3.611 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{10}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac{2 d \left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{315 c^4 x^3}-\frac{\left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{105 c^3 x^5}-\frac{2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{21 c^2 x^7} \]

[Out]

-(a^2*(c + d*x^2)^(3/2))/(9*c*x^9) - (2*a*(3*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c
^2*x^7) - ((21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(c + d*x^2)^(3/2))/(105*c^3*x^5) +
(2*d*(21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(c + d*x^2)^(3/2))/(315*c^4*x^3)

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Rubi [A]  time = 0.337251, antiderivative size = 144, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right )}{105 c^3 x^5}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac{2 d \left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{315 c^4 x^3}-\frac{2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{21 c^2 x^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]

[Out]

-(a^2*(c + d*x^2)^(3/2))/(9*c*x^9) - (2*a*(3*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c
^2*x^7) - ((21*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*(c + d*x^2)^(3/2))/(105*c^3*x^5
) + (2*d*(21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(c + d*x^2)^(3/2))/(315*c^4*x^3)

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Rubi in Sympy [A]  time = 26.9728, size = 134, normalized size = 0.94 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{9 c x^{9}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 3 b c\right )}{21 c^{2} x^{7}} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (8 a d \left (a d - 3 b c\right ) + 21 b^{2} c^{2}\right )}{105 c^{3} x^{5}} + \frac{2 d \left (c + d x^{2}\right )^{\frac{3}{2}} \left (8 a d \left (a d - 3 b c\right ) + 21 b^{2} c^{2}\right )}{315 c^{4} x^{3}} \]

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**10,x)

[Out]

-a**2*(c + d*x**2)**(3/2)/(9*c*x**9) + 2*a*(c + d*x**2)**(3/2)*(a*d - 3*b*c)/(21
*c**2*x**7) - (c + d*x**2)**(3/2)*(8*a*d*(a*d - 3*b*c) + 21*b**2*c**2)/(105*c**3
*x**5) + 2*d*(c + d*x**2)**(3/2)*(8*a*d*(a*d - 3*b*c) + 21*b**2*c**2)/(315*c**4*
x**3)

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Mathematica [A]  time = 0.1055, size = 108, normalized size = 0.76 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (a^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+6 a b c x^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+21 b^2 c^2 x^4 \left (3 c-2 d x^2\right )\right )}{315 c^4 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]

[Out]

-((c + d*x^2)^(3/2)*(21*b^2*c^2*x^4*(3*c - 2*d*x^2) + 6*a*b*c*x^2*(15*c^2 - 12*c
*d*x^2 + 8*d^2*x^4) + a^2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6)))/
(315*c^4*x^9)

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Maple [A]  time = 0.011, size = 117, normalized size = 0.8 \[ -{\frac{-16\,{x}^{6}{a}^{2}{d}^{3}+48\,{x}^{6}abc{d}^{2}-42\,{x}^{6}{b}^{2}{c}^{2}d+24\,{x}^{4}{a}^{2}c{d}^{2}-72\,{x}^{4}ab{c}^{2}d+63\,{x}^{4}{b}^{2}{c}^{3}-30\,{x}^{2}{a}^{2}{c}^{2}d+90\,{x}^{2}ab{c}^{3}+35\,{a}^{2}{c}^{3}}{315\,{x}^{9}{c}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x)

[Out]

-1/315*(d*x^2+c)^(3/2)*(-16*a^2*d^3*x^6+48*a*b*c*d^2*x^6-42*b^2*c^2*d*x^6+24*a^2
*c*d^2*x^4-72*a*b*c^2*d*x^4+63*b^2*c^3*x^4-30*a^2*c^2*d*x^2+90*a*b*c^3*x^2+35*a^
2*c^3)/x^9/c^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((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.476799, size = 198, normalized size = 1.38 \[ \frac{{\left (2 \,{\left (21 \, b^{2} c^{2} d^{2} - 24 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{8} -{\left (21 \, b^{2} c^{3} d - 24 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{6} - 35 \, a^{2} c^{4} - 3 \,{\left (21 \, b^{2} c^{4} + 6 \, a b c^{3} d - 2 \, a^{2} c^{2} d^{2}\right )} x^{4} - 5 \,{\left (18 \, a b c^{4} + a^{2} c^{3} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{315 \, c^{4} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^10,x, algorithm="fricas")

[Out]

1/315*(2*(21*b^2*c^2*d^2 - 24*a*b*c*d^3 + 8*a^2*d^4)*x^8 - (21*b^2*c^3*d - 24*a*
b*c^2*d^2 + 8*a^2*c*d^3)*x^6 - 35*a^2*c^4 - 3*(21*b^2*c^4 + 6*a*b*c^3*d - 2*a^2*
c^2*d^2)*x^4 - 5*(18*a*b*c^4 + a^2*c^3*d)*x^2)*sqrt(d*x^2 + c)/(c^4*x^9)

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Sympy [A]  time = 15.8475, size = 1061, normalized size = 7.42 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**10,x)

[Out]

-35*a**2*c**7*d**(19/2)*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) - 110*a**2*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) - 114*a**2*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) - 40*a**2*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)
+ 5*a**2*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) + 30*a**2*c**2*d**(2
9/2)*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 + 315*c**4*d**12*x**14) + 40*a**2*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 + 3
15*c**4*d**12*x**14) + 16*a**2*d**(33/2)*x**14*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) -
30*a*b*c**5*d**(9/2)*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) - 66*a*b*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) - 34*a*b*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) - 6*a*b*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) - 24*a*b*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) -
16*a*b*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) - b**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(5*x**4) - b**2*
d**(3/2)*sqrt(c/(d*x**2) + 1)/(15*c*x**2) + 2*b**2*d**(5/2)*sqrt(c/(d*x**2) + 1)
/(15*c**2)

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GIAC/XCAS [A]  time = 0.268721, size = 782, normalized size = 5.47 \[ \frac{4 \,{\left (315 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{14} b^{2} d^{\frac{5}{2}} - 1155 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{12} b^{2} c d^{\frac{5}{2}} + 1680 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{12} a b d^{\frac{7}{2}} + 1575 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} b^{2} c^{2} d^{\frac{5}{2}} - 2520 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} a b c d^{\frac{7}{2}} + 2520 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} a^{2} d^{\frac{9}{2}} - 1071 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{3} d^{\frac{5}{2}} + 504 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac{7}{2}} + 1512 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} c d^{\frac{9}{2}} + 609 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{4} d^{\frac{5}{2}} - 336 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac{7}{2}} + 672 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac{9}{2}} - 441 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{5} d^{\frac{5}{2}} + 864 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac{7}{2}} - 288 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac{9}{2}} + 189 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{6} d^{\frac{5}{2}} - 216 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac{7}{2}} + 72 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac{9}{2}} - 21 \, b^{2} c^{7} d^{\frac{5}{2}} + 24 \, a b c^{6} d^{\frac{7}{2}} - 8 \, a^{2} c^{5} d^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^10,x, algorithm="giac")

[Out]

4/315*(315*(sqrt(d)*x - sqrt(d*x^2 + c))^14*b^2*d^(5/2) - 1155*(sqrt(d)*x - sqrt
(d*x^2 + c))^12*b^2*c*d^(5/2) + 1680*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a*b*d^(7/2
) + 1575*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^2*d^(5/2) - 2520*(sqrt(d)*x - sq
rt(d*x^2 + c))^10*a*b*c*d^(7/2) + 2520*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a^2*d^(9
/2) - 1071*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^3*d^(5/2) + 504*(sqrt(d)*x - sq
rt(d*x^2 + c))^8*a*b*c^2*d^(7/2) + 1512*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*c*d^
(9/2) + 609*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^4*d^(5/2) - 336*(sqrt(d)*x - s
qrt(d*x^2 + c))^6*a*b*c^3*d^(7/2) + 672*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c^2*
d^(9/2) - 441*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^5*d^(5/2) + 864*(sqrt(d)*x -
sqrt(d*x^2 + c))^4*a*b*c^4*d^(7/2) - 288*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c^
3*d^(9/2) + 189*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^6*d^(5/2) - 216*(sqrt(d)*x
- sqrt(d*x^2 + c))^2*a*b*c^5*d^(7/2) + 72*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c
^4*d^(9/2) - 21*b^2*c^7*d^(5/2) + 24*a*b*c^6*d^(7/2) - 8*a^2*c^5*d^(9/2))/((sqrt
(d)*x - sqrt(d*x^2 + c))^2 - c)^9