3.1200 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^5} \, dx\)

Optimal. Leaf size=123 \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt{b^2-4 a c}}-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4} \]

[Out]

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

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Rubi [A]  time = 0.211237, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt{b^2-4 a c}}-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.39, size = 117, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{8 c d^{5} \left (b + 2 c x\right )^{4}} - \frac{3 \sqrt{a + b x + c x^{2}}}{64 c^{2} d^{5} \left (b + 2 c x\right )^{2}} + \frac{3 \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{128 c^{\frac{5}{2}} d^{5} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x + c*x**2)**(3/2)/(8*c*d**5*(b + 2*c*x)**4) - 3*sqrt(a + b*x + c*x**2)/
(64*c**2*d**5*(b + 2*c*x)**2) + 3*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*
a*c + b**2))/(128*c**(5/2)*d**5*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.619681, size = 154, normalized size = 1.25 \[ \frac{-\frac{3 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\sqrt{4 a c-b^2}}-\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 c \left (2 a+5 c x^2\right )+3 b^2+20 b c x\right )}{(b+2 c x)^4}+\frac{3 \log (b+2 c x)}{\sqrt{4 a c-b^2}}}{128 c^{5/2} d^5} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(3*b^2 + 20*b*c*x + 4*c*(2*a + 5*c*x^2)))/(b
+ 2*c*x)^4 + (3*Log[b + 2*c*x])/Sqrt[-b^2 + 4*a*c] - (3*Log[-(b^2*Sqrt[c]) + 4*a
*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])/Sqrt[-b^2 + 4*a*c])/(1
28*c^(5/2)*d^5)

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Maple [B]  time = 0.021, size = 622, normalized size = 5.1 \[ -{\frac{1}{32\,{c}^{4}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-4}}-{\frac{1}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{16\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a}{32\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{b}^{2}}{128\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{a}^{2}}{8\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{3\,a{b}^{2}}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{3\,{b}^{4}}{128\,{c}^{3}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(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)^(5/2
)-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)^(
5/2)+1/16/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)+3/32/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-3/128/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)*b^2-3/8/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^2+3/16/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*b^2-3/128/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))*b^4

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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)^(3/2)/(2*c*d*x + b*d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.46116, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (20 \, c^{2} x^{2} + 20 \, b c x + 3 \, b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} - 3 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{256 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}}}, -\frac{2 \,{\left (20 \, c^{2} x^{2} + 20 \, b c x + 3 \, b^{2} + 8 \, a c\right )} \sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} + 3 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{128 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )} \sqrt{b^{2} c - 4 \, a c^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^5,x, algorithm="fricas")

[Out]

[-1/256*(4*(20*c^2*x^2 + 20*b*c*x + 3*b^2 + 8*a*c)*sqrt(-b^2*c + 4*a*c^2)*sqrt(c
*x^2 + b*x + a) - 3*(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*c^6*d^5*x^4
+ 32*b*c^5*d^5*x^3 + 24*b^2*c^4*d^5*x^2 + 8*b^3*c^3*d^5*x + b^4*c^2*d^5)*sqrt(-b
^2*c + 4*a*c^2)), -1/128*(2*(20*c^2*x^2 + 20*b*c*x + 3*b^2 + 8*a*c)*sqrt(b^2*c -
4*a*c^2)*sqrt(c*x^2 + b*x + a) + 3*(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*c^6*d^5*x^4 + 32*b*c^5*d^5*x^3 + 24*b^2*c^4*d^5*x^2 + 8*b^3*c^3*d^5*x + 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. \[ \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(Integral(a*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*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) + Integral(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))/d**5

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GIAC/XCAS [A]  time = 0.370445, size = 911, normalized size = 7.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^5,x, algorithm="giac")

[Out]

-1/48*(3*sqrt(-b^2*c + 4*a*c^2)*ln(sqrt(c))*sign(1/(2*c*d*x + b*d))*sign(c)*sign
(d)/(b^6*c^2*d^8*abs(c) - 12*a*b^4*c^3*d^8*abs(c) + 48*a^2*b^2*c^4*d^8*abs(c) -
64*a^3*c^5*d^8*abs(c)) + 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)*(5*(b^4*c^10*d^10*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d) - 8*a*
b^2*c^11*d^10*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d) + 16*a^2*c^12*d^10*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^16) - 2*(b^6*c^12*d^14*sig
n(1/(2*c*d*x + b*d))*sign(c)*sign(d) - 12*a*b^4*c^13*d^14*sign(1/(2*c*d*x + b*d)
)*sign(c)*sign(d) + 48*a^2*b^2*c^14*d^14*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d)
- 64*a^3*c^15*d^14*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d))/((b^8*c^12*d^16 - 1
6*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) - 3*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^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^8*
abs(c)))*d^2*abs(c)