3.1222 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx\)

Optimal. Leaf size=118 \[ \frac{16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac{8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (8*(a + b*x
+ c*x^2)^(7/2))/(99*(b^2 - 4*a*c)^2*d^12*(b + 2*c*x)^9) + (16*(a + b*x + c*x^2)
^(7/2))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^7)

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Rubi [A]  time = 0.175583, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac{8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (8*(a + b*x
+ c*x^2)^(7/2))/(99*(b^2 - 4*a*c)^2*d^12*(b + 2*c*x)^9) + (16*(a + b*x + c*x^2)
^(7/2))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^7)

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Rubi in Sympy [A]  time = 43.8642, size = 114, normalized size = 0.97 \[ \frac{16 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{693 d^{12} \left (b + 2 c x\right )^{7} \left (- 4 a c + b^{2}\right )^{3}} + \frac{8 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{99 d^{12} \left (b + 2 c x\right )^{9} \left (- 4 a c + b^{2}\right )^{2}} + \frac{2 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{11 d^{12} \left (b + 2 c x\right )^{11} \left (- 4 a c + b^{2}\right )} \]

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)**12,x)

[Out]

16*(a + b*x + c*x**2)**(7/2)/(693*d**12*(b + 2*c*x)**7*(-4*a*c + b**2)**3) + 8*(
a + b*x + c*x**2)**(7/2)/(99*d**12*(b + 2*c*x)**9*(-4*a*c + b**2)**2) + 2*(a + b
*x + c*x**2)**(7/2)/(11*d**12*(b + 2*c*x)**11*(-4*a*c + b**2))

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Mathematica [A]  time = 0.388888, size = 110, normalized size = 0.93 \[ \frac{2 (a+x (b+c x))^{7/2} \left (16 c^2 \left (63 a^2-28 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (38 c x^2-77 a\right )+64 b c^2 x \left (4 c x^2-7 a\right )+99 b^4+176 b^3 c x\right )}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]

[Out]

(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c*x^2) +
8*b^2*c*(-77*a + 38*c*x^2) + 16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^4)))/(693*(b
^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)

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Maple [A]  time = 0.013, size = 133, normalized size = 1.1 \[ -{\frac{256\,{c}^{4}{x}^{4}+512\,b{c}^{3}{x}^{3}-896\,a{c}^{3}{x}^{2}+608\,{b}^{2}{c}^{2}{x}^{2}-896\,ab{c}^{2}x+352\,{b}^{3}cx+2016\,{a}^{2}{c}^{2}-1232\,ac{b}^{2}+198\,{b}^{4}}{693\, \left ( 2\,cx+b \right ) ^{11}{d}^{12} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \]

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)^12,x)

[Out]

-2/693*(128*c^4*x^4+256*b*c^3*x^3-448*a*c^3*x^2+304*b^2*c^2*x^2-448*a*b*c^2*x+17
6*b^3*c*x+1008*a^2*c^2-616*a*b^2*c+99*b^4)*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^11/d^12
/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*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)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 19.1893, size = 1149, normalized size = 9.74 \[ \text{result too large to display} \]

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)^12,x, algorithm="fricas")

[Out]

2/693*(128*c^7*x^10 + 640*b*c^6*x^9 + 16*(91*b^2*c^5 - 4*a*c^6)*x^8 + 64*(31*b^3
*c^4 - 4*a*b*c^5)*x^7 + 99*a^3*b^4 - 616*a^4*b^2*c + 1008*a^5*c^2 + (1795*b^4*c^
3 - 472*a*b^2*c^4 + 48*a^2*c^5)*x^6 + (1129*b^5*c^2 - 520*a*b^3*c^3 + 144*a^2*b*
c^4)*x^5 + (473*b^6*c - 31*a*b^4*c^2 - 1176*a^2*b^2*c^3 + 1808*a^3*c^4)*x^4 + (9
9*b^7 + 506*a*b^5*c - 2592*a^2*b^3*c^2 + 3616*a^3*b*c^3)*x^3 + (297*a*b^6 - 1023
*a^2*b^4*c + 136*a^3*b^2*c^2 + 2576*a^4*c^3)*x^2 + (297*a^2*b^5 - 1672*a^3*b^3*c
+ 2576*a^4*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(2048*(b^6*c^11 - 12*a*b^4*c^12 + 48
*a^2*b^2*c^13 - 64*a^3*c^14)*d^12*x^11 + 11264*(b^7*c^10 - 12*a*b^5*c^11 + 48*a^
2*b^3*c^12 - 64*a^3*b*c^13)*d^12*x^10 + 28160*(b^8*c^9 - 12*a*b^6*c^10 + 48*a^2*
b^4*c^11 - 64*a^3*b^2*c^12)*d^12*x^9 + 42240*(b^9*c^8 - 12*a*b^7*c^9 + 48*a^2*b^
5*c^10 - 64*a^3*b^3*c^11)*d^12*x^8 + 42240*(b^10*c^7 - 12*a*b^8*c^8 + 48*a^2*b^6
*c^9 - 64*a^3*b^4*c^10)*d^12*x^7 + 29568*(b^11*c^6 - 12*a*b^9*c^7 + 48*a^2*b^7*c
^8 - 64*a^3*b^5*c^9)*d^12*x^6 + 14784*(b^12*c^5 - 12*a*b^10*c^6 + 48*a^2*b^8*c^7
- 64*a^3*b^6*c^8)*d^12*x^5 + 5280*(b^13*c^4 - 12*a*b^11*c^5 + 48*a^2*b^9*c^6 -
64*a^3*b^7*c^7)*d^12*x^4 + 1320*(b^14*c^3 - 12*a*b^12*c^4 + 48*a^2*b^10*c^5 - 64
*a^3*b^8*c^6)*d^12*x^3 + 220*(b^15*c^2 - 12*a*b^13*c^3 + 48*a^2*b^11*c^4 - 64*a^
3*b^9*c^5)*d^12*x^2 + 22*(b^16*c - 12*a*b^14*c^2 + 48*a^2*b^12*c^3 - 64*a^3*b^10
*c^4)*d^12*x + (b^17 - 12*a*b^15*c + 48*a^2*b^13*c^2 - 64*a^3*b^11*c^3)*d^12)

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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)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 4.38747, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

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)^12,x, algorithm="giac")

[Out]

sage0*x