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

Optimal. Leaf size=67 \[ \frac{(d x-c)^{3/2} (c+d x)^{3/2} \left (a d^2+b c^2\right )}{3 d^4}+\frac{b (d x-c)^{5/2} (c+d x)^{5/2}}{5 d^4} \]

[Out]

((b*c^2 + a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^4) + (b*(-c + d*x)^(5/2)
*(c + d*x)^(5/2))/(5*d^4)

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Rubi [A]  time = 0.151357, antiderivative size = 72, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(d x-c)^{3/2} (c+d x)^{3/2} \left (5 a d^2+2 b c^2\right )}{15 d^4}+\frac{b x^2 (d x-c)^{3/2} (c+d x)^{3/2}}{5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.89443, size = 61, normalized size = 0.91 \[ \frac{b x^{2} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{5 d^{2}} + \frac{\left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d^{2} + 2 b c^{2}\right )}{15 d^{4}} \]

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*(-c + d*x)**(3/2)*(c + d*x)**(3/2)/(5*d**2) + (-c + d*x)**(3/2)*(c + d*x)
**(3/2)*(5*a*d**2 + 2*b*c**2)/(15*d**4)

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Mathematica [A]  time = 0.0582542, size = 62, normalized size = 0.93 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (d^2 x^2-c^2\right ) \left (5 a d^2+2 b c^2+3 b d^2 x^2\right )}{15 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x*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)*(2*b*c^2 + 5*a*d^2 + 3*b*d^2*x^2)
)/(15*d^4)

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Maple [A]  time = 0.006, size = 44, normalized size = 0.7 \[{\frac{3\,b{d}^{2}{x}^{2}+5\,a{d}^{2}+2\,b{c}^{2}}{15\,{d}^{4}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( dx-c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37831, size = 95, normalized size = 1.42 \[ \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x^{2}}{5 \, d^{2}} + \frac{2 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{2}}{15 \, d^{4}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a}{3 \, d^{2}} \]

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.279759, size = 421, normalized size = 6.28 \[ -\frac{48 \, b d^{10} x^{10} + 2 \, b c^{10} + 5 \, a c^{8} d^{2} - 20 \,{\left (5 \, b c^{2} d^{8} - 4 \, a d^{10}\right )} x^{8} + 5 \,{\left (7 \, b c^{4} d^{6} - 44 \, a c^{2} d^{8}\right )} x^{6} + 5 \,{\left (8 \, b c^{6} d^{4} + 41 \, a c^{4} d^{6}\right )} x^{4} - 5 \,{\left (5 \, b c^{8} d^{2} + 14 \, a c^{6} d^{4}\right )} x^{2} -{\left (48 \, b d^{9} x^{9} - 4 \,{\left (19 \, b c^{2} d^{7} - 20 \, a d^{9}\right )} x^{7} + 3 \,{\left (b c^{4} d^{5} - 60 \, a c^{2} d^{7}\right )} x^{5} + 5 \,{\left (7 \, b c^{6} d^{3} + 25 \, a c^{4} d^{5}\right )} x^{3} - 5 \,{\left (2 \, b c^{8} d + 5 \, a c^{6} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{15 \,{\left (16 \, d^{9} x^{5} - 20 \, c^{2} d^{7} x^{3} + 5 \, c^{4} d^{5} x -{\left (16 \, d^{8} x^{4} - 12 \, c^{2} d^{6} x^{2} + c^{4} d^{4}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(48*b*d^10*x^10 + 2*b*c^10 + 5*a*c^8*d^2 - 20*(5*b*c^2*d^8 - 4*a*d^10)*x^8
+ 5*(7*b*c^4*d^6 - 44*a*c^2*d^8)*x^6 + 5*(8*b*c^6*d^4 + 41*a*c^4*d^6)*x^4 - 5*(
5*b*c^8*d^2 + 14*a*c^6*d^4)*x^2 - (48*b*d^9*x^9 - 4*(19*b*c^2*d^7 - 20*a*d^9)*x^
7 + 3*(b*c^4*d^5 - 60*a*c^2*d^7)*x^5 + 5*(7*b*c^6*d^3 + 25*a*c^4*d^5)*x^3 - 5*(2
*b*c^8*d + 5*a*c^6*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))/(16*d^9*x^5 - 20*c^2*d^7
*x^3 + 5*c^4*d^5*x - (16*d^8*x^4 - 12*c^2*d^6*x^2 + c^4*d^4)*sqrt(d*x + c)*sqrt(
d*x - c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217173, size = 126, normalized size = 1.88 \[ \frac{{\left ({\left (d x + c\right )}{\left (3 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{3}} - \frac{4 \, c}{d^{3}}\right )} + \frac{17 \, c^{2}}{d^{3}}\right )} - \frac{10 \, c^{3}}{d^{3}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} b + \frac{5 \,{\left (d x + c\right )}^{\frac{3}{2}}{\left (d x - c\right )}^{\frac{3}{2}} a}{d}}{15 \, d} \]

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,x, algorithm="giac")

[Out]

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