3.1204 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx\)
Optimal. Leaf size=207 \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}+\frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]
[Out]
-Sqrt[a + b*x + c*x^2]/(128*c^2*d^9*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(512*
c^2*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^4) + (3*Sqrt[a + b*x + c*x^2])/(1024*c^2*(b^2
- 4*a*c)^2*d^9*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(16*c*d^9*(b + 2*c*x)^8)
+ (3*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(2048*c^(5/2)
*(b^2 - 4*a*c)^(5/2)*d^9)
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Rubi [A] time = 0.386108, antiderivative size = 207, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154
\[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}+\frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]
[Out]
-Sqrt[a + b*x + c*x^2]/(128*c^2*d^9*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(512*
c^2*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^4) + (3*Sqrt[a + b*x + c*x^2])/(1024*c^2*(b^2
- 4*a*c)^2*d^9*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(16*c*d^9*(b + 2*c*x)^8)
+ (3*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(2048*c^(5/2)
*(b^2 - 4*a*c)^(5/2)*d^9)
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Rubi in Sympy [A] time = 97.3037, size = 197, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{16 c d^{9} \left (b + 2 c x\right )^{8}} + \frac{3 \sqrt{a + b x + c x^{2}}}{1024 c^{2} d^{9} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{\sqrt{a + b x + c x^{2}}}{512 c^{2} d^{9} \left (b + 2 c x\right )^{4} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x + c x^{2}}}{128 c^{2} d^{9} \left (b + 2 c x\right )^{6}} + \frac{3 \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2048 c^{\frac{5}{2}} d^{9} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]
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)**9,x)
[Out]
-(a + b*x + c*x**2)**(3/2)/(16*c*d**9*(b + 2*c*x)**8) + 3*sqrt(a + b*x + c*x**2)
/(1024*c**2*d**9*(b + 2*c*x)**2*(-4*a*c + b**2)**2) + sqrt(a + b*x + c*x**2)/(51
2*c**2*d**9*(b + 2*c*x)**4*(-4*a*c + b**2)) - sqrt(a + b*x + c*x**2)/(128*c**2*d
**9*(b + 2*c*x)**6) + 3*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2
))/(2048*c**(5/2)*d**9*(-4*a*c + b**2)**(5/2))
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Mathematica [A] time = 0.559137, size = 207, normalized size = 1. \[ \frac{-3 (b+2 c x)^8 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)} \left (2 \left (b^2-4 a c\right ) (b+2 c x)^4-24 \left (b^2-4 a c\right )^2 (b+2 c x)^2+16 \left (b^2-4 a c\right )^3+3 (b+2 c x)^6\right )+3 (b+2 c x)^8 \log (b+2 c x)}{2048 c^{5/2} d^9 \left (4 a c-b^2\right )^{5/2} (b+2 c x)^8} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]
[Out]
(2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]*(16*(b^2 - 4*a*c)^3 - 24*(b^
2 - 4*a*c)^2*(b + 2*c*x)^2 + 2*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 3*(b + 2*c*x)^6) +
3*(b + 2*c*x)^8*Log[b + 2*c*x] - 3*(b + 2*c*x)^8*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2
) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])/(2048*c^(5/2)*(-b^2 + 4*a*c)^
(5/2)*d^9*(b + 2*c*x)^8)
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Maple [B] time = 0.062, size = 742, normalized size = 3.6 \[ \text{result too large to display} \]
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)^9,x)
[Out]
-1/1024/d^9/c^8/(4*a*c-b^2)/(x+1/2*b/c)^8*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5
/2)+1/512/d^9/c^6/(4*a*c-b^2)^2/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c
)^(5/2)-1/512/d^9/c^4/(4*a*c-b^2)^3/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^
2)/c)^(5/2)-1/256/d^9/c^2/(4*a*c-b^2)^4/(x+1/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*
c-b^2)/c)^(5/2)+1/256/d^9/c/(4*a*c-b^2)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3
/2)+3/512/d^9/c/(4*a*c-b^2)^4*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a-3/2048/d
^9/c^2/(4*a*c-b^2)^4*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^2-3/128/d^9/c/(4*
a*c-b^2)^4/((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+3/256/d^9/c^2/(4*a*c-
b^2)^4/((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^2-3/2048/d^9/c^3/(4*a*c-b
^2)^4/((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)^9,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 4.56443, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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)^9,x, algorithm="fricas")
[Out]
[1/4096*(4*(192*c^6*x^6 + 576*b*c^5*x^5 - 3*b^6 - 8*a*b^4*c + 384*a^2*b^2*c^2 -
1024*a^3*c^3 + 16*(47*b^2*c^4 - 8*a*c^5)*x^4 + 32*(17*b^3*c^3 - 8*a*b*c^4)*x^3 +
12*(11*b^4*c^2 + 48*a*b^2*c^3 - 128*a^2*c^4)*x^2 - 4*(11*b^5*c - 176*a*b^3*c^2
+ 384*a^2*b*c^3)*x)*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a) + 3*(256*c^8*x^
8 + 1024*b*c^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c^5*x^5 + 1120*b^4*c^4*x^4 + 44
8*b^5*c^3*x^3 + 112*b^6*c^2*x^2 + 16*b^7*c*x + b^8)*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)))/((256*(b^4*c^10 - 8*a*b^2*c^11 + 16*a^2*c^12)*d^
9*x^8 + 1024*(b^5*c^9 - 8*a*b^3*c^10 + 16*a^2*b*c^11)*d^9*x^7 + 1792*(b^6*c^8 -
8*a*b^4*c^9 + 16*a^2*b^2*c^10)*d^9*x^6 + 1792*(b^7*c^7 - 8*a*b^5*c^8 + 16*a^2*b^
3*c^9)*d^9*x^5 + 1120*(b^8*c^6 - 8*a*b^6*c^7 + 16*a^2*b^4*c^8)*d^9*x^4 + 448*(b^
9*c^5 - 8*a*b^7*c^6 + 16*a^2*b^5*c^7)*d^9*x^3 + 112*(b^10*c^4 - 8*a*b^8*c^5 + 16
*a^2*b^6*c^6)*d^9*x^2 + 16*(b^11*c^3 - 8*a*b^9*c^4 + 16*a^2*b^7*c^5)*d^9*x + (b^
12*c^2 - 8*a*b^10*c^3 + 16*a^2*b^8*c^4)*d^9)*sqrt(-b^2*c + 4*a*c^2)), 1/2048*(2*
(192*c^6*x^6 + 576*b*c^5*x^5 - 3*b^6 - 8*a*b^4*c + 384*a^2*b^2*c^2 - 1024*a^3*c^
3 + 16*(47*b^2*c^4 - 8*a*c^5)*x^4 + 32*(17*b^3*c^3 - 8*a*b*c^4)*x^3 + 12*(11*b^4
*c^2 + 48*a*b^2*c^3 - 128*a^2*c^4)*x^2 - 4*(11*b^5*c - 176*a*b^3*c^2 + 384*a^2*b
*c^3)*x)*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a) - 3*(256*c^8*x^8 + 1024*b*c
^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c^5*x^5 + 1120*b^4*c^4*x^4 + 448*b^5*c^3*x^
3 + 112*b^6*c^2*x^2 + 16*b^7*c*x + b^8)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)/(sqrt(c
*x^2 + b*x + a)*c)))/((256*(b^4*c^10 - 8*a*b^2*c^11 + 16*a^2*c^12)*d^9*x^8 + 102
4*(b^5*c^9 - 8*a*b^3*c^10 + 16*a^2*b*c^11)*d^9*x^7 + 1792*(b^6*c^8 - 8*a*b^4*c^9
+ 16*a^2*b^2*c^10)*d^9*x^6 + 1792*(b^7*c^7 - 8*a*b^5*c^8 + 16*a^2*b^3*c^9)*d^9*
x^5 + 1120*(b^8*c^6 - 8*a*b^6*c^7 + 16*a^2*b^4*c^8)*d^9*x^4 + 448*(b^9*c^5 - 8*a
*b^7*c^6 + 16*a^2*b^5*c^7)*d^9*x^3 + 112*(b^10*c^4 - 8*a*b^8*c^5 + 16*a^2*b^6*c^
6)*d^9*x^2 + 16*(b^11*c^3 - 8*a*b^9*c^4 + 16*a^2*b^7*c^5)*d^9*x + (b^12*c^2 - 8*
a*b^10*c^3 + 16*a^2*b^8*c^4)*d^9)*sqrt(b^2*c - 4*a*c^2))]
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Sympy [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)**(3/2)/(2*c*d*x+b*d)**9,x)
[Out]
Timed out
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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)^(3/2)/(2*c*d*x + b*d)^9,x, algorithm="giac")
[Out]
Exception raised: TypeError