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

Optimal. Leaf size=74 \[ -\frac{2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]

[Out]

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

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Rubi [A]  time = 0.163477, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.1117, size = 65, normalized size = 0.88 \[ \frac{b^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d^{3}} + \frac{2 b \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{3}} + \frac{\sqrt{c + d x^{2}} \left (a d - b c\right )^{2}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0683084, size = 66, normalized size = 0.89 \[ \frac{\sqrt{c+d x^2} \left (15 a^2 d^2+10 a b d \left (d x^2-2 c\right )+b^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )\right )}{15 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 69, normalized size = 0.9 \[{\frac{3\,{b}^{2}{d}^{2}{x}^{4}+10\,ab{d}^{2}{x}^{2}-4\,{b}^{2}cd{x}^{2}+15\,{a}^{2}{d}^{2}-20\,cabd+8\,{b}^{2}{c}^{2}}{15\,{d}^{3}}\sqrt{d{x}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(d*x^2+c)^(1/2)*(3*b^2*d^2*x^4+10*a*b*d^2*x^2-4*b^2*c*d*x^2+15*a^2*d^2-20*a
*b*c*d+8*b^2*c^2)/d^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*x/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244498, size = 92, normalized size = 1.24 \[ \frac{{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*b^2*d^2*x^4 + 8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 - 2*(2*b^2*c*d - 5*a*b
*d^2)*x^2)*sqrt(d*x^2 + c)/d^3

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Sympy [A]  time = 3.10816, size = 158, normalized size = 2.14 \[ \begin{cases} \frac{a^{2} \sqrt{c + d x^{2}}}{d} - \frac{4 a b c \sqrt{c + d x^{2}}}{3 d^{2}} + \frac{2 a b x^{2} \sqrt{c + d x^{2}}}{3 d} + \frac{8 b^{2} c^{2} \sqrt{c + d x^{2}}}{15 d^{3}} - \frac{4 b^{2} c x^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{b^{2} x^{4} \sqrt{c + d x^{2}}}{5 d} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{\sqrt{c}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**2*sqrt(c + d*x**2)/d - 4*a*b*c*sqrt(c + d*x**2)/(3*d**2) + 2*a*b*x
**2*sqrt(c + d*x**2)/(3*d) + 8*b**2*c**2*sqrt(c + d*x**2)/(15*d**3) - 4*b**2*c*x
**2*sqrt(c + d*x**2)/(15*d**2) + b**2*x**4*sqrt(c + d*x**2)/(5*d), Ne(d, 0)), ((
a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)/sqrt(c), True))

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GIAC/XCAS [A]  time = 0.232259, size = 132, normalized size = 1.78 \[ \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} - 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c + 15 \, \sqrt{d x^{2} + c} b^{2} c^{2} + 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d - 30 \, \sqrt{d x^{2} + c} a b c d + 15 \, \sqrt{d x^{2} + c} a^{2} d^{2}}{15 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*(d*x^2 + c)^(5/2)*b^2 - 10*(d*x^2 + c)^(3/2)*b^2*c + 15*sqrt(d*x^2 + c)*
b^2*c^2 + 10*(d*x^2 + c)^(3/2)*a*b*d - 30*sqrt(d*x^2 + c)*a*b*c*d + 15*sqrt(d*x^
2 + c)*a^2*d^2)/d^3