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

Optimal. Leaf size=68 \[ \frac{\left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^3}+\frac{b x \sqrt{d x-c} \sqrt{c+d x}}{2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.115793, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^3}+\frac{b x \sqrt{d x-c} \sqrt{c+d x}}{2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.259, size = 75, normalized size = 1.1 \[ \frac{2 a \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d} + \frac{b c^{2} \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d^{3}} + \frac{b x \sqrt{- c + d x} \sqrt{c + d x}}{2 d^{2}} \]

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]

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

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Mathematica [A]  time = 0.0665225, size = 71, normalized size = 1.04 \[ \frac{\left (2 a d^2+b c^2\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )+b d x \sqrt{d x-c} \sqrt{c+d x}}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.025, size = 124, normalized size = 1.8 \[{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}}\sqrt{dx-c}\sqrt{dx+c} \left ( bx\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d+b{c}^{2}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) +2\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) a{d}^{2} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

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]

1/2*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x*(d^2*x^2-c^2)^(1/2)*csgn(d)*d+b*c^2*ln((csg
n(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))+2*ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*cs
gn(d))*a*d^2)/(d^2*x^2-c^2)^(1/2)/d^3*csgn(d)

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Maxima [A]  time = 1.39622, size = 140, normalized size = 2.06 \[ \frac{a \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{\sqrt{d^{2}}} + \frac{b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} b x}{2 \, d^{2}} \]

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]

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

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Fricas [A]  time = 0.237688, size = 261, normalized size = 3.84 \[ -\frac{2 \, b d^{4} x^{4} - 2 \, b c^{2} d^{2} x^{2} -{\left (2 \, b d^{3} x^{3} - b c^{2} d x\right )} \sqrt{d x + c} \sqrt{d x - c} -{\left (b c^{4} + 2 \, a c^{2} d^{2} + 2 \,{\left (b c^{2} d + 2 \, a d^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} x - 2 \,{\left (b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{2 \,{\left (2 \, d^{5} x^{2} - 2 \, \sqrt{d x + c} \sqrt{d x - c} d^{4} x - c^{2} d^{3}\right )}} \]

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]

-1/2*(2*b*d^4*x^4 - 2*b*c^2*d^2*x^2 - (2*b*d^3*x^3 - b*c^2*d*x)*sqrt(d*x + c)*sq
rt(d*x - c) - (b*c^4 + 2*a*c^2*d^2 + 2*(b*c^2*d + 2*a*d^3)*sqrt(d*x + c)*sqrt(d*
x - c)*x - 2*(b*c^2*d^2 + 2*a*d^4)*x^2)*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))
/(2*d^5*x^2 - 2*sqrt(d*x + c)*sqrt(d*x - c)*d^4*x - c^2*d^3)

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Sympy [A]  time = 43.6661, size = 199, normalized size = 2.93 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{b c^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i b c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} \]

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]

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

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GIAC/XCAS [A]  time = 0.255644, size = 107, normalized size = 1.57 \[ \frac{{\left ({\left (d x + c\right )} b d^{4} - b c d^{4}\right )} \sqrt{d x + c} \sqrt{d x - c} - 2 \,{\left (b c^{2} d^{4} + 2 \, a d^{6}\right )}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{384 \, d} \]

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]

1/384*(((d*x + c)*b*d^4 - b*c*d^4)*sqrt(d*x + c)*sqrt(d*x - c) - 2*(b*c^2*d^4 +
2*a*d^6)*ln(abs(-sqrt(d*x + c) + sqrt(d*x - c))))/d