3.1213 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{7/2} d^3}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{64 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2} \]

[Out]

(-5*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])/(64*c^3*d^3) + (5*(a + b*x + c*x^2)^(3/
2))/(48*c^2*d^3) - (a + b*x + c*x^2)^(5/2)/(4*c*d^3*(b + 2*c*x)^2) + (5*(b^2 - 4
*a*c)^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(128*c^
(7/2)*d^3)

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Rubi [A]  time = 0.290239, 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 \[ \frac{5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{7/2} d^3}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{64 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^3,x]

[Out]

(-5*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])/(64*c^3*d^3) + (5*(a + b*x + c*x^2)^(3/
2))/(48*c^2*d^3) - (a + b*x + c*x^2)^(5/2)/(4*c*d^3*(b + 2*c*x)^2) + (5*(b^2 - 4
*a*c)^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(128*c^
(7/2)*d^3)

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Rubi in Sympy [A]  time = 68.2208, size = 141, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{4 c d^{3} \left (b + 2 c x\right )^{2}} + \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{48 c^{2} d^{3}} - \frac{5 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{3} d^{3}} + \frac{5 \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{128 c^{\frac{7}{2}} d^{3}} \]

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)**3,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(4*c*d**3*(b + 2*c*x)**2) + 5*(a + b*x + c*x**2)**(3/
2)/(48*c**2*d**3) - 5*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)/(64*c**3*d**3) + 5*
(-4*a*c + b**2)**(3/2)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2)
)/(128*c**(7/2)*d**3)

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Mathematica [A]  time = 0.603423, size = 165, normalized size = 1.12 \[ \frac{-15 \left (4 a c-b^2\right )^{3/2} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-\frac{3 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}+56 a c-12 b^2+8 b c x+8 c^2 x^2\right )+15 \left (4 a c-b^2\right )^{3/2} \log (b+2 c x)}{384 c^{7/2} d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^3,x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-12*b^2 + 56*a*c + 8*b*c*x + 8*c^2*x^2 - (3*(b
^2 - 4*a*c)^2)/(b + 2*c*x)^2) + 15*(-b^2 + 4*a*c)^(3/2)*Log[b + 2*c*x] - 15*(-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)]])/(384*c^(7/2)*d^3)

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Maple [B]  time = 0.016, size = 840, normalized size = 5.7 \[ \text{result too large to display} \]

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)^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)^(7/2)
+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)^(5/2)+5/12/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)*a-5/48/d^3/c^2/(4*a*c-b^2)*((x
+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^2+5/8/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^2-5/16/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)*a*b^2+5/128/d^3/c^3/(4*a*c-b^2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^
(1/2)*b^4-5/2/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^3+
15/8/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^2*b^2-15/
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))*a*b^4+5/128/d
^3/c^4/(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^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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.424852, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \,{\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 80 \, a b^{2} c - 48 \, a^{2} c^{2} - 8 \,{\left (b^{2} c^{2} - 28 \, a c^{3}\right )} x^{2} - 8 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}}, \frac{15 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) + 2 \,{\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 80 \, a b^{2} c - 48 \, a^{2} c^{2} - 8 \,{\left (b^{2} c^{2} - 28 \, a c^{3}\right )} x^{2} - 8 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[-1/768*(15*(b^4 - 4*a*b^2*c + 4*(b^2*c^2 - 4*a*c^3)*x^2 + 4*(b^3*c - 4*a*b*c^2)
*x)*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*(32*c^4*
x^4 + 64*b*c^3*x^3 - 15*b^4 + 80*a*b^2*c - 48*a^2*c^2 - 8*(b^2*c^2 - 28*a*c^3)*x
^2 - 8*(5*b^3*c - 28*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a))/(4*c^5*d^3*x^2 + 4*b*c^4
*d^3*x + b^2*c^3*d^3), 1/384*(15*(b^4 - 4*a*b^2*c + 4*(b^2*c^2 - 4*a*c^3)*x^2 +
4*(b^3*c - 4*a*b*c^2)*x)*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*(32*c^4*x^4 + 64*b*c^3*x^3 - 15*b^
4 + 80*a*b^2*c - 48*a^2*c^2 - 8*(b^2*c^2 - 28*a*c^3)*x^2 - 8*(5*b^3*c - 28*a*b*c
^2)*x)*sqrt(c*x^2 + b*x + a))/(4*c^5*d^3*x^2 + 4*b*c^4*d^3*x + b^2*c^3*d^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**3,x)

[Out]

(Integral(a**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) + Integral(b**2*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) + Integral(c**2*x**4*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(2*a*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(2
*a*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) + Integral(2*b*c*x**3*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

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GIAC/XCAS [A]  time = 0.669429, 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)^(5/2)/(2*c*d*x + b*d)^3,x, algorithm="giac")

[Out]

sage0*x