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

Optimal. Leaf size=114 \[ \frac{x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+b c^2\right )}{8 d^2}-\frac{c^2 \left (4 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^3}+\frac{b x (d x-c)^{3/2} (c+d x)^{3/2}}{4 d^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.161227, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{8} x \sqrt{d x-c} \sqrt{c+d x} \left (4 a+\frac{b c^2}{d^2}\right )-\frac{c^2 \left (4 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^3}+\frac{b x (d x-c)^{3/2} (c+d x)^{3/2}}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.3635, size = 129, normalized size = 1.13 \[ - \frac{a c^{2} \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d} + \frac{a x \sqrt{- c + d x} \sqrt{c + d x}}{2} - \frac{b c^{4} \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{4 d^{3}} + \frac{b c^{2} x \sqrt{- c + d x} \sqrt{c + d x}}{8 d^{2}} + \frac{b x \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*c**2*atanh(sqrt(-c + d*x)/sqrt(c + d*x))/d + a*x*sqrt(-c + d*x)*sqrt(c + d*x)
/2 - b*c**4*atanh(sqrt(-c + d*x)/sqrt(c + d*x))/(4*d**3) + b*c**2*x*sqrt(-c + d*
x)*sqrt(c + d*x)/(8*d**2) + b*x*(-c + d*x)**(3/2)*(c + d*x)**(3/2)/(4*d**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0825454, size = 96, normalized size = 0.84 \[ \frac{d x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2-b c^2+2 b d^2 x^2\right )-\left (4 a c^2 d^2+b c^4\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{8 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.016, size = 182, normalized size = 1.6 \[{\frac{{\it csgn} \left ( d \right ) }{8\,{d}^{3}}\sqrt{dx-c}\sqrt{dx+c} \left ( 2\,{\it csgn} \left ( d \right ){x}^{3}b{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+4\,ax\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) -b{c}^{2}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d-4\,a{c}^{2}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){d}^{2}-b{c}^{4}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(2*csgn(d)*x^3*b*d^3*(d^2*x^2-c^2)^(1/2)+4*a*x*(
d^2*x^2-c^2)^(1/2)*d^3*csgn(d)-b*c^2*x*(d^2*x^2-c^2)^(1/2)*csgn(d)*d-4*a*c^2*ln(
(csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))*d^2-b*c^4*ln((csgn(d)*(d^2*x^2-c^2)^(
1/2)+d*x)*csgn(d)))*csgn(d)/(d^2*x^2-c^2)^(1/2)/d^3

_______________________________________________________________________________________

Maxima [A]  time = 1.43454, size = 205, normalized size = 1.8 \[ -\frac{a c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}}} - \frac{b c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{8 \, \sqrt{d^{2}} d^{2}} + \frac{1}{2} \, \sqrt{d^{2} x^{2} - c^{2}} a x + \frac{\sqrt{d^{2} x^{2} - c^{2}} b c^{2} x}{8 \, d^{2}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x}{4 \, d^{2}} \]

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, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.259909, size = 517, normalized size = 4.54 \[ -\frac{16 \, b d^{8} x^{8} - 32 \,{\left (b c^{2} d^{6} - a d^{8}\right )} x^{6} + 4 \,{\left (5 \, b c^{4} d^{4} - 12 \, a c^{2} d^{6}\right )} x^{4} - 4 \,{\left (b c^{6} d^{2} - 4 \, a c^{4} d^{4}\right )} x^{2} -{\left (16 \, b d^{7} x^{7} - 8 \,{\left (3 \, b c^{2} d^{5} - 4 \, a d^{7}\right )} x^{5} + 2 \,{\left (5 \, b c^{4} d^{3} - 16 \, a c^{2} d^{5}\right )} x^{3} -{\left (b c^{6} d - 4 \, a c^{4} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} -{\left (b c^{8} + 4 \, a c^{6} d^{2} + 8 \,{\left (b c^{4} d^{4} + 4 \, a c^{2} d^{6}\right )} x^{4} - 8 \,{\left (b c^{6} d^{2} + 4 \, a c^{4} d^{4}\right )} x^{2} - 4 \,{\left (2 \,{\left (b c^{4} d^{3} + 4 \, a c^{2} d^{5}\right )} x^{3} -{\left (b c^{6} d + 4 \, a c^{4} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{8 \,{\left (8 \, d^{7} x^{4} - 8 \, c^{2} d^{5} x^{2} + c^{4} d^{3} - 4 \,{\left (2 \, d^{6} x^{3} - c^{2} d^{4} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

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, algorithm="fricas")

[Out]

-1/8*(16*b*d^8*x^8 - 32*(b*c^2*d^6 - a*d^8)*x^6 + 4*(5*b*c^4*d^4 - 12*a*c^2*d^6)
*x^4 - 4*(b*c^6*d^2 - 4*a*c^4*d^4)*x^2 - (16*b*d^7*x^7 - 8*(3*b*c^2*d^5 - 4*a*d^
7)*x^5 + 2*(5*b*c^4*d^3 - 16*a*c^2*d^5)*x^3 - (b*c^6*d - 4*a*c^4*d^3)*x)*sqrt(d*
x + c)*sqrt(d*x - c) - (b*c^8 + 4*a*c^6*d^2 + 8*(b*c^4*d^4 + 4*a*c^2*d^6)*x^4 -
8*(b*c^6*d^2 + 4*a*c^4*d^4)*x^2 - 4*(2*(b*c^4*d^3 + 4*a*c^2*d^5)*x^3 - (b*c^6*d
+ 4*a*c^4*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))*log(-d*x + sqrt(d*x + c)*sqrt(d*x
- c)))/(8*d^7*x^4 - 8*c^2*d^5*x^2 + c^4*d^3 - 4*(2*d^6*x^3 - c^2*d^4*x)*sqrt(d*
x + c)*sqrt(d*x - c))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.279106, size = 204, normalized size = 1.79 \[ \frac{4 \,{\left (\sqrt{d x + c} \sqrt{d x - c} d x + 2 \, c^{2}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )\right )} a +{\left (\frac{2 \, c^{4}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{d^{2}} +{\left ({\left (d x + c\right )}{\left (2 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{2}} - \frac{3 \, c}{d^{2}}\right )} + \frac{5 \, c^{2}}{d^{2}}\right )} - \frac{c^{3}}{d^{2}}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} b}{8 \, d} \]

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, algorithm="giac")

[Out]

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