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3.1221 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx\)

Optimal. Leaf size=239 \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16384 c^{7/2} d^{11} \left (b^2-4 a c\right )^{5/2}}+\frac{3 \sqrt{a+b x+c x^2}}{8192 c^3 d^{11} \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{4096 c^3 d^{11} \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac{\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(1024*c^3*d^11*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(40
96*c^3*(b^2 - 4*a*c)*d^11*(b + 2*c*x)^4) + (3*Sqrt[a + b*x + c*x^2])/(8192*c^3*(
b^2 - 4*a*c)^2*d^11*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(128*c^2*d^11*(b +
2*c*x)^8) - (a + b*x + c*x^2)^(5/2)/(20*c*d^11*(b + 2*c*x)^10) + (3*ArcTan[(2*Sq
rt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(16384*c^(7/2)*(b^2 - 4*a*c)^(5
/2)*d^11)

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Rubi [A]  time = 0.501883, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16384 c^{7/2} d^{11} \left (b^2-4 a c\right )^{5/2}}+\frac{3 \sqrt{a+b x+c x^2}}{8192 c^3 d^{11} \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{4096 c^3 d^{11} \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac{\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(1024*c^3*d^11*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(40
96*c^3*(b^2 - 4*a*c)*d^11*(b + 2*c*x)^4) + (3*Sqrt[a + b*x + c*x^2])/(8192*c^3*(
b^2 - 4*a*c)^2*d^11*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(128*c^2*d^11*(b +
2*c*x)^8) - (a + b*x + c*x^2)^(5/2)/(20*c*d^11*(b + 2*c*x)^10) + (3*ArcTan[(2*Sq
rt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(16384*c^(7/2)*(b^2 - 4*a*c)^(5
/2)*d^11)

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Rubi in Sympy [A]  time = 118.804, size = 228, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{20 c d^{11} \left (b + 2 c x\right )^{10}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{128 c^{2} d^{11} \left (b + 2 c x\right )^{8}} + \frac{3 \sqrt{a + b x + c x^{2}}}{8192 c^{3} d^{11} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{\sqrt{a + b x + c x^{2}}}{4096 c^{3} d^{11} \left (b + 2 c x\right )^{4} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x + c x^{2}}}{1024 c^{3} d^{11} \left (b + 2 c x\right )^{6}} + \frac{3 \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{16384 c^{\frac{7}{2}} d^{11} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**11,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(20*c*d**11*(b + 2*c*x)**10) - (a + b*x + c*x**2)**(3
/2)/(128*c**2*d**11*(b + 2*c*x)**8) + 3*sqrt(a + b*x + c*x**2)/(8192*c**3*d**11*
(b + 2*c*x)**2*(-4*a*c + b**2)**2) + sqrt(a + b*x + c*x**2)/(4096*c**3*d**11*(b
+ 2*c*x)**4*(-4*a*c + b**2)) - sqrt(a + b*x + c*x**2)/(1024*c**3*d**11*(b + 2*c*
x)**6) + 3*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(16384*c**
(7/2)*d**11*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 0.740265, size = 227, normalized size = 0.95 \[ \frac{-15 (b+2 c x)^{10} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )-2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)} \left (-10 \left (b^2-4 a c\right ) (b+2 c x)^6+248 \left (b^2-4 a c\right )^2 (b+2 c x)^4-336 \left (b^2-4 a c\right )^3 (b+2 c x)^2+128 \left (b^2-4 a c\right )^4-15 (b+2 c x)^8\right )+15 (b+2 c x)^{10} \log (b+2 c x)}{81920 c^{7/2} d^{11} \left (4 a c-b^2\right )^{5/2} (b+2 c x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x]

[Out]

(-2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]*(128*(b^2 - 4*a*c)^4 - 336*
(b^2 - 4*a*c)^3*(b + 2*c*x)^2 + 248*(b^2 - 4*a*c)^2*(b + 2*c*x)^4 - 10*(b^2 - 4*
a*c)*(b + 2*c*x)^6 - 15*(b + 2*c*x)^8) + 15*(b + 2*c*x)^10*Log[b + 2*c*x] - 15*(
b + 2*c*x)^10*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a +
 x*(b + c*x)]])/(81920*c^(7/2)*(-b^2 + 4*a*c)^(5/2)*d^11*(b + 2*c*x)^10)

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Maple [B]  time = 0.148, size = 1080, normalized size = 4.5 \[ \text{result too large to display} \]

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)^11,x)

[Out]

-1/5120/d^11/c^10/(4*a*c-b^2)/(x+1/2*b/c)^10*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)
^(7/2)+3/10240/d^11/c^8/(4*a*c-b^2)^2/(x+1/2*b/c)^8*((x+1/2*b/c)^2*c+1/4*(4*a*c-
b^2)/c)^(7/2)-1/5120/d^11/c^6/(4*a*c-b^2)^3/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(
4*a*c-b^2)/c)^(7/2)-1/5120/d^11/c^4/(4*a*c-b^2)^4/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c
+1/4*(4*a*c-b^2)/c)^(7/2)-3/2560/d^11/c^2/(4*a*c-b^2)^5/(x+1/2*b/c)^2*((x+1/2*b/
c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+3/2560/d^11/c/(4*a*c-b^2)^5*((x+1/2*b/c)^2*c+1/4
*(4*a*c-b^2)/c)^(5/2)+1/512/d^11/c/(4*a*c-b^2)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2
)/c)^(3/2)*a-1/2048/d^11/c^2/(4*a*c-b^2)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(
3/2)*b^2+3/1024/d^11/c/(4*a*c-b^2)^5*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a^2
-3/2048/d^11/c^2/(4*a*c-b^2)^5*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a*b^2+3/1
6384/d^11/c^3/(4*a*c-b^2)^5*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^4-3/256/d^
11/c/(4*a*c-b^2)^5/((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+9/1024/d^11/c
^2/(4*a*c-b^2)^5/((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^2-9/4096/d^11
/c^3/(4*a*c-b^2)^5/((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+3/16384/d^1
1/c^4/(4*a*c-b^2)^5/((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)^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 12.489, 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)^(5/2)/(2*c*d*x + b*d)^11,x, algorithm="fricas")

[Out]

[1/163840*(4*(3840*c^8*x^8 + 15360*b*c^7*x^7 - 15*b^8 - 40*a*b^6*c - 128*a^2*b^4
*c^2 + 11264*a^3*b^2*c^3 - 32768*a^4*c^4 + 640*(43*b^2*c^6 - 4*a*c^7)*x^6 + 1920
*(15*b^3*c^5 - 4*a*b*c^6)*x^5 + 128*(119*b^4*c^4 + 173*a*b^2*c^5 - 496*a^2*c^6)*
x^4 + 128*(3*b^5*c^3 + 446*a*b^3*c^4 - 992*a^2*b*c^5)*x^3 - 24*(97*b^6*c^2 - 121
2*a*b^4*c^3 + 1280*a^2*b^2*c^4 + 3584*a^3*c^5)*x^2 - 8*(35*b^7*c + 92*a*b^5*c^2
- 4096*a^2*b^3*c^3 + 10752*a^3*b*c^4)*x)*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x
 + a) + 15*(1024*c^10*x^10 + 5120*b*c^9*x^9 + 11520*b^2*c^8*x^8 + 15360*b^3*c^7*
x^7 + 13440*b^4*c^6*x^6 + 8064*b^5*c^5*x^5 + 3360*b^6*c^4*x^4 + 960*b^7*c^3*x^3
+ 180*b^8*c^2*x^2 + 20*b^9*c*x + b^10)*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)))/((1024*(b^4*c^13 - 8*a*b^2*c^14 + 16*a^2*c^15)*d^11*x^10 + 51
20*(b^5*c^12 - 8*a*b^3*c^13 + 16*a^2*b*c^14)*d^11*x^9 + 11520*(b^6*c^11 - 8*a*b^
4*c^12 + 16*a^2*b^2*c^13)*d^11*x^8 + 15360*(b^7*c^10 - 8*a*b^5*c^11 + 16*a^2*b^3
*c^12)*d^11*x^7 + 13440*(b^8*c^9 - 8*a*b^6*c^10 + 16*a^2*b^4*c^11)*d^11*x^6 + 80
64*(b^9*c^8 - 8*a*b^7*c^9 + 16*a^2*b^5*c^10)*d^11*x^5 + 3360*(b^10*c^7 - 8*a*b^8
*c^8 + 16*a^2*b^6*c^9)*d^11*x^4 + 960*(b^11*c^6 - 8*a*b^9*c^7 + 16*a^2*b^7*c^8)*
d^11*x^3 + 180*(b^12*c^5 - 8*a*b^10*c^6 + 16*a^2*b^8*c^7)*d^11*x^2 + 20*(b^13*c^
4 - 8*a*b^11*c^5 + 16*a^2*b^9*c^6)*d^11*x + (b^14*c^3 - 8*a*b^12*c^4 + 16*a^2*b^
10*c^5)*d^11)*sqrt(-b^2*c + 4*a*c^2)), 1/81920*(2*(3840*c^8*x^8 + 15360*b*c^7*x^
7 - 15*b^8 - 40*a*b^6*c - 128*a^2*b^4*c^2 + 11264*a^3*b^2*c^3 - 32768*a^4*c^4 +
640*(43*b^2*c^6 - 4*a*c^7)*x^6 + 1920*(15*b^3*c^5 - 4*a*b*c^6)*x^5 + 128*(119*b^
4*c^4 + 173*a*b^2*c^5 - 496*a^2*c^6)*x^4 + 128*(3*b^5*c^3 + 446*a*b^3*c^4 - 992*
a^2*b*c^5)*x^3 - 24*(97*b^6*c^2 - 1212*a*b^4*c^3 + 1280*a^2*b^2*c^4 + 3584*a^3*c
^5)*x^2 - 8*(35*b^7*c + 92*a*b^5*c^2 - 4096*a^2*b^3*c^3 + 10752*a^3*b*c^4)*x)*sq
rt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a) - 15*(1024*c^10*x^10 + 5120*b*c^9*x^9
+ 11520*b^2*c^8*x^8 + 15360*b^3*c^7*x^7 + 13440*b^4*c^6*x^6 + 8064*b^5*c^5*x^5 +
 3360*b^6*c^4*x^4 + 960*b^7*c^3*x^3 + 180*b^8*c^2*x^2 + 20*b^9*c*x + b^10)*arcta
n(1/2*sqrt(b^2*c - 4*a*c^2)/(sqrt(c*x^2 + b*x + a)*c)))/((1024*(b^4*c^13 - 8*a*b
^2*c^14 + 16*a^2*c^15)*d^11*x^10 + 5120*(b^5*c^12 - 8*a*b^3*c^13 + 16*a^2*b*c^14
)*d^11*x^9 + 11520*(b^6*c^11 - 8*a*b^4*c^12 + 16*a^2*b^2*c^13)*d^11*x^8 + 15360*
(b^7*c^10 - 8*a*b^5*c^11 + 16*a^2*b^3*c^12)*d^11*x^7 + 13440*(b^8*c^9 - 8*a*b^6*
c^10 + 16*a^2*b^4*c^11)*d^11*x^6 + 8064*(b^9*c^8 - 8*a*b^7*c^9 + 16*a^2*b^5*c^10
)*d^11*x^5 + 3360*(b^10*c^7 - 8*a*b^8*c^8 + 16*a^2*b^6*c^9)*d^11*x^4 + 960*(b^11
*c^6 - 8*a*b^9*c^7 + 16*a^2*b^7*c^8)*d^11*x^3 + 180*(b^12*c^5 - 8*a*b^10*c^6 + 1
6*a^2*b^8*c^7)*d^11*x^2 + 20*(b^13*c^4 - 8*a*b^11*c^5 + 16*a^2*b^9*c^6)*d^11*x +
 (b^14*c^3 - 8*a*b^12*c^4 + 16*a^2*b^10*c^5)*d^11)*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)**11,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)^11,x, algorithm="giac")

[Out]

Exception raised: TypeError

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