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

Optimal. Leaf size=72 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+2 b c^2\right )}{3 d^4}+\frac{b x^2 \sqrt{d x-c} \sqrt{c+d x}}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.158959, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+2 b c^2\right )}{3 d^4}+\frac{b x^2 \sqrt{d x-c} \sqrt{c+d x}}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.97023, size = 61, normalized size = 0.85 \[ \frac{b x^{2} \sqrt{- c + d x} \sqrt{c + d x}}{3 d^{2}} + \frac{\sqrt{- c + d x} \sqrt{c + d x} \left (3 a d^{2} + 2 b c^{2}\right )}{3 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0556434, size = 48, normalized size = 0.67 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+2 b c^2+b d^2 x^2\right )}{3 d^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 43, normalized size = 0.6 \[{\frac{b{d}^{2}{x}^{2}+3\,a{d}^{2}+2\,b{c}^{2}}{3\,{d}^{4}}\sqrt{dx+c}\sqrt{dx-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(d*x+c)^(1/2)*(b*d^2*x^2+3*a*d^2+2*b*c^2)/d^4*(d*x-c)^(1/2)

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Maxima [A]  time = 1.38412, size = 93, normalized size = 1.29 \[ \frac{\sqrt{d^{2} x^{2} - c^{2}} b x^{2}}{3 \, d^{2}} + \frac{2 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{2}}{3 \, d^{4}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.240768, size = 262, normalized size = 3.64 \[ -\frac{4 \, b d^{6} x^{6} + 2 \, b c^{6} + 3 \, a c^{4} d^{2} + 3 \,{\left (b c^{2} d^{4} + 4 \, a d^{6}\right )} x^{4} - 3 \,{\left (3 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{2} -{\left (4 \, b d^{5} x^{5} +{\left (5 \, b c^{2} d^{3} + 12 \, a d^{5}\right )} x^{3} - 3 \,{\left (2 \, b c^{4} d + 3 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \,{\left (4 \, d^{7} x^{3} - 3 \, c^{2} d^{5} x -{\left (4 \, d^{6} x^{2} - c^{2} d^{4}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(4*b*d^6*x^6 + 2*b*c^6 + 3*a*c^4*d^2 + 3*(b*c^2*d^4 + 4*a*d^6)*x^4 - 3*(3*b
*c^4*d^2 + 5*a*c^2*d^4)*x^2 - (4*b*d^5*x^5 + (5*b*c^2*d^3 + 12*a*d^5)*x^3 - 3*(2
*b*c^4*d + 3*a*c^2*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))/(4*d^7*x^3 - 3*c^2*d^5*x
- (4*d^6*x^2 - c^2*d^4)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [A]  time = 76.0226, size = 223, normalized size = 3.1 \[ \frac{a c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i a c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{b c^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{i b c^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), c
**2/(d**2*x**2))/(4*pi**(3/2)*d**2) + I*a*c*meijerg(((-1, -3/4, -1/2, -1/4, 0, 1
), ()), ((-3/4, -1/4), (-1, -1/2, -1/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))
/(4*pi**(3/2)*d**2) + b*c**3*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3/2,
-5/4, -1, -3/4, -1/2, 0), ()), c**2/(d**2*x**2))/(4*pi**(3/2)*d**4) + I*b*c**3*m
eijerg(((-2, -7/4, -3/2, -5/4, -1, 1), ()), ((-7/4, -5/4), (-2, -3/2, -3/2, 0)),
c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**4)

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GIAC/XCAS [A]  time = 0.217631, size = 82, normalized size = 1.14 \[ \frac{{\left (3 \, b c^{2} d^{9} + 3 \, a d^{11} +{\left ({\left (d x + c\right )} b d^{9} - 2 \, b c d^{9}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c}}{1920 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(3*b*c^2*d^9 + 3*a*d^11 + ((d*x + c)*b*d^9 - 2*b*c*d^9)*(d*x + c))*sqrt(d
*x + c)*sqrt(d*x - c)/d