3.401 \(\int \frac{1}{a c+b c x^n+d \sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=135 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2}-\frac{d x \sqrt{\frac{b x^n}{a}+1} F_1\left (\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) \sqrt{a+b x^n}} \]

[Out]

-((d*x*Sqrt[1 + (b*x^n)/a]*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -(
(b*c^2*x^n)/(a*c^2 - d^2))])/((a*c^2 - d^2)*Sqrt[a + b*x^n])) + (c*x*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), -((b*c^2*x^n)/(a*c^2 - d^2))])/(a*c^2 - d^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.216469, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2}-\frac{d x \sqrt{\frac{b x^n}{a}+1} F_1\left (\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + b*c*x^n + d*Sqrt[a + b*x^n])^(-1),x]

[Out]

-((d*x*Sqrt[1 + (b*x^n)/a]*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -(
(b*c^2*x^n)/(a*c^2 - d^2))])/((a*c^2 - d^2)*Sqrt[a + b*x^n])) + (c*x*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), -((b*c^2*x^n)/(a*c^2 - d^2))])/(a*c^2 - d^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.4942, size = 105, normalized size = 0.78 \[ - \frac{c x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b c^{2} x^{n}}{a c^{2} - d^{2}}} \right )}}{- a c^{2} + d^{2}} + \frac{d x \sqrt{a + b x^{n}} \operatorname{appellf_{1}}{\left (\frac{1}{n},\frac{1}{2},1,1 + \frac{1}{n},- \frac{b x^{n}}{a},- \frac{b c^{2} x^{n}}{a c^{2} - d^{2}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (- a c^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*x*hyper((1, 1/n), (1 + 1/n,), -b*c**2*x**n/(a*c**2 - d**2))/(-a*c**2 + d**2)
+ d*x*sqrt(a + b*x**n)*appellf1(1/n, 1/2, 1, 1 + 1/n, -b*x**n/a, -b*c**2*x**n/(a
*c**2 - d**2))/(a*sqrt(1 + b*x**n/a)*(-a*c**2 + d**2))

_______________________________________________________________________________________

Mathematica [B]  time = 0.895518, size = 320, normalized size = 2.37 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2}-\frac{2 a d (n+1) x \left (a c^2-d^2\right ) F_1\left (\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )}{\sqrt{a+b x^n} \left (a c^2+b c^2 x^n-d^2\right ) \left (\left (a c^2-d^2\right ) \left (2 a (n+1) F_1\left (\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )-b n x^n F_1\left (1+\frac{1}{n};\frac{3}{2},1;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )\right )-2 a b c^2 n x^n F_1\left (1+\frac{1}{n};\frac{1}{2},2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a*c + b*c*x^n + d*Sqrt[a + b*x^n])^(-1),x]

[Out]

(-2*a*d*(a*c^2 - d^2)*(1 + n)*x*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a
), -((b*c^2*x^n)/(a*c^2 - d^2))])/(Sqrt[a + b*x^n]*(a*c^2 - d^2 + b*c^2*x^n)*(-2
*a*b*c^2*n*x^n*AppellF1[1 + n^(-1), 1/2, 2, 2 + n^(-1), -((b*x^n)/a), -((b*c^2*x
^n)/(a*c^2 - d^2))] + (a*c^2 - d^2)*(-(b*n*x^n*AppellF1[1 + n^(-1), 3/2, 1, 2 +
n^(-1), -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))]) + 2*a*(1 + n)*AppellF1[n^(-
1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))]))) + (c*x*Hy
pergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*c^2*x^n)/(a*c^2 - d^2))])/(a*c^2 - d
^2)

_______________________________________________________________________________________

Maple [F]  time = 0.014, size = 0, normalized size = 0. \[ \int \left ( ac+bc{x}^{n}+d\sqrt{a+b{x}^{n}} \right ) ^{-1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{b c x^{n} + a c + \sqrt{b x^{n} + a} d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),x, algorithm="maxima")

[Out]

integrate(1/(b*c*x^n + a*c + sqrt(b*x^n + a)*d), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b c x^{n} + a c + \sqrt{b x^{n} + a} d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),x, algorithm="fricas")

[Out]

integral(1/(b*c*x^n + a*c + sqrt(b*x^n + a)*d), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{a c + b c x^{n} + d \sqrt{a + b x^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(a*c + b*c*x**n + d*sqrt(a + b*x**n)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{b c x^{n} + a c + \sqrt{b x^{n} + a} d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),x, algorithm="giac")

[Out]

integrate(1/(b*c*x^n + a*c + sqrt(b*x^n + a)*d), x)