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

Optimal. Leaf size=118 \[ \frac{2 c^2 \sqrt{d x-c} \sqrt{c+d x} \left (5 a d^2+4 b c^2\right )}{15 d^6}+\frac{x^2 \sqrt{d x-c} \sqrt{c+d x} \left (5 a d^2+4 b c^2\right )}{15 d^4}+\frac{b x^4 \sqrt{d x-c} \sqrt{c+d x}}{5 d^2} \]

[Out]

(2*c^2*(4*b*c^2 + 5*a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(15*d^6) + ((4*b*c^2 +
5*a*d^2)*x^2*Sqrt[-c + d*x]*Sqrt[c + d*x])/(15*d^4) + (b*x^4*Sqrt[-c + d*x]*Sqrt
[c + d*x])/(5*d^2)

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Rubi [A]  time = 0.305834, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{2 c^2 \sqrt{d x-c} \sqrt{c+d x} \left (5 a d^2+4 b c^2\right )}{15 d^6}+\frac{x^2 \sqrt{d x-c} \sqrt{c+d x} \left (5 a d^2+4 b c^2\right )}{15 d^4}+\frac{b x^4 \sqrt{d x-c} \sqrt{c+d x}}{5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*(4*b*c^2 + 5*a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(15*d^6) + ((4*b*c^2 +
5*a*d^2)*x^2*Sqrt[-c + d*x]*Sqrt[c + d*x])/(15*d^4) + (b*x^4*Sqrt[-c + d*x]*Sqrt
[c + d*x])/(5*d^2)

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Rubi in Sympy [A]  time = 17.574, size = 105, normalized size = 0.89 \[ \frac{b x^{4} \sqrt{- c + d x} \sqrt{c + d x}}{5 d^{2}} + \frac{2 c^{2} \sqrt{- c + d x} \sqrt{c + d x} \left (5 a d^{2} + 4 b c^{2}\right )}{15 d^{6}} + \frac{x^{2} \sqrt{- c + d x} \sqrt{c + d x} \left (5 a d^{2} + 4 b c^{2}\right )}{15 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

b*x**4*sqrt(-c + d*x)*sqrt(c + d*x)/(5*d**2) + 2*c**2*sqrt(-c + d*x)*sqrt(c + d*
x)*(5*a*d**2 + 4*b*c**2)/(15*d**6) + x**2*sqrt(-c + d*x)*sqrt(c + d*x)*(5*a*d**2
+ 4*b*c**2)/(15*d**4)

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Mathematica [A]  time = 0.079503, size = 74, normalized size = 0.63 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (5 a d^2 \left (2 c^2+d^2 x^2\right )+b \left (8 c^4+4 c^2 d^2 x^2+3 d^4 x^4\right )\right )}{15 d^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-c + d*x]*Sqrt[c + d*x]*(5*a*d^2*(2*c^2 + d^2*x^2) + b*(8*c^4 + 4*c^2*d^2*
x^2 + 3*d^4*x^4)))/(15*d^6)

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Maple [A]  time = 0.009, size = 68, normalized size = 0.6 \[{\frac{3\,b{d}^{4}{x}^{4}+5\,a{d}^{4}{x}^{2}+4\,b{c}^{2}{d}^{2}{x}^{2}+10\,a{c}^{2}{d}^{2}+8\,b{c}^{4}}{15\,{d}^{6}}\sqrt{dx+c}\sqrt{dx-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)

[Out]

1/15*(d*x+c)^(1/2)*(3*b*d^4*x^4+5*a*d^4*x^2+4*b*c^2*d^2*x^2+10*a*c^2*d^2+8*b*c^4
)/d^6*(d*x-c)^(1/2)

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Maxima [A]  time = 1.37497, size = 167, normalized size = 1.42 \[ \frac{\sqrt{d^{2} x^{2} - c^{2}} b x^{4}}{5 \, d^{2}} + \frac{4 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{2} x^{2}}{15 \, d^{4}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a x^{2}}{3 \, d^{2}} + \frac{8 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{4}}{15 \, d^{6}} + \frac{2 \, \sqrt{d^{2} x^{2} - c^{2}} a c^{2}}{3 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(d^2*x^2 - c^2)*b*x^4/d^2 + 4/15*sqrt(d^2*x^2 - c^2)*b*c^2*x^2/d^4 + 1/3
*sqrt(d^2*x^2 - c^2)*a*x^2/d^2 + 8/15*sqrt(d^2*x^2 - c^2)*b*c^4/d^6 + 2/3*sqrt(d
^2*x^2 - c^2)*a*c^2/d^4

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Fricas [A]  time = 0.2754, size = 420, normalized size = 3.56 \[ -\frac{48 \, b d^{10} x^{10} - 8 \, b c^{10} - 10 \, a c^{8} d^{2} - 20 \,{\left (b c^{2} d^{8} - 4 \, a d^{10}\right )} x^{8} + 5 \,{\left (11 \, b c^{4} d^{6} + 4 \, a c^{2} d^{8}\right )} x^{6} - 5 \,{\left (35 \, b c^{6} d^{4} + 43 \, a c^{4} d^{6}\right )} x^{4} + 25 \,{\left (4 \, b c^{8} d^{2} + 5 \, a c^{6} d^{4}\right )} x^{2} -{\left (48 \, b d^{9} x^{9} + 4 \,{\left (b c^{2} d^{7} + 20 \, a d^{9}\right )} x^{7} + 3 \,{\left (21 \, b c^{4} d^{5} + 20 \, a c^{2} d^{7}\right )} x^{5} - 35 \,{\left (4 \, b c^{6} d^{3} + 5 \, a c^{4} d^{5}\right )} x^{3} + 10 \,{\left (4 \, b c^{8} d + 5 \, a c^{6} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{15 \,{\left (16 \, d^{11} x^{5} - 20 \, c^{2} d^{9} x^{3} + 5 \, c^{4} d^{7} x -{\left (16 \, d^{10} x^{4} - 12 \, c^{2} d^{8} x^{2} + c^{4} d^{6}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(48*b*d^10*x^10 - 8*b*c^10 - 10*a*c^8*d^2 - 20*(b*c^2*d^8 - 4*a*d^10)*x^8
+ 5*(11*b*c^4*d^6 + 4*a*c^2*d^8)*x^6 - 5*(35*b*c^6*d^4 + 43*a*c^4*d^6)*x^4 + 25*
(4*b*c^8*d^2 + 5*a*c^6*d^4)*x^2 - (48*b*d^9*x^9 + 4*(b*c^2*d^7 + 20*a*d^9)*x^7 +
3*(21*b*c^4*d^5 + 20*a*c^2*d^7)*x^5 - 35*(4*b*c^6*d^3 + 5*a*c^4*d^5)*x^3 + 10*(
4*b*c^8*d + 5*a*c^6*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))/(16*d^11*x^5 - 20*c^2*d
^9*x^3 + 5*c^4*d^7*x - (16*d^10*x^4 - 12*c^2*d^8*x^2 + c^4*d^6)*sqrt(d*x + c)*sq
rt(d*x - c))

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Sympy [A]  time = 136.219, size = 240, normalized size = 2.03 \[ \frac{a c^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{i a c^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{b c^{5}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{9}{4}, - \frac{7}{4} & -2, -2, - \frac{3}{2}, 1 \\- \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{6}} + \frac{i b c^{5}{G_{6, 6}^{2, 6}\left (\begin{matrix} -3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 1 & \\- \frac{11}{4}, - \frac{9}{4} & -3, - \frac{5}{2}, - \frac{5}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

a*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*a*c**3*meijerg(((-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) + b*c**5*meijerg(((-9/4, -7/4), (-2, -2, -3/2,
1)), ((-5/2, -9/4, -2, -7/4, -3/2, 0), ()), c**2/(d**2*x**2))/(4*pi**(3/2)*d**6
) + I*b*c**5*meijerg(((-3, -11/4, -5/2, -9/4, -2, 1), ()), ((-11/4, -9/4), (-3,
-5/2, -5/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**6)

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GIAC/XCAS [A]  time = 0.221716, size = 151, normalized size = 1.28 \[ \frac{{\left (15 \, b c^{4} d^{25} + 15 \, a c^{2} d^{27} -{\left (20 \, b c^{3} d^{25} + 10 \, a c d^{27} -{\left (22 \, b c^{2} d^{25} + 5 \, a d^{27} + 3 \,{\left ({\left (d x + c\right )} b d^{25} - 4 \, b c d^{25}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c}}{276480 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/276480*(15*b*c^4*d^25 + 15*a*c^2*d^27 - (20*b*c^3*d^25 + 10*a*c*d^27 - (22*b*c
^2*d^25 + 5*a*d^27 + 3*((d*x + c)*b*d^25 - 4*b*c*d^25)*(d*x + c))*(d*x + c))*(d*
x + c))*sqrt(d*x + c)*sqrt(d*x - c)/d