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3.1219 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx\)

Optimal. Leaf size=197 \[ \frac{5 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{8192 c^{7/2} d^9 \left (b^2-4 a c\right )^{3/2}}+\frac{5 \sqrt{a+b x+c x^2}}{4096 c^3 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{5 \sqrt{a+b x+c x^2}}{2048 c^3 d^9 (b+2 c x)^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{384 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8} \]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(2048*c^3*d^9*(b + 2*c*x)^4) + (5*Sqrt[a + b*x + c*x^
2])/(4096*c^3*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^2) - (5*(a + b*x + c*x^2)^(3/2))/(38
4*c^2*d^9*(b + 2*c*x)^6) - (a + b*x + c*x^2)^(5/2)/(16*c*d^9*(b + 2*c*x)^8) + (5
*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(8192*c^(7/2)*(b^2
 - 4*a*c)^(3/2)*d^9)

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Rubi [A]  time = 0.402246, antiderivative size = 197, 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 \[ \frac{5 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{8192 c^{7/2} d^9 \left (b^2-4 a c\right )^{3/2}}+\frac{5 \sqrt{a+b x+c x^2}}{4096 c^3 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{5 \sqrt{a+b x+c x^2}}{2048 c^3 d^9 (b+2 c x)^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{384 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^9,x]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(2048*c^3*d^9*(b + 2*c*x)^4) + (5*Sqrt[a + b*x + c*x^
2])/(4096*c^3*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^2) - (5*(a + b*x + c*x^2)^(3/2))/(38
4*c^2*d^9*(b + 2*c*x)^6) - (a + b*x + c*x^2)^(5/2)/(16*c*d^9*(b + 2*c*x)^8) + (5
*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(8192*c^(7/2)*(b^2
 - 4*a*c)^(3/2)*d^9)

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Rubi in Sympy [A]  time = 93.9952, size = 190, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{16 c d^{9} \left (b + 2 c x\right )^{8}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{2} d^{9} \left (b + 2 c x\right )^{6}} + \frac{5 \sqrt{a + b x + c x^{2}}}{4096 c^{3} d^{9} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} - \frac{5 \sqrt{a + b x + c x^{2}}}{2048 c^{3} d^{9} \left (b + 2 c x\right )^{4}} + \frac{5 \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{8192 c^{\frac{7}{2}} d^{9} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

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)**9,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(16*c*d**9*(b + 2*c*x)**8) - 5*(a + b*x + c*x**2)**(3
/2)/(384*c**2*d**9*(b + 2*c*x)**6) + 5*sqrt(a + b*x + c*x**2)/(4096*c**3*d**9*(b
 + 2*c*x)**2*(-4*a*c + b**2)) - 5*sqrt(a + b*x + c*x**2)/(2048*c**3*d**9*(b + 2*
c*x)**4) + 5*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(8192*c*
*(7/2)*d**9*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.572183, size = 207, normalized size = 1.05 \[ -\frac{-15 (b+2 c x)^8 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )-2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)} \left (118 \left (b^2-4 a c\right ) (b+2 c x)^4-136 \left (b^2-4 a c\right )^2 (b+2 c x)^2+48 \left (b^2-4 a c\right )^3-15 (b+2 c x)^6\right )+15 (b+2 c x)^8 \log (b+2 c x)}{24576 c^{7/2} d^9 \left (4 a c-b^2\right )^{3/2} (b+2 c x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^9,x]

[Out]

-(-2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]*(48*(b^2 - 4*a*c)^3 - 136*
(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 118*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 15*(b + 2*c*x)
^6) + 15*(b + 2*c*x)^8*Log[b + 2*c*x] - 15*(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)]])/(24576*c^(7/2)*(-b^2
+ 4*a*c)^(3/2)*d^9*(b + 2*c*x)^8)

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Maple [B]  time = 0.065, size = 1020, normalized size = 5.2 \[ \text{result too large to display} \]

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)^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)^(7
/2)+1/1536/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)^(7/2)+1/1536/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)^(7/2)+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)^(7/2)-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)^
(5/2)-5/768/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)*a+5/30
72/d^9/c^2/(4*a*c-b^2)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^2-5/512/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^2+5/1024/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)*a*b^2-5/8192/d^9/c^3/(4*a*c-b^
2)^4*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^4+5/128/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^3-15/512/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^2*b^2+15/2048/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))*a*b^4-5/8192/d^9/c^4/(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^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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.53947, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^9,x, algorithm="fricas")

[Out]

[1/49152*(4*(960*c^6*x^6 + 2880*b*c^5*x^5 - 15*b^6 - 40*a*b^4*c - 128*a^2*b^2*c^
2 + 3072*a^3*c^3 + 16*(107*b^2*c^4 + 472*a*c^5)*x^4 - 32*(43*b^3*c^3 - 472*a*b*c
^4)*x^3 - 4*(347*b^4*c^2 - 1744*a*b^2*c^3 - 2176*a^2*c^4)*x^2 - 4*(55*b^5*c + 14
4*a*b^3*c^2 - 2176*a^2*b*c^3)*x)*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a) -
15*(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)*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^2*c^11 - 4*a*c^12)*d^9*x^
8 + 1024*(b^3*c^10 - 4*a*b*c^11)*d^9*x^7 + 1792*(b^4*c^9 - 4*a*b^2*c^10)*d^9*x^6
 + 1792*(b^5*c^8 - 4*a*b^3*c^9)*d^9*x^5 + 1120*(b^6*c^7 - 4*a*b^4*c^8)*d^9*x^4 +
 448*(b^7*c^6 - 4*a*b^5*c^7)*d^9*x^3 + 112*(b^8*c^5 - 4*a*b^6*c^6)*d^9*x^2 + 16*
(b^9*c^4 - 4*a*b^7*c^5)*d^9*x + (b^10*c^3 - 4*a*b^8*c^4)*d^9)*sqrt(-b^2*c + 4*a*
c^2)), 1/24576*(2*(960*c^6*x^6 + 2880*b*c^5*x^5 - 15*b^6 - 40*a*b^4*c - 128*a^2*
b^2*c^2 + 3072*a^3*c^3 + 16*(107*b^2*c^4 + 472*a*c^5)*x^4 - 32*(43*b^3*c^3 - 472
*a*b*c^4)*x^3 - 4*(347*b^4*c^2 - 1744*a*b^2*c^3 - 2176*a^2*c^4)*x^2 - 4*(55*b^5*
c + 144*a*b^3*c^2 - 2176*a^2*b*c^3)*x)*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x +
a) - 15*(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 + 11
20*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^2*c^11 - 4*a*c^12)*
d^9*x^8 + 1024*(b^3*c^10 - 4*a*b*c^11)*d^9*x^7 + 1792*(b^4*c^9 - 4*a*b^2*c^10)*d
^9*x^6 + 1792*(b^5*c^8 - 4*a*b^3*c^9)*d^9*x^5 + 1120*(b^6*c^7 - 4*a*b^4*c^8)*d^9
*x^4 + 448*(b^7*c^6 - 4*a*b^5*c^7)*d^9*x^3 + 112*(b^8*c^5 - 4*a*b^6*c^6)*d^9*x^2
 + 16*(b^9*c^4 - 4*a*b^7*c^5)*d^9*x + (b^10*c^3 - 4*a*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)**(5/2)/(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)^(5/2)/(2*c*d*x + b*d)^9,x, algorithm="giac")

[Out]

Exception raised: TypeError

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