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

Optimal. Leaf size=53 \[ \sqrt{b x-a} \sqrt{a+b x} \left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \]

[Out]

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

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Rubi [A]  time = 0.297772, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018 \[ \sqrt{b x-a} \sqrt{a+b x} \left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.0874, size = 42, normalized size = 0.79 \[ x^{- \frac{b^{2} c}{a^{2} d + b^{2} c}} \sqrt{- a + b x} \sqrt{a + b x} \left (\frac{d}{b^{2}} + \frac{c}{a^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 4.26321, size = 1424, normalized size = 26.87 \[ -\frac{d \left (d a^2+b^2 c\right ) x^{-\frac{b^2 c}{d a^2+b^2 c}} \left (\frac{d (a-b x)^2 \sqrt{\frac{b x}{a}+1} F_1\left (-\frac{b^2 c}{d a^2+b^2 c};-\frac{1}{2},\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right ) a^3}{c \sqrt{1-\frac{b x}{a}} \left (2 a^3 d F_1\left (-\frac{b^2 c}{d a^2+b^2 c};-\frac{1}{2},\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right )-b \left (d a^2+b^2 c\right ) x \left (F_1\left (\frac{a^2 d}{d a^2+b^2 c};-\frac{1}{2},\frac{3}{2};\frac{2 d a^2+b^2 c}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right )+\, _2F_1\left (\frac{1}{2},\frac{a^2 d}{2 \left (d a^2+b^2 c\right )};\frac{3 d a^2}{2 \left (d a^2+b^2 c\right )}+\frac{b^2 c}{d a^2+b^2 c};\frac{b^2 x^2}{a^2}\right )\right )\right )}+\frac{d (a+b x)^2 \sqrt{1-\frac{b x}{a}} F_1\left (-\frac{b^2 c}{d a^2+b^2 c};\frac{1}{2},-\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right ) a^3}{c \sqrt{\frac{b x}{a}+1} \left (2 d F_1\left (-\frac{b^2 c}{d a^2+b^2 c};\frac{1}{2},-\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right ) a^3+b \left (d a^2+b^2 c\right ) x \left (F_1\left (\frac{a^2 d}{d a^2+b^2 c};\frac{3}{2},-\frac{1}{2};\frac{2 d a^2+b^2 c}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right )+\, _2F_1\left (\frac{1}{2},\frac{a^2 d}{2 \left (d a^2+b^2 c\right )};\frac{3 d a^2}{2 \left (d a^2+b^2 c\right )}+\frac{b^2 c}{d a^2+b^2 c};\frac{b^2 x^2}{a^2}\right )\right )\right )}+\frac{b^2 (a-b x)^2 \sqrt{\frac{b x}{a}+1} F_1\left (-\frac{b^2 c}{d a^2+b^2 c};-\frac{1}{2},\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right ) a}{\sqrt{1-\frac{b x}{a}} \left (2 a^3 d F_1\left (-\frac{b^2 c}{d a^2+b^2 c};-\frac{1}{2},\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right )-b \left (d a^2+b^2 c\right ) x \left (F_1\left (\frac{a^2 d}{d a^2+b^2 c};-\frac{1}{2},\frac{3}{2};\frac{2 d a^2+b^2 c}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right )+\, _2F_1\left (\frac{1}{2},\frac{a^2 d}{2 \left (d a^2+b^2 c\right )};\frac{3 d a^2}{2 \left (d a^2+b^2 c\right )}+\frac{b^2 c}{d a^2+b^2 c};\frac{b^2 x^2}{a^2}\right )\right )\right )}+\frac{b^2 (a+b x)^2 \sqrt{1-\frac{b x}{a}} F_1\left (-\frac{b^2 c}{d a^2+b^2 c};\frac{1}{2},-\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right ) a}{\sqrt{\frac{b x}{a}+1} \left (2 d F_1\left (-\frac{b^2 c}{d a^2+b^2 c};\frac{1}{2},-\frac{1}{2};\frac{a^2 d}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right ) a^3+b \left (d a^2+b^2 c\right ) x \left (F_1\left (\frac{a^2 d}{d a^2+b^2 c};\frac{3}{2},-\frac{1}{2};\frac{2 d a^2+b^2 c}{d a^2+b^2 c};\frac{b x}{a},-\frac{b x}{a}\right )+\, _2F_1\left (\frac{1}{2},\frac{a^2 d}{2 \left (d a^2+b^2 c\right )};\frac{3 d a^2}{2 \left (d a^2+b^2 c\right )}+\frac{b^2 c}{d a^2+b^2 c};\frac{b^2 x^2}{a^2}\right )\right )\right )}-\frac{(a-b x) (a+b x) \, _2F_1\left (-\frac{1}{2},-\frac{b^2 c}{2 \left (d a^2+b^2 c\right )};1-\frac{b^2 c}{2 \left (d a^2+b^2 c\right )};\frac{b^2 x^2}{a^2}\right )}{c}\right )}{b^4 \sqrt{b x-a} \sqrt{a+b x} \sqrt{1-\frac{b^2 x^2}{a^2}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((d*(b^2*c + a^2*d)*(-(((a - b*x)*(a + b*x)*Hypergeometric2F1[-1/2, -(b^2*c)/(2
*(b^2*c + a^2*d)), 1 - (b^2*c)/(2*(b^2*c + a^2*d)), (b^2*x^2)/a^2])/c) + (a*b^2*
(a - b*x)^2*Sqrt[1 + (b*x)/a]*AppellF1[-((b^2*c)/(b^2*c + a^2*d)), -1/2, 1/2, (a
^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)])/(Sqrt[1 - (b*x)/a]*(2*a^3*d*AppellF
1[-((b^2*c)/(b^2*c + a^2*d)), -1/2, 1/2, (a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*
x)/a)] - b*(b^2*c + a^2*d)*x*(AppellF1[(a^2*d)/(b^2*c + a^2*d), -1/2, 3/2, (b^2*
c + 2*a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)] + HypergeometricPFQ[{1/2, (a^
2*d)/(2*(b^2*c + a^2*d))}, {(b^2*c)/(b^2*c + a^2*d) + (3*a^2*d)/(2*(b^2*c + a^2*
d))}, (b^2*x^2)/a^2]))) + (a^3*d*(a - b*x)^2*Sqrt[1 + (b*x)/a]*AppellF1[-((b^2*c
)/(b^2*c + a^2*d)), -1/2, 1/2, (a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)])/(c
*Sqrt[1 - (b*x)/a]*(2*a^3*d*AppellF1[-((b^2*c)/(b^2*c + a^2*d)), -1/2, 1/2, (a^2
*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)] - b*(b^2*c + a^2*d)*x*(AppellF1[(a^2*d
)/(b^2*c + a^2*d), -1/2, 3/2, (b^2*c + 2*a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x
)/a)] + HypergeometricPFQ[{1/2, (a^2*d)/(2*(b^2*c + a^2*d))}, {(b^2*c)/(b^2*c +
a^2*d) + (3*a^2*d)/(2*(b^2*c + a^2*d))}, (b^2*x^2)/a^2]))) + (a*b^2*(a + b*x)^2*
Sqrt[1 - (b*x)/a]*AppellF1[-((b^2*c)/(b^2*c + a^2*d)), 1/2, -1/2, (a^2*d)/(b^2*c
+ a^2*d), (b*x)/a, -((b*x)/a)])/(Sqrt[1 + (b*x)/a]*(2*a^3*d*AppellF1[-((b^2*c)/
(b^2*c + a^2*d)), 1/2, -1/2, (a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)] + b*(
b^2*c + a^2*d)*x*(AppellF1[(a^2*d)/(b^2*c + a^2*d), 3/2, -1/2, (b^2*c + 2*a^2*d)
/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)] + HypergeometricPFQ[{1/2, (a^2*d)/(2*(b^2
*c + a^2*d))}, {(b^2*c)/(b^2*c + a^2*d) + (3*a^2*d)/(2*(b^2*c + a^2*d))}, (b^2*x
^2)/a^2]))) + (a^3*d*(a + b*x)^2*Sqrt[1 - (b*x)/a]*AppellF1[-((b^2*c)/(b^2*c + a
^2*d)), 1/2, -1/2, (a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)])/(c*Sqrt[1 + (b
*x)/a]*(2*a^3*d*AppellF1[-((b^2*c)/(b^2*c + a^2*d)), 1/2, -1/2, (a^2*d)/(b^2*c +
a^2*d), (b*x)/a, -((b*x)/a)] + b*(b^2*c + a^2*d)*x*(AppellF1[(a^2*d)/(b^2*c + a
^2*d), 3/2, -1/2, (b^2*c + 2*a^2*d)/(b^2*c + a^2*d), (b*x)/a, -((b*x)/a)] + Hype
rgeometricPFQ[{1/2, (a^2*d)/(2*(b^2*c + a^2*d))}, {(b^2*c)/(b^2*c + a^2*d) + (3*
a^2*d)/(2*(b^2*c + a^2*d))}, (b^2*x^2)/a^2])))))/(b^4*x^((b^2*c)/(b^2*c + a^2*d)
)*Sqrt[-a + b*x]*Sqrt[a + b*x]*Sqrt[1 - (b^2*x^2)/a^2]))

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Maple [A]  time = 0.014, size = 66, normalized size = 1.3 \[{\frac{x \left ({a}^{2}d+{b}^{2}c \right ) }{{a}^{2}{b}^{2}}\sqrt{bx+a}\sqrt{bx-a} \left ({x}^{{\frac{{a}^{2}d+2\,{b}^{2}c}{{a}^{2}d+{b}^{2}c}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.85853, size = 107, normalized size = 2.02 \[ \frac{{\left (b^{2} c + a^{2} d\right )} \sqrt{b x + a} \sqrt{b x - a} x e^{\left (-\frac{2 \, b^{2} c \log \left (x\right )}{b^{2} c + a^{2} d} - \frac{a^{2} d \log \left (x\right )}{b^{2} c + a^{2} d}\right )}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.262433, size = 88, normalized size = 1.66 \[ \frac{{\left (b^{2} c + a^{2} d\right )} \sqrt{b x + a} \sqrt{b x - a} x}{a^{2} b^{2} x^{\frac{2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{\sqrt{b x + a} \sqrt{b x - a} x^{\frac{2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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