3.1212 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^2} \, dx\)

Optimal. Leaf size=153 \[ \frac{15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{7/2} d^2}-\frac{15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^3 d^2}+\frac{5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)} \]

[Out]

(-15*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^3*d^2) + (5*(b + 2*
c*x)*(a + b*x + c*x^2)^(3/2))/(32*c^2*d^2) - (a + b*x + c*x^2)^(5/2)/(2*c*d^2*(b
+ 2*c*x)) + (15*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c
*x^2])])/(512*c^(7/2)*d^2)

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Rubi [A]  time = 0.171289, antiderivative size = 153, 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 \[ \frac{15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{7/2} d^2}-\frac{15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^3 d^2}+\frac{5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)} \]

Antiderivative was successfully verified.

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

[Out]

(-15*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^3*d^2) + (5*(b + 2*
c*x)*(a + b*x + c*x^2)^(3/2))/(32*c^2*d^2) - (a + b*x + c*x^2)^(5/2)/(2*c*d^2*(b
+ 2*c*x)) + (15*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c
*x^2])])/(512*c^(7/2)*d^2)

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Rubi in Sympy [A]  time = 26.1736, size = 146, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{2 c d^{2} \left (b + 2 c x\right )} + \frac{5 \left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{32 c^{2} d^{2}} - \frac{15 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{256 c^{3} d^{2}} + \frac{15 \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{512 c^{\frac{7}{2}} d^{2}} \]

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)**2,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(2*c*d**2*(b + 2*c*x)) + 5*(b + 2*c*x)*(a + b*x + c*x
**2)**(3/2)/(32*c**2*d**2) - 15*(b + 2*c*x)*(-4*a*c + b**2)*sqrt(a + b*x + c*x**
2)/(256*c**3*d**2) + 15*(-4*a*c + b**2)**2*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a +
b*x + c*x**2)))/(512*c**(7/2)*d**2)

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Mathematica [A]  time = 0.272607, size = 151, normalized size = 0.99 \[ \frac{\frac{\sqrt{a+x (b+c x)} \left (16 c^2 \left (-8 a^2+9 a c x^2+2 c^2 x^4\right )+4 b^2 c \left (25 a+3 c x^2\right )+16 b c^2 x \left (9 a+4 c x^2\right )-15 b^4-20 b^3 c x\right )}{256 c^3 (b+2 c x)}+\frac{15 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{512 c^{7/2}}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a + x*(b + c*x)]*(-15*b^4 - 20*b^3*c*x + 4*b^2*c*(25*a + 3*c*x^2) + 16*b*
c^2*x*(9*a + 4*c*x^2) + 16*c^2*(-8*a^2 + 9*a*c*x^2 + 2*c^2*x^4)))/(256*c^3*(b +
2*c*x)) + (15*(b^2 - 4*a*c)^2*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/
(512*c^(7/2)))/d^2

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Maple [B]  time = 0.017, size = 961, normalized size = 6.3 \[ \text{result too large to display} \]

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)^2,x)

[Out]

-1/c/d^2/(4*a*c-b^2)/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+1/d^2
/(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*x+1/2/c/d^2/(4*a*c-b^2)*(
(x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*b+5/4/d^2/(4*a*c-b^2)*((x+1/2*b/c)^2*c+
1/4*(4*a*c-b^2)/c)^(3/2)*x*a-5/16/c/d^2/(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-
b^2)/c)^(3/2)*x*b^2+5/8/c/d^2/(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3
/2)*b*a-5/32/c^2/d^2/(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^3+1
5/8/d^2/(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a^2-15/16/c/d^2/
(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a*b^2+15/128/c^2/d^2/(4*
a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*b^4+15/16/c/d^2/(4*a*c-b^2)
*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b*a^2-15/32/c^2/d^2/(4*a*c-b^2)*((x+1
/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^3*a+15/256/c^3/d^2/(4*a*c-b^2)*((x+1/2*b/
c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^5+15/8/c^(1/2)/d^2/(4*a*c-b^2)*ln(c^(1/2)*(x+1
/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*a^3-45/32/c^(3/2)/d^2/(4*a*c-
b^2)*ln(c^(1/2)*(x+1/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^2*a^2+4
5/128/c^(5/2)/d^2/(4*a*c-b^2)*ln(c^(1/2)*(x+1/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c
-b^2)/c)^(1/2))*b^4*a-15/512/c^(7/2)/d^2/(4*a*c-b^2)*ln(c^(1/2)*(x+1/2*b/c)+((x+
1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.379096, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 100 \, a b^{2} c - 128 \, a^{2} c^{2} + 12 \,{\left (b^{2} c^{2} + 12 \, a c^{3}\right )} x^{2} - 4 \,{\left (5 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{1024 \,{\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )} \sqrt{c}}, \frac{2 \,{\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 100 \, a b^{2} c - 128 \, a^{2} c^{2} + 12 \,{\left (b^{2} c^{2} + 12 \, a c^{3}\right )} x^{2} - 4 \,{\left (5 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{512 \,{\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )} \sqrt{-c}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[1/1024*(4*(32*c^4*x^4 + 64*b*c^3*x^3 - 15*b^4 + 100*a*b^2*c - 128*a^2*c^2 + 12*
(b^2*c^2 + 12*a*c^3)*x^2 - 4*(5*b^3*c - 36*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqr
t(c) + 15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)
*x)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 +
4*a*c)*sqrt(c)))/((2*c^4*d^2*x + b*c^3*d^2)*sqrt(c)), 1/512*(2*(32*c^4*x^4 + 64*
b*c^3*x^3 - 15*b^4 + 100*a*b^2*c - 128*a^2*c^2 + 12*(b^2*c^2 + 12*a*c^3)*x^2 - 4
*(5*b^3*c - 36*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 15*(b^5 - 8*a*b^3*c
+ 16*a^2*b*c^2 + 2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x)*arctan(1/2*(2*c*x + b)*
sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((2*c^4*d^2*x + b*c^3*d^2)*sqrt(-c))]

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

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)**2,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integ
ral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integr
al(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integra
l(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integral(2
*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integral(2
*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x))/d**2

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

[Out]

Exception raised: TypeError