3.612 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{12}} \, dx\)
Optimal. Leaf size=189 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac{8 d^2 \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3465 c^5 x^3}+\frac{4 d \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{1155 c^4 x^5}-\frac{\left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{231 c^3 x^7}-\frac{2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{99 c^2 x^9} \]
[Out]
-(a^2*(c + d*x^2)^(3/2))/(11*c*x^11) - (2*a*(11*b*c - 4*a*d)*(c + d*x^2)^(3/2))/
(99*c^2*x^9) - ((33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(231*c^
3*x^7) + (4*d*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(1155*c^4
*x^5) - (8*d^2*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(3465*c^
5*x^3)
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Rubi [A] time = 0.421208, antiderivative size = 190, normalized size of antiderivative =
1.01, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ -\frac{\left (c+d x^2\right )^{3/2} \left (16 a^2 d^2-44 a b c d+33 b^2 c^2\right )}{231 c^3 x^7}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac{8 d^2 \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3465 c^5 x^3}+\frac{4 d \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{1155 c^4 x^5}-\frac{2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{99 c^2 x^9} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^12,x]
[Out]
-(a^2*(c + d*x^2)^(3/2))/(11*c*x^11) - (2*a*(11*b*c - 4*a*d)*(c + d*x^2)^(3/2))/
(99*c^2*x^9) - ((33*b^2*c^2 - 44*a*b*c*d + 16*a^2*d^2)*(c + d*x^2)^(3/2))/(231*c
^3*x^7) + (4*d*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(1155*c^
4*x^5) - (8*d^2*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(3465*c
^5*x^3)
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Rubi in Sympy [A] time = 32.6139, size = 187, normalized size = 0.99 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{11 c x^{11}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d - 11 b c\right )}{99 c^{2} x^{9}} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d \left (4 a d - 11 b c\right ) + 33 b^{2} c^{2}\right )}{231 c^{3} x^{7}} + \frac{4 d \left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d \left (4 a d - 11 b c\right ) + 33 b^{2} c^{2}\right )}{1155 c^{4} x^{5}} - \frac{8 d^{2} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d \left (4 a d - 11 b c\right ) + 33 b^{2} c^{2}\right )}{3465 c^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**12,x)
[Out]
-a**2*(c + d*x**2)**(3/2)/(11*c*x**11) + 2*a*(c + d*x**2)**(3/2)*(4*a*d - 11*b*c
)/(99*c**2*x**9) - (c + d*x**2)**(3/2)*(4*a*d*(4*a*d - 11*b*c) + 33*b**2*c**2)/(
231*c**3*x**7) + 4*d*(c + d*x**2)**(3/2)*(4*a*d*(4*a*d - 11*b*c) + 33*b**2*c**2)
/(1155*c**4*x**5) - 8*d**2*(c + d*x**2)**(3/2)*(4*a*d*(4*a*d - 11*b*c) + 33*b**2
*c**2)/(3465*c**5*x**3)
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Mathematica [A] time = 0.122385, size = 141, normalized size = 0.75 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (a^2 \left (315 c^4-280 c^3 d x^2+240 c^2 d^2 x^4-192 c d^3 x^6+128 d^4 x^8\right )+22 a b c x^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+33 b^2 c^2 x^4 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )\right )}{3465 c^5 x^{11}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^12,x]
[Out]
-((c + d*x^2)^(3/2)*(33*b^2*c^2*x^4*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^4) + 22*a*b*c
*x^2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6) + a^2*(315*c^4 - 280*c^
3*d*x^2 + 240*c^2*d^2*x^4 - 192*c*d^3*x^6 + 128*d^4*x^8)))/(3465*c^5*x^11)
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Maple [A] time = 0.012, size = 158, normalized size = 0.8 \[ -{\frac{128\,{a}^{2}{d}^{4}{x}^{8}-352\,abc{d}^{3}{x}^{8}+264\,{b}^{2}{c}^{2}{d}^{2}{x}^{8}-192\,{a}^{2}c{d}^{3}{x}^{6}+528\,ab{c}^{2}{d}^{2}{x}^{6}-396\,{b}^{2}{c}^{3}d{x}^{6}+240\,{a}^{2}{c}^{2}{d}^{2}{x}^{4}-660\,ab{c}^{3}d{x}^{4}+495\,{b}^{2}{c}^{4}{x}^{4}-280\,{a}^{2}{c}^{3}d{x}^{2}+770\,ab{c}^{4}{x}^{2}+315\,{a}^{2}{c}^{4}}{3465\,{x}^{11}{c}^{5}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x)
[Out]
-1/3465*(d*x^2+c)^(3/2)*(128*a^2*d^4*x^8-352*a*b*c*d^3*x^8+264*b^2*c^2*d^2*x^8-1
92*a^2*c*d^3*x^6+528*a*b*c^2*d^2*x^6-396*b^2*c^3*d*x^6+240*a^2*c^2*d^2*x^4-660*a
*b*c^3*d*x^4+495*b^2*c^4*x^4-280*a^2*c^3*d*x^2+770*a*b*c^4*x^2+315*a^2*c^4)/x^11
/c^5
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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((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^12,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.799553, size = 250, normalized size = 1.32 \[ -\frac{{\left (8 \,{\left (33 \, b^{2} c^{2} d^{3} - 44 \, a b c d^{4} + 16 \, a^{2} d^{5}\right )} x^{10} - 4 \,{\left (33 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 16 \, a^{2} c d^{4}\right )} x^{8} + 315 \, a^{2} c^{5} + 3 \,{\left (33 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 16 \, a^{2} c^{2} d^{3}\right )} x^{6} + 5 \,{\left (99 \, b^{2} c^{5} + 22 \, a b c^{4} d - 8 \, a^{2} c^{3} d^{2}\right )} x^{4} + 35 \,{\left (22 \, a b c^{5} + a^{2} c^{4} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3465 \, c^{5} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^12,x, algorithm="fricas")
[Out]
-1/3465*(8*(33*b^2*c^2*d^3 - 44*a*b*c*d^4 + 16*a^2*d^5)*x^10 - 4*(33*b^2*c^3*d^2
- 44*a*b*c^2*d^3 + 16*a^2*c*d^4)*x^8 + 315*a^2*c^5 + 3*(33*b^2*c^4*d - 44*a*b*c
^3*d^2 + 16*a^2*c^2*d^3)*x^6 + 5*(99*b^2*c^5 + 22*a*b*c^4*d - 8*a^2*c^3*d^2)*x^4
+ 35*(22*a*b*c^5 + a^2*c^4*d)*x^2)*sqrt(d*x^2 + c)/(c^5*x^11)
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Sympy [A] time = 24.0671, size = 1856, normalized size = 9.82 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**12,x)
[Out]
-315*a**2*c**9*d**(33/2)*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**
8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**2
0*x**18) - 1295*a**2*c**8*d**(35/2)*x**2*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x
**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16
+ 3465*c**5*d**20*x**18) - 1990*a**2*c**7*d**(37/2)*x**4*sqrt(c/(d*x**2) + 1)/(3
465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c
**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 1358*a**2*c**6*d**(39/2)*x**6*sqrt(c/
(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18
*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 343*a**2*c**5*d**(41/
2)*x**8*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 2
0790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 35*a**
2*c**4*d**(43/2)*x**10*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*
d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*
x**18) - 280*a**2*c**3*d**(45/2)*x**12*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**
10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 +
3465*c**5*d**20*x**18) - 560*a**2*c**2*d**(47/2)*x**14*sqrt(c/(d*x**2) + 1)/(346
5*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**
6*d**19*x**16 + 3465*c**5*d**20*x**18) - 448*a**2*c*d**(49/2)*x**16*sqrt(c/(d*x*
*2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**1
4 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 128*a**2*d**(51/2)*x**18*s
qrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7
*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 70*a*b*c**7*d**
(19/2)*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**
5*d**11*x**12 + 315*c**4*d**12*x**14) - 220*a*b*c**6*d**(21/2)*x**2*sqrt(c/(d*x*
*2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315
*c**4*d**12*x**14) - 228*a*b*c**5*d**(23/2)*x**4*sqrt(c/(d*x**2) + 1)/(315*c**7*
d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14)
- 80*a*b*c**4*d**(25/2)*x**6*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6
*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) + 10*a*b*c**3*d**(27
/2)*x**8*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c
**5*d**11*x**12 + 315*c**4*d**12*x**14) + 60*a*b*c**2*d**(29/2)*x**10*sqrt(c/(d*
x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 3
15*c**4*d**12*x**14) + 80*a*b*c*d**(31/2)*x**12*sqrt(c/(d*x**2) + 1)/(315*c**7*d
**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) +
32*a*b*d**(33/2)*x**14*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**1
0*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) - 15*b**2*c**5*d**(9/2)*s
qrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**
10) - 33*b**2*c**4*d**(11/2)*x**2*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210
*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 17*b**2*c**3*d**(13/2)*x**4*sqrt(c/(d*x
**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 3*b*
*2*c**2*d**(15/2)*x**6*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*
x**8 + 105*c**3*d**6*x**10) - 12*b**2*c*d**(17/2)*x**8*sqrt(c/(d*x**2) + 1)/(105
*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 8*b**2*d**(19/2)*x
**10*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d*
*6*x**10)
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GIAC/XCAS [A] time = 0.266392, size = 902, normalized size = 4.77 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^12,x, algorithm="giac")
[Out]
16/3465*(2310*(sqrt(d)*x - sqrt(d*x^2 + c))^16*b^2*d^(7/2) - 8085*(sqrt(d)*x - s
qrt(d*x^2 + c))^14*b^2*c*d^(7/2) + 13860*(sqrt(d)*x - sqrt(d*x^2 + c))^14*a*b*d^
(9/2) + 9933*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*c^2*d^(7/2) - 19404*(sqrt(d)*x
- sqrt(d*x^2 + c))^12*a*b*c*d^(9/2) + 22176*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a^
2*d^(11/2) - 5313*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^3*d^(7/2) + 924*(sqrt(d
)*x - sqrt(d*x^2 + c))^10*a*b*c^2*d^(9/2) + 14784*(sqrt(d)*x - sqrt(d*x^2 + c))^
10*a^2*c*d^(11/2) + 2805*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^4*d^(7/2) - 660*(
sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^3*d^(9/2) + 5280*(sqrt(d)*x - sqrt(d*x^2 +
c))^8*a^2*c^2*d^(11/2) - 3135*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^5*d^(7/2) +
7260*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^4*d^(9/2) - 2640*(sqrt(d)*x - sqrt(d*
x^2 + c))^6*a^2*c^3*d^(11/2) + 1815*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^6*d^(7
/2) - 2420*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^5*d^(9/2) + 880*(sqrt(d)*x - sq
rt(d*x^2 + c))^4*a^2*c^4*d^(11/2) - 363*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^7*
d^(7/2) + 484*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^6*d^(9/2) - 176*(sqrt(d)*x -
sqrt(d*x^2 + c))^2*a^2*c^5*d^(11/2) + 33*b^2*c^8*d^(7/2) - 44*a*b*c^7*d^(9/2) +
16*a^2*c^6*d^(11/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^11