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

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 = 0.0907204, 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, Rules used = {450} \[ \sqrt{b x-a} \sqrt{a+b x} \left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \]

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))

Rule 450

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*
(m + 1)), x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && EqQ
[a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \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 &=\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{b^2 c+a^2 d}} \sqrt{-a+b x} \sqrt{a+b x}\\ \end{align*}

Mathematica [C]  time = 0.258403, size = 244, normalized size = 4.6 \[ \frac{\sqrt{1-\frac{b^2 x^2}{a^2}} \left (a^2 d+b^2 c\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \left (b^2 d x^2 \, _2F_1\left (\frac{1}{2},\frac{2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac{4 d a^2+3 b^2 c}{2 d a^2+2 b^2 c};\frac{b^2 x^2}{a^2}\right )-\left (2 a^2 d+b^2 c\right ) \, _2F_1\left (\frac{1}{2},-\frac{b^2 c}{2 \left (d a^2+b^2 c\right )};\frac{2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac{b^2 x^2}{a^2}\right )\right )}{b^2 \sqrt{b x-a} \sqrt{a+b x} \left (2 a^2 d+b^2 c\right )} \]

Antiderivative was successfully verified.

[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]

((b^2*c + a^2*d)*Sqrt[1 - (b^2*x^2)/a^2]*(-((b^2*c + 2*a^2*d)*Hypergeometric2F1[1/2, -(b^2*c)/(2*(b^2*c + a^2*
d)), (b^2*c + 2*a^2*d)/(2*b^2*c + 2*a^2*d), (b^2*x^2)/a^2]) + b^2*d*x^2*Hypergeometric2F1[1/2, (b^2*c + 2*a^2*
d)/(2*b^2*c + 2*a^2*d), (3*b^2*c + 4*a^2*d)/(2*b^2*c + 2*a^2*d), (b^2*x^2)/a^2]))/(b^2*(b^2*c + 2*a^2*d)*x^((b
^2*c)/(b^2*c + a^2*d))*Sqrt[-a + b*x]*Sqrt[a + b*x])

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Maple [A]  time = 0.008, size = 66, normalized size = 1.3 \begin{align*}{\frac{x \left ({a}^{2}d+{b}^{2}c \right ) }{{b}^{2}{a}^{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}} \end{align*}

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)/b^2/a^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 = 1.61435, size = 107, normalized size = 2.02 \begin{align*} \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}} \end{align*}

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, 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 = 1.79243, size = 128, normalized size = 2.42 \begin{align*} \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}}} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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)