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

Optimal. Leaf size=109 \[ \frac{(d x-c)^{5/2} (c+d x)^{5/2} \left (a d^2+2 b c^2\right )}{5 d^6}+\frac{c^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (a d^2+b c^2\right )}{3 d^6}+\frac{b (d x-c)^{7/2} (c+d x)^{7/2}}{7 d^6} \]

[Out]

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

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Rubi [A]  time = 0.291181, antiderivative size = 118, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{2 c^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (7 a d^2+4 b c^2\right )}{105 d^6}+\frac{x^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (7 a d^2+4 b c^2\right )}{35 d^4}+\frac{b x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*(4*b*c^2 + 7*a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(105*d^6) + ((4*b*c
^2 + 7*a*d^2)*x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(35*d^4) + (b*x^4*(-c + d*x)
^(3/2)*(c + d*x)^(3/2))/(7*d^2)

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Rubi in Sympy [A]  time = 17.5402, size = 105, normalized size = 0.96 \[ \frac{b x^{4} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{7 d^{2}} + \frac{2 c^{2} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (7 a d^{2} + 4 b c^{2}\right )}{105 d^{6}} + \frac{x^{2} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (7 a d^{2} + 4 b c^{2}\right )}{35 d^{4}} \]

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*(-c + d*x)**(3/2)*(c + d*x)**(3/2)/(7*d**2) + 2*c**2*(-c + d*x)**(3/2)*(c
+ d*x)**(3/2)*(7*a*d**2 + 4*b*c**2)/(105*d**6) + x**2*(-c + d*x)**(3/2)*(c + d*
x)**(3/2)*(7*a*d**2 + 4*b*c**2)/(35*d**4)

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Mathematica [A]  time = 0.0750872, size = 88, normalized size = 0.81 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (d^2 x^2-c^2\right ) \left (7 a d^2 \left (2 c^2+3 d^2 x^2\right )+b \left (8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right )\right )}{105 d^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-c^2 + d^2*x^2)*(7*a*d^2*(2*c^2 + 3*d^2*x^2) + b*
(8*c^4 + 12*c^2*d^2*x^2 + 15*d^4*x^4)))/(105*d^6)

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Maple [A]  time = 0.009, size = 68, normalized size = 0.6 \[{\frac{15\,b{d}^{4}{x}^{4}+21\,a{d}^{4}{x}^{2}+12\,b{c}^{2}{d}^{2}{x}^{2}+14\,a{c}^{2}{d}^{2}+8\,b{c}^{4}}{105\,{d}^{6}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( dx-c \right ) ^{{\frac{3}{2}}}} \]

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]

1/105*(d*x+c)^(3/2)*(15*b*d^4*x^4+21*a*d^4*x^2+12*b*c^2*d^2*x^2+14*a*c^2*d^2+8*b
*c^4)*(d*x-c)^(3/2)/d^6

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Maxima [A]  time = 1.38088, size = 167, normalized size = 1.53 \[ \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x^{4}}{7 \, d^{2}} + \frac{4 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{2} x^{2}}{35 \, d^{4}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a x^{2}}{5 \, d^{2}} + \frac{8 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{4}}{105 \, d^{6}} + \frac{2 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a c^{2}}{15 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*(d^2*x^2 - c^2)^(3/2)*b*x^4/d^2 + 4/35*(d^2*x^2 - c^2)^(3/2)*b*c^2*x^2/d^4 +
1/5*(d^2*x^2 - c^2)^(3/2)*a*x^2/d^2 + 8/105*(d^2*x^2 - c^2)^(3/2)*b*c^4/d^6 + 2
/15*(d^2*x^2 - c^2)^(3/2)*a*c^2/d^4

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Fricas [A]  time = 0.413169, size = 582, normalized size = 5.34 \[ -\frac{960 \, b d^{14} x^{14} - 8 \, b c^{14} - 14 \, a c^{12} d^{2} - 336 \,{\left (7 \, b c^{2} d^{12} - 4 \, a d^{14}\right )} x^{12} + 1736 \,{\left (b c^{4} d^{10} - 2 \, a c^{2} d^{12}\right )} x^{10} - 7 \,{\left (89 \, b c^{6} d^{8} - 328 \, a c^{4} d^{10}\right )} x^{8} + 7 \,{\left (118 \, b c^{8} d^{6} + 109 \, a c^{6} d^{8}\right )} x^{6} - 105 \,{\left (7 \, b c^{10} d^{4} + 12 \, a c^{8} d^{6}\right )} x^{4} + 49 \,{\left (4 \, b c^{12} d^{2} + 7 \, a c^{10} d^{4}\right )} x^{2} -{\left (960 \, b d^{13} x^{13} - 48 \,{\left (39 \, b c^{2} d^{11} - 28 \, a d^{13}\right )} x^{11} + 40 \,{\left (23 \, b c^{4} d^{9} - 70 \, a c^{2} d^{11}\right )} x^{9} -{\left (337 \, b c^{6} d^{7} - 1064 \, a c^{4} d^{9}\right )} x^{7} + 21 \,{\left (33 \, b c^{8} d^{5} + 49 \, a c^{6} d^{7}\right )} x^{5} - 105 \,{\left (4 \, b c^{10} d^{3} + 7 \, a c^{8} d^{5}\right )} x^{3} + 14 \,{\left (4 \, b c^{12} d + 7 \, a c^{10} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{105 \,{\left (64 \, d^{13} x^{7} - 112 \, c^{2} d^{11} x^{5} + 56 \, c^{4} d^{9} x^{3} - 7 \, c^{6} d^{7} x -{\left (64 \, d^{12} x^{6} - 80 \, c^{2} d^{10} x^{4} + 24 \, c^{4} d^{8} x^{2} - c^{6} d^{6}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(960*b*d^14*x^14 - 8*b*c^14 - 14*a*c^12*d^2 - 336*(7*b*c^2*d^12 - 4*a*d^1
4)*x^12 + 1736*(b*c^4*d^10 - 2*a*c^2*d^12)*x^10 - 7*(89*b*c^6*d^8 - 328*a*c^4*d^
10)*x^8 + 7*(118*b*c^8*d^6 + 109*a*c^6*d^8)*x^6 - 105*(7*b*c^10*d^4 + 12*a*c^8*d
^6)*x^4 + 49*(4*b*c^12*d^2 + 7*a*c^10*d^4)*x^2 - (960*b*d^13*x^13 - 48*(39*b*c^2
*d^11 - 28*a*d^13)*x^11 + 40*(23*b*c^4*d^9 - 70*a*c^2*d^11)*x^9 - (337*b*c^6*d^7
- 1064*a*c^4*d^9)*x^7 + 21*(33*b*c^8*d^5 + 49*a*c^6*d^7)*x^5 - 105*(4*b*c^10*d^
3 + 7*a*c^8*d^5)*x^3 + 14*(4*b*c^12*d + 7*a*c^10*d^3)*x)*sqrt(d*x + c)*sqrt(d*x
- c))/(64*d^13*x^7 - 112*c^2*d^11*x^5 + 56*c^4*d^9*x^3 - 7*c^6*d^7*x - (64*d^12*
x^6 - 80*c^2*d^10*x^4 + 24*c^4*d^8*x^2 - c^6*d^6)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \left (a + b x^{2}\right ) \sqrt{- c + d x} \sqrt{c + d x}\, dx \]

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]

Integral(x**3*(a + b*x**2)*sqrt(-c + d*x)*sqrt(c + d*x), x)

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GIAC/XCAS [A]  time = 0.230614, size = 227, normalized size = 2.08 \[ \frac{7 \,{\left ({\left (d x + c\right )}{\left (3 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{3}} - \frac{4 \, c}{d^{3}}\right )} + \frac{17 \, c^{2}}{d^{3}}\right )} - \frac{10 \, c^{3}}{d^{3}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} a +{\left ({\left (3 \,{\left ({\left (d x + c\right )}{\left (5 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{5}} - \frac{6 \, c}{d^{5}}\right )} + \frac{74 \, c^{2}}{d^{5}}\right )} - \frac{96 \, c^{3}}{d^{5}}\right )}{\left (d x + c\right )} + \frac{203 \, c^{4}}{d^{5}}\right )}{\left (d x + c\right )} - \frac{70 \, c^{5}}{d^{5}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} b}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(7*((d*x + c)*(3*(d*x + c)*((d*x + c)/d^3 - 4*c/d^3) + 17*c^2/d^3) - 10*c^
3/d^3)*(d*x + c)^(3/2)*sqrt(d*x - c)*a + ((3*((d*x + c)*(5*(d*x + c)*((d*x + c)/
d^5 - 6*c/d^5) + 74*c^2/d^5) - 96*c^3/d^5)*(d*x + c) + 203*c^4/d^5)*(d*x + c) -
70*c^5/d^5)*(d*x + c)^(3/2)*sqrt(d*x - c)*b)/d