3.610 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^8} \, dx\)

Optimal. Leaf size=99 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac{\left (c+d x^2\right )^{3/2} \left (35 b^2 c^2-4 a d (7 b c-2 a d)\right )}{105 c^3 x^3}-\frac{2 a \left (c+d x^2\right )^{3/2} (7 b c-2 a d)}{35 c^2 x^5} \]

[Out]

-(a^2*(c + d*x^2)^(3/2))/(7*c*x^7) - (2*a*(7*b*c - 2*a*d)*(c + d*x^2)^(3/2))/(35
*c^2*x^5) - ((35*b^2*c^2 - 4*a*d*(7*b*c - 2*a*d))*(c + d*x^2)^(3/2))/(105*c^3*x^
3)

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Rubi [A]  time = 0.214255, antiderivative size = 100, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (8 a^2 d^2-28 a b c d+35 b^2 c^2\right )}{105 c^3 x^3}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac{2 a \left (c+d x^2\right )^{3/2} (7 b c-2 a d)}{35 c^2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^8,x]

[Out]

-(a^2*(c + d*x^2)^(3/2))/(7*c*x^7) - (2*a*(7*b*c - 2*a*d)*(c + d*x^2)^(3/2))/(35
*c^2*x^5) - ((35*b^2*c^2 - 28*a*b*c*d + 8*a^2*d^2)*(c + d*x^2)^(3/2))/(105*c^3*x
^3)

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Rubi in Sympy [A]  time = 22.0912, size = 94, normalized size = 0.95 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{7 c x^{7}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (2 a d - 7 b c\right )}{35 c^{2} x^{5}} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d \left (2 a d - 7 b c\right ) + 35 b^{2} c^{2}\right )}{105 c^{3} x^{3}} \]

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**8,x)

[Out]

-a**2*(c + d*x**2)**(3/2)/(7*c*x**7) + 2*a*(c + d*x**2)**(3/2)*(2*a*d - 7*b*c)/(
35*c**2*x**5) - (c + d*x**2)**(3/2)*(4*a*d*(2*a*d - 7*b*c) + 35*b**2*c**2)/(105*
c**3*x**3)

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Mathematica [A]  time = 0.0832567, size = 76, normalized size = 0.77 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (a^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+14 a b c x^2 \left (3 c-2 d x^2\right )+35 b^2 c^2 x^4\right )}{105 c^3 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^8,x]

[Out]

-((c + d*x^2)^(3/2)*(35*b^2*c^2*x^4 + 14*a*b*c*x^2*(3*c - 2*d*x^2) + a^2*(15*c^2
- 12*c*d*x^2 + 8*d^2*x^4)))/(105*c^3*x^7)

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Maple [A]  time = 0.01, size = 78, normalized size = 0.8 \[ -{\frac{8\,{x}^{4}{a}^{2}{d}^{2}-28\,{x}^{4}abcd+35\,{x}^{4}{b}^{2}{c}^{2}-12\,{x}^{2}{a}^{2}cd+42\,a{c}^{2}b{x}^{2}+15\,{a}^{2}{c}^{2}}{105\,{x}^{7}{c}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^8,x)

[Out]

-1/105*(d*x^2+c)^(3/2)*(8*a^2*d^2*x^4-28*a*b*c*d*x^4+35*b^2*c^2*x^4-12*a^2*c*d*x
^2+42*a*b*c^2*x^2+15*a^2*c^2)/x^7/c^3

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307874, size = 144, normalized size = 1.45 \[ -\frac{{\left ({\left (35 \, b^{2} c^{2} d - 28 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{6} + 15 \, a^{2} c^{3} +{\left (35 \, b^{2} c^{3} + 14 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x^{4} + 3 \,{\left (14 \, a b c^{3} + a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, c^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^8,x, algorithm="fricas")

[Out]

-1/105*((35*b^2*c^2*d - 28*a*b*c*d^2 + 8*a^2*d^3)*x^6 + 15*a^2*c^3 + (35*b^2*c^3
+ 14*a*b*c^2*d - 4*a^2*c*d^2)*x^4 + 3*(14*a*b*c^3 + a^2*c^2*d)*x^2)*sqrt(d*x^2
+ c)/(c^3*x^7)

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Sympy [A]  time = 10.6966, size = 510, normalized size = 5.15 \[ - \frac{15 a^{2} c^{5} d^{\frac{9}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{33 a^{2} c^{4} d^{\frac{11}{2}} x^{2} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{17 a^{2} c^{3} d^{\frac{13}{2}} x^{4} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{3 a^{2} c^{2} d^{\frac{15}{2}} x^{6} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{12 a^{2} c d^{\frac{17}{2}} x^{8} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{8 a^{2} d^{\frac{19}{2}} x^{10} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{2 a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{5 x^{4}} - \frac{2 a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c x^{2}} + \frac{4 a b d^{\frac{5}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{2}} - \frac{b^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{b^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**8,x)

[Out]

-15*a**2*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) - 33*a**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*a**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*a**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*a**2*c*d**(17/2)*x**8*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) - 8*a**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) - 2*a*b*sqrt(d)*sqrt(c/(d*x**2) + 1)/(5*x**4)
- 2*a*b*d**(3/2)*sqrt(c/(d*x**2) + 1)/(15*c*x**2) + 4*a*b*d**(5/2)*sqrt(c/(d*x**
2) + 1)/(15*c**2) - b**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(3*x**2) - b**2*d**(3/2)*s
qrt(c/(d*x**2) + 1)/(3*c)

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GIAC/XCAS [A]  time = 0.253373, size = 662, normalized size = 6.69 \[ \frac{2 \,{\left (105 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{12} b^{2} d^{\frac{3}{2}} - 420 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} b^{2} c d^{\frac{3}{2}} + 420 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} a b d^{\frac{5}{2}} + 665 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{2} d^{\frac{3}{2}} - 700 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c d^{\frac{5}{2}} + 560 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} d^{\frac{7}{2}} - 560 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{3} d^{\frac{3}{2}} + 280 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac{5}{2}} + 280 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} c d^{\frac{7}{2}} + 315 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{4} d^{\frac{3}{2}} - 168 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac{5}{2}} + 168 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{7}{2}} - 140 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{5} d^{\frac{3}{2}} + 196 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac{5}{2}} - 56 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac{7}{2}} + 35 \, b^{2} c^{6} d^{\frac{3}{2}} - 28 \, a b c^{5} d^{\frac{5}{2}} + 8 \, a^{2} c^{4} d^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^8,x, algorithm="giac")

[Out]

2/105*(105*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*d^(3/2) - 420*(sqrt(d)*x - sqrt(
d*x^2 + c))^10*b^2*c*d^(3/2) + 420*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*d^(5/2)
+ 665*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^2*d^(3/2) - 700*(sqrt(d)*x - sqrt(d*
x^2 + c))^8*a*b*c*d^(5/2) + 560*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*d^(7/2) - 56
0*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^3*d^(3/2) + 280*(sqrt(d)*x - sqrt(d*x^2
+ c))^6*a*b*c^2*d^(5/2) + 280*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c*d^(7/2) + 31
5*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^4*d^(3/2) - 168*(sqrt(d)*x - sqrt(d*x^2
+ c))^4*a*b*c^3*d^(5/2) + 168*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c^2*d^(7/2) -
140*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^5*d^(3/2) + 196*(sqrt(d)*x - sqrt(d*x^
2 + c))^2*a*b*c^4*d^(5/2) - 56*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^3*d^(7/2) +
35*b^2*c^6*d^(3/2) - 28*a*b*c^5*d^(5/2) + 8*a^2*c^4*d^(7/2))/((sqrt(d)*x - sqrt
(d*x^2 + c))^2 - c)^7