
tegral(2*a*c*x**2*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) + Integral(2*b*c*x**3*
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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out