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

Optimal. Leaf size=80 \[ a \sqrt{d x-c} \sqrt{c+d x}-a c \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )+\frac{b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.286411, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ a \sqrt{d x-c} \sqrt{c+d x}-a c \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )+\frac{b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.3838, size = 65, normalized size = 0.81 \[ - a c \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )} + a \sqrt{- c + d x} \sqrt{c + d x} + \frac{b \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{2}} \]

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,x)

[Out]

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

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Mathematica [A]  time = 0.142888, size = 75, normalized size = 0.94 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2-b c^2+b d^2 x^2\right )}{3 d^2}+a c \tan ^{-1}\left (\frac{c}{\sqrt{d x-c} \sqrt{c+d x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 174, normalized size = 2.2 \[{\frac{1}{3\,{d}^{2}}\sqrt{dx-c}\sqrt{dx+c} \left ({x}^{2}b{d}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+3\,a{c}^{2}\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){d}^{2}+3\,a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{2}\sqrt{-{c}^{2}}-b{c}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

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,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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]

Exception raised: ValueError

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Fricas [A]  time = 0.250421, size = 383, normalized size = 4.79 \[ -\frac{4 \, b d^{6} x^{6} - b c^{6} + 3 \, a c^{4} d^{2} - 3 \,{\left (3 \, b c^{2} d^{4} - 4 \, a d^{6}\right )} x^{4} + 3 \,{\left (2 \, b c^{4} d^{2} - 5 \, a c^{2} d^{4}\right )} x^{2} -{\left (4 \, b d^{5} x^{5} -{\left (7 \, b c^{2} d^{3} - 12 \, a d^{5}\right )} x^{3} + 3 \,{\left (b c^{4} d - 3 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} + 6 \,{\left (4 \, a c d^{5} x^{3} - 3 \, a c^{3} d^{3} x -{\left (4 \, a c d^{4} x^{2} - a c^{3} d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{3 \,{\left (4 \, d^{5} x^{3} - 3 \, c^{2} d^{3} x -{\left (4 \, d^{4} x^{2} - c^{2} d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

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]

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

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

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,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224712, size = 109, normalized size = 1.36 \[ 2 \, a c \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right ) + \frac{1}{1920} \,{\left (3 \, a d^{6} +{\left ({\left (d x + c\right )} b d^{4} - 2 \, b c d^{4}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c} \]

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]

2*a*c*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c) + 1/1920*(3*a*d^6 + ((d*x
+ c)*b*d^4 - 2*b*c*d^4)*(d*x + c))*sqrt(d*x + c)*sqrt(d*x - c)