3.649 \(\int \frac{x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{b \left (c+d x^2\right )^{3/2} (3 b c-2 a d)}{3 d^4}+\frac{\sqrt{c+d x^2} (b c-a d) (3 b c-a d)}{d^4}+\frac{c (b c-a d)^2}{d^4 \sqrt{c+d x^2}}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^4} \]

[Out]

(c*(b*c - a*d)^2)/(d^4*Sqrt[c + d*x^2]) + ((b*c - a*d)*(3*b*c - a*d)*Sqrt[c + d*
x^2])/d^4 - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(3/2))/(3*d^4) + (b^2*(c + d*x^2)^(5/
2))/(5*d^4)

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Rubi [A]  time = 0.262291, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{b \left (c+d x^2\right )^{3/2} (3 b c-2 a d)}{3 d^4}+\frac{\sqrt{c+d x^2} (b c-a d) (3 b c-a d)}{d^4}+\frac{c (b c-a d)^2}{d^4 \sqrt{c+d x^2}}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

(c*(b*c - a*d)^2)/(d^4*Sqrt[c + d*x^2]) + ((b*c - a*d)*(3*b*c - a*d)*Sqrt[c + d*
x^2])/d^4 - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(3/2))/(3*d^4) + (b^2*(c + d*x^2)^(5/
2))/(5*d^4)

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Rubi in Sympy [A]  time = 32.0789, size = 97, normalized size = 0.9 \[ \frac{b^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \left (2 a d - 3 b c\right )}{3 d^{4}} + \frac{c \left (a d - b c\right )^{2}}{d^{4} \sqrt{c + d x^{2}}} + \frac{\sqrt{c + d x^{2}} \left (a d - 3 b c\right ) \left (a d - b c\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

b**2*(c + d*x**2)**(5/2)/(5*d**4) + b*(c + d*x**2)**(3/2)*(2*a*d - 3*b*c)/(3*d**
4) + c*(a*d - b*c)**2/(d**4*sqrt(c + d*x**2)) + sqrt(c + d*x**2)*(a*d - 3*b*c)*(
a*d - b*c)/d**4

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Mathematica [A]  time = 0.102846, size = 97, normalized size = 0.9 \[ \frac{15 a^2 d^2 \left (2 c+d x^2\right )+10 a b d \left (-8 c^2-4 c d x^2+d^2 x^4\right )+3 b^2 \left (16 c^3+8 c^2 d x^2-2 c d^2 x^4+d^3 x^6\right )}{15 d^4 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

(15*a^2*d^2*(2*c + d*x^2) + 10*a*b*d*(-8*c^2 - 4*c*d*x^2 + d^2*x^4) + 3*b^2*(16*
c^3 + 8*c^2*d*x^2 - 2*c*d^2*x^4 + d^3*x^6))/(15*d^4*Sqrt[c + d*x^2])

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Maple [A]  time = 0.01, size = 108, normalized size = 1. \[{\frac{3\,{b}^{2}{x}^{6}{d}^{3}+10\,ab{d}^{3}{x}^{4}-6\,{b}^{2}c{d}^{2}{x}^{4}+15\,{a}^{2}{d}^{3}{x}^{2}-40\,abc{d}^{2}{x}^{2}+24\,{b}^{2}{c}^{2}d{x}^{2}+30\,{a}^{2}c{d}^{2}-80\,ab{c}^{2}d+48\,{b}^{2}{c}^{3}}{15\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^2/(d*x^2+c)^(3/2),x)

[Out]

1/15*(3*b^2*d^3*x^6+10*a*b*d^3*x^4-6*b^2*c*d^2*x^4+15*a^2*d^3*x^2-40*a*b*c*d^2*x
^2+24*b^2*c^2*d*x^2+30*a^2*c*d^2-80*a*b*c^2*d+48*b^2*c^3)/(d*x^2+c)^(1/2)/d^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*x^3/(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227164, size = 155, normalized size = 1.44 \[ \frac{{\left (3 \, b^{2} d^{3} x^{6} + 48 \, b^{2} c^{3} - 80 \, a b c^{2} d + 30 \, a^{2} c d^{2} - 2 \,{\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )} x^{4} +{\left (24 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 15 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \,{\left (d^{5} x^{2} + c d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^3/(d*x^2 + c)^(3/2),x, algorithm="fricas")

[Out]

1/15*(3*b^2*d^3*x^6 + 48*b^2*c^3 - 80*a*b*c^2*d + 30*a^2*c*d^2 - 2*(3*b^2*c*d^2
- 5*a*b*d^3)*x^4 + (24*b^2*c^2*d - 40*a*b*c*d^2 + 15*a^2*d^3)*x^2)*sqrt(d*x^2 +
c)/(d^5*x^2 + c*d^4)

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Sympy [A]  time = 5.67156, size = 236, normalized size = 2.19 \[ \begin{cases} \frac{2 a^{2} c}{d^{2} \sqrt{c + d x^{2}}} + \frac{a^{2} x^{2}}{d \sqrt{c + d x^{2}}} - \frac{16 a b c^{2}}{3 d^{3} \sqrt{c + d x^{2}}} - \frac{8 a b c x^{2}}{3 d^{2} \sqrt{c + d x^{2}}} + \frac{2 a b x^{4}}{3 d \sqrt{c + d x^{2}}} + \frac{16 b^{2} c^{3}}{5 d^{4} \sqrt{c + d x^{2}}} + \frac{8 b^{2} c^{2} x^{2}}{5 d^{3} \sqrt{c + d x^{2}}} - \frac{2 b^{2} c x^{4}}{5 d^{2} \sqrt{c + d x^{2}}} + \frac{b^{2} x^{6}}{5 d \sqrt{c + d x^{2}}} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

Piecewise((2*a**2*c/(d**2*sqrt(c + d*x**2)) + a**2*x**2/(d*sqrt(c + d*x**2)) - 1
6*a*b*c**2/(3*d**3*sqrt(c + d*x**2)) - 8*a*b*c*x**2/(3*d**2*sqrt(c + d*x**2)) +
2*a*b*x**4/(3*d*sqrt(c + d*x**2)) + 16*b**2*c**3/(5*d**4*sqrt(c + d*x**2)) + 8*b
**2*c**2*x**2/(5*d**3*sqrt(c + d*x**2)) - 2*b**2*c*x**4/(5*d**2*sqrt(c + d*x**2)
) + b**2*x**6/(5*d*sqrt(c + d*x**2)), Ne(d, 0)), ((a**2*x**4/4 + a*b*x**6/3 + b*
*2*x**8/8)/c**(3/2), True))

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GIAC/XCAS [A]  time = 0.232378, size = 180, normalized size = 1.67 \[ \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} - 15 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c + 45 \, \sqrt{d x^{2} + c} b^{2} c^{2} + 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d - 60 \, \sqrt{d x^{2} + c} a b c d + 15 \, \sqrt{d x^{2} + c} a^{2} d^{2} + \frac{15 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}}{\sqrt{d x^{2} + c}}}{15 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^3/(d*x^2 + c)^(3/2),x, algorithm="giac")

[Out]

1/15*(3*(d*x^2 + c)^(5/2)*b^2 - 15*(d*x^2 + c)^(3/2)*b^2*c + 45*sqrt(d*x^2 + c)*
b^2*c^2 + 10*(d*x^2 + c)^(3/2)*a*b*d - 60*sqrt(d*x^2 + c)*a*b*c*d + 15*sqrt(d*x^
2 + c)*a^2*d^2 + 15*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)/sqrt(d*x^2 + c))/d^4