3.1214 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx\)

Optimal. Leaf size=145 \[ -\frac{5 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} d^4}+\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3} \]

[Out]

(5*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^3*d^4) - (5*(a + b*x + c*x^2)^(3/2))
/(24*c^2*d^4*(b + 2*c*x)) - (a + b*x + c*x^2)^(5/2)/(6*c*d^4*(b + 2*c*x)^3) - (5
*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7
/2)*d^4)

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Rubi [A]  time = 0.193352, antiderivative size = 145, 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 \[ -\frac{5 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} d^4}+\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4,x]

[Out]

(5*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^3*d^4) - (5*(a + b*x + c*x^2)^(3/2))
/(24*c^2*d^4*(b + 2*c*x)) - (a + b*x + c*x^2)^(5/2)/(6*c*d^4*(b + 2*c*x)^3) - (5
*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7
/2)*d^4)

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Rubi in Sympy [A]  time = 36.1658, size = 138, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{6 c d^{4} \left (b + 2 c x\right )^{3}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2} d^{4} \left (b + 2 c x\right )} + \frac{5 \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{64 c^{3} d^{4}} - \frac{5 \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}} d^{4}} \]

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)**4,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(6*c*d**4*(b + 2*c*x)**3) - 5*(a + b*x + c*x**2)**(3/
2)/(24*c**2*d**4*(b + 2*c*x)) + 5*(b + 2*c*x)*sqrt(a + b*x + c*x**2)/(64*c**3*d*
*4) - 5*(-4*a*c + b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(1
28*c**(7/2)*d**4)

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Mathematica [A]  time = 0.411637, size = 115, normalized size = 0.79 \[ \frac{\frac{\sqrt{a+x (b+c x)} \left (-\frac{2 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{14 \left (b^2-4 a c\right )}{b+2 c x}+3 b+6 c x\right )}{192 c^3}-\frac{5 \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{128 c^{7/2}}}{d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4,x]

[Out]

((Sqrt[a + x*(b + c*x)]*(3*b + 6*c*x - (2*(b^2 - 4*a*c)^2)/(b + 2*c*x)^3 + (14*(
b^2 - 4*a*c))/(b + 2*c*x)))/(192*c^3) - (5*(b^2 - 4*a*c)*Log[b + 2*c*x + 2*Sqrt[
c]*Sqrt[a + x*(b + c*x)]])/(128*c^(7/2)))/d^4

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Maple [B]  time = 0.018, size = 1022, normalized size = 7.1 \[ \text{result too large to display} \]

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)^4,x)

[Out]

-1/12/d^4/c^3/(4*a*c-b^2)/(x+1/2*b/c)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2
)-4/3/d^4/c/(4*a*c-b^2)^2/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+
4/3/d^4/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*x+2/3/d^4/c/(4*a
*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*b+5/3/d^4/(4*a*c-b^2)^2*((x+
1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*x*a-5/12/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^
2*c+1/4*(4*a*c-b^2)/c)^(3/2)*x*b^2+5/6/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*
(4*a*c-b^2)/c)^(3/2)*b*a-5/24/d^4/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^3+5/2/d^4/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2
)*x*a^2-5/4/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a*b^
2+5/32/d^4/c^2/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*b^4+5/4
/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b*a^2-5/8/d^4/c^2
/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^3*a+5/64/d^4/c^3/(4*a
*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^5+5/2/d^4/c^(1/2)/(4*a*c-b
^2)^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^3-15/8
/d^4/c^(3/2)/(4*a*c-b^2)^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+15/32/d^4/c^(5/2)/(4*a*c-b^2)^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-5/128/d^4/c^(7/2)/(4*a*c-b^2)^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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.6269, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 15 \, b^{4} - 40 \, a b^{2} c - 32 \, a^{2} c^{2} + 32 \,{\left (4 \, b^{2} c^{2} - 7 \, a c^{3}\right )} x^{2} + 16 \,{\left (5 \, b^{3} c - 14 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (b^{5} - 4 \, a b^{3} c + 8 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \,{\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )} \sqrt{c}}, \frac{2 \,{\left (48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 15 \, b^{4} - 40 \, a b^{2} c - 32 \, a^{2} c^{2} + 32 \,{\left (4 \, b^{2} c^{2} - 7 \, a c^{3}\right )} x^{2} + 16 \,{\left (5 \, b^{3} c - 14 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 15 \,{\left (b^{5} - 4 \, a b^{3} c + 8 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \,{\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{384 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )} \sqrt{-c}}\right ] \]

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)^4,x, algorithm="fricas")

[Out]

[1/768*(4*(48*c^4*x^4 + 96*b*c^3*x^3 + 15*b^4 - 40*a*b^2*c - 32*a^2*c^2 + 32*(4*
b^2*c^2 - 7*a*c^3)*x^2 + 16*(5*b^3*c - 14*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt
(c) - 15*(b^5 - 4*a*b^3*c + 8*(b^2*c^3 - 4*a*c^4)*x^3 + 12*(b^3*c^2 - 4*a*b*c^3)
*x^2 + 6*(b^4*c - 4*a*b^2*c^2)*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)))/((8*c^6*d^4*x^3 + 12*b*c^5*d^4*x^
2 + 6*b^2*c^4*d^4*x + b^3*c^3*d^4)*sqrt(c)), 1/384*(2*(48*c^4*x^4 + 96*b*c^3*x^3
+ 15*b^4 - 40*a*b^2*c - 32*a^2*c^2 + 32*(4*b^2*c^2 - 7*a*c^3)*x^2 + 16*(5*b^3*c
- 14*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 15*(b^5 - 4*a*b^3*c + 8*(b^2*
c^3 - 4*a*c^4)*x^3 + 12*(b^3*c^2 - 4*a*b*c^3)*x^2 + 6*(b^4*c - 4*a*b^2*c^2)*x)*a
rctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((8*c^6*d^4*x^3 + 12*
b*c^5*d^4*x^2 + 6*b^2*c^4*d^4*x + b^3*c^3*d^4)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{c^{2} x^{4} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{2 a b x \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{2 a c x^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{2 b c x^{3} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \]

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)**4,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 3
2*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b
**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x) + Inte
gral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 3
2*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**
4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x) + Integr
al(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32
*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b
**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x))/d**4

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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)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError