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

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Rubi [A]  time = 0.09, antiderivative size = 53, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.31, size = 244, normalized size = 4.60 \[ \frac {\sqrt {1-\frac {b^2 x^2}{a^2}} \left (a^2 d+b^2 c\right ) x^{-\frac {b^2 c}{a^2 d+b^2 c}} \left (b^2 d x^2 \, _2F_1\left (\frac {1}{2},\frac {2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac {4 d a^2+3 b^2 c}{2 d a^2+2 b^2 c};\frac {b^2 x^2}{a^2}\right )-\left (2 a^2 d+b^2 c\right ) \, _2F_1\left (\frac {1}{2},-\frac {b^2 c}{2 \left (d a^2+b^2 c\right )};\frac {2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac {b^2 x^2}{a^2}\right )\right )}{b^2 \sqrt {b x-a} \sqrt {a+b x} \left (2 a^2 d+b^2 c\right )} \]

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, -1/2*(b^2*c)/(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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fricas [A]  time = 1.01, size = 65, normalized size = 1.23 \[ \frac {{\left (b^{2} c + a^{2} d\right )} \sqrt {b x + a} \sqrt {b x - a} x}{a^{2} b^{2} x^{\frac {2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x^{2} + c}{\sqrt {b x + a} \sqrt {b x - a} x^{\frac {2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(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)

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maple [A]  time = 0.05, size = 66, normalized size = 1.25 \[ \frac {\left (a^{2} d +b^{2} c \right ) \sqrt {b x +a}\, \sqrt {b x -a}\, x \,x^{-\frac {a^{2} d +2 b^{2} c}{a^{2} d +b^{2} c}}}{a^{2} b^{2}} \]

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 = 1.24, size = 79, normalized size = 1.49 \[ \frac {{\left (b^{2} c + a^{2} d\right )} \sqrt {b x + a} \sqrt {b x - a} x e^{\left (-\frac {2 \, b^{2} c \log \relax (x)}{b^{2} c + a^{2} d} - \frac {a^{2} d \log \relax (x)}{b^{2} c + a^{2} d}\right )}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(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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mupad [B]  time = 3.27, size = 96, normalized size = 1.81 \[ -\frac {\frac {x\,\left (d\,a^4+c\,a^2\,b^2\right )}{a^2\,b^2}-\frac {x^3\,\left (d\,a^2\,b^2+c\,b^4\right )}{a^2\,b^2}}{x^{\frac {d\,a^2+2\,c\,b^2}{d\,a^2+c\,b^2}}\,\sqrt {a+b\,x}\,\sqrt {b\,x-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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