3.4.79 \(\int \frac {x^{-\frac {2 b^2 c+a^2 d}{b^2 c+a^2 d}} (c+d x^2)}{\sqrt {-a+b x} \sqrt {a+b x}} \, dx\) [379]

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

[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 = 0.06, 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 = {461} \begin {gather*} \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}} \end {gather*}

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 461

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] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 10.33, size = 296, normalized size = 5.58 \begin {gather*} \frac {\left (b^2 c+a^2 d\right ) x^{-\frac {b^2 c}{b^2 c+a^2 d}} \sqrt {-a+b x} \sqrt {a+b x} \sqrt {1-\frac {b^2 x^2}{a^2}} \left (\left (b^2 c+a^2 d\right ) F_1\left (-\frac {b^2 c}{b^2 c+a^2 d};-\frac {1}{2},\frac {1}{2};\frac {a^2 d}{b^2 c+a^2 d};\frac {b x}{a},-\frac {b x}{a}\right )+\left (b^2 c+a^2 d\right ) F_1\left (-\frac {b^2 c}{b^2 c+a^2 d};\frac {1}{2},-\frac {1}{2};\frac {a^2 d}{b^2 c+a^2 d};\frac {b x}{a},-\frac {b x}{a}\right )-2 a^2 d \, _2F_1\left (-\frac {1}{2},-\frac {b^2 c}{2 \left (b^2 c+a^2 d\right )};1-\frac {b^2 c}{2 \left (b^2 c+a^2 d\right )};\frac {b^2 x^2}{a^2}\right )\right )}{2 b^4 c \left (a^2-b^2 x^2\right )} \end {gather*}

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]

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

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Maple [A]
time = 0.29, size = 66, normalized size = 1.25

method	result	size
gosper	\(\frac {x \left (a^{2} d +b^{2} c \right ) \sqrt {b x +a}\, x^{-\frac {a^{2} d +2 b^{2} c}{a^{2} d +b^{2} c}} \sqrt {b x -a}}{a^{2} b^{2}}\)	\(66\)

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,method=_RETURNVERBOSE)

[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 = 0.37, size = 79, normalized size = 1.49 \begin {gather*} \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 {gather*}

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 = 2.72, size = 65, normalized size = 1.23 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3437 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [B]
time = 3.27, size = 96, normalized size = 1.81 \begin {gather*} -\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}} \end {gather*}

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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