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

Optimal. Leaf size=77 \[ -\frac{2 b \left (c+d x^2\right )^{5/2} (b c-a d)}{5 d^3}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^3}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^3} \]

[Out]

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

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Rubi [A]  time = 0.158102, antiderivative size = 77, 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 \[ -\frac{2 b \left (c+d x^2\right )^{5/2} (b c-a d)}{5 d^3}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^3}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.0662, size = 66, normalized size = 0.86 \[ \frac{b^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{2 b \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 d^{3}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{3 d^{3}} \]

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

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Mathematica [A]  time = 0.0652225, size = 67, normalized size = 0.87 \[ \frac{\left (c+d x^2\right )^{3/2} \left (35 a^2 d^2+14 a b d \left (3 d x^2-2 c\right )+b^2 \left (8 c^2-12 c d x^2+15 d^2 x^4\right )\right )}{105 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 69, normalized size = 0.9 \[{\frac{15\,{b}^{2}{d}^{2}{x}^{4}+42\,ab{d}^{2}{x}^{2}-12\,{b}^{2}cd{x}^{2}+35\,{a}^{2}{d}^{2}-28\,cabd+8\,{b}^{2}{c}^{2}}{105\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]

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]

1/105*(d*x^2+c)^(3/2)*(15*b^2*d^2*x^4+42*a*b*d^2*x^2-12*b^2*c*d*x^2+35*a^2*d^2-2
8*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*sqrt(d*x^2 + c)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215168, size = 139, normalized size = 1.81 \[ \frac{{\left (15 \, b^{2} d^{3} x^{6} + 8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2} + 3 \,{\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{4} -{\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.35255, size = 226, normalized size = 2.94 \[ \begin{cases} \frac{a^{2} c \sqrt{c + d x^{2}}}{3 d} + \frac{a^{2} x^{2} \sqrt{c + d x^{2}}}{3} - \frac{4 a b c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{2 a b c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{2 a b x^{4} \sqrt{c + d x^{2}}}{5} + \frac{8 b^{2} c^{3} \sqrt{c + d x^{2}}}{105 d^{3}} - \frac{4 b^{2} c^{2} x^{2} \sqrt{c + d x^{2}}}{105 d^{2}} + \frac{b^{2} c x^{4} \sqrt{c + d x^{2}}}{35 d} + \frac{b^{2} x^{6} \sqrt{c + d x^{2}}}{7} & \text{for}\: d \neq 0 \\\sqrt{c} \left (\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}\right ) & \text{otherwise} \end{cases} \]

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*c*sqrt(c + d*x**2)/(3*d) + a**2*x**2*sqrt(c + d*x**2)/3 - 4*a*b*
c**2*sqrt(c + d*x**2)/(15*d**2) + 2*a*b*c*x**2*sqrt(c + d*x**2)/(15*d) + 2*a*b*x
**4*sqrt(c + d*x**2)/5 + 8*b**2*c**3*sqrt(c + d*x**2)/(105*d**3) - 4*b**2*c**2*x
**2*sqrt(c + d*x**2)/(105*d**2) + b**2*c*x**4*sqrt(c + d*x**2)/(35*d) + b**2*x**
6*sqrt(c + d*x**2)/7, Ne(d, 0)), (sqrt(c)*(a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/
6), True))

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GIAC/XCAS [A]  time = 0.227028, size = 130, normalized size = 1.69 \[ \frac{35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} + \frac{14 \,{\left (3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c\right )} a b}{d} + \frac{{\left (15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2}\right )} b^{2}}{d^{2}}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(35*(d*x^2 + c)^(3/2)*a^2 + 14*(3*(d*x^2 + c)^(5/2) - 5*(d*x^2 + c)^(3/2)*
c)*a*b/d + (15*(d*x^2 + c)^(7/2) - 42*(d*x^2 + c)^(5/2)*c + 35*(d*x^2 + c)^(3/2)
*c^2)*b^2/d^2)/d