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

Optimal. Leaf size=57 \[ \frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.227326, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.5265, size = 46, normalized size = 0.81 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{c^{2} x} + \frac{2 b \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d x}} \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0726016, size = 60, normalized size = 1.05 \[ \frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{b \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.029, size = 97, normalized size = 1.7 \[{\frac{{\it csgn} \left ( d \right ) }{{c}^{2}xd}\sqrt{dx-c}\sqrt{dx+c} \left ( \ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) xb{c}^{2}+a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237346, size = 117, normalized size = 2.05 \[ \frac{a d -{\left (b d x^{2} - \sqrt{d x + c} \sqrt{d x - c} b x\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{d^{2} x^{2} - \sqrt{d x + c} \sqrt{d x - c} d x} \]

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^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 51.5059, size = 165, normalized size = 2.89 \[ - \frac{a d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} - \frac{i a d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), c**2
/(d**2*x**2))/(4*pi**(3/2)*c**2) - I*a*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()
), ((3/4, 5/4), (1/2, 1, 1, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2
)*c**2) + b*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*b*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*p
i**(3/2)*d)

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GIAC/XCAS [A]  time = 0.224909, size = 89, normalized size = 1.56 \[ \frac{\frac{16 \, a d^{2}}{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}} - b{\rm ln}\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4}\right )}{2 \, d} \]

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^2),x, algorithm="giac")

[Out]

1/2*(16*a*d^2/((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2) - b*ln((sqrt(d*x + c)
- sqrt(d*x - c))^4))/d