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

Optimal. Leaf size=167 \[ \frac{c x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right )}-\frac{d x^{m+1} \sqrt{\frac{b x^n}{a}+1} F_1\left (\frac{m+1}{n};\frac{1}{2},1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt{a+b x^n}} \]

[Out]

-((d*x^(1 + m)*Sqrt[1 + (b*x^n)/a]*AppellF1[(1 + m)/n, 1/2, 1, (1 + m + n)/n, -(
(b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))])/((a*c^2 - d^2)*(1 + m)*Sqrt[a + b*x^n
])) + (c*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*c^2*x^n)/
(a*c^2 - d^2))])/((a*c^2 - d^2)*(1 + m))

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Rubi [A]  time = 0.455837, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{c x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right )}-\frac{d x^{m+1} \sqrt{\frac{b x^n}{a}+1} F_1\left (\frac{m+1}{n};\frac{1}{2},1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

-((d*x^(1 + m)*Sqrt[1 + (b*x^n)/a]*AppellF1[(1 + m)/n, 1/2, 1, (1 + m + n)/n, -(
(b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))])/((a*c^2 - d^2)*(1 + m)*Sqrt[a + b*x^n
])) + (c*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*c^2*x^n)/
(a*c^2 - d^2))])/((a*c^2 - d^2)*(1 + m))

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Rubi in Sympy [A]  time = 43.1399, size = 126, normalized size = 0.75 \[ - \frac{c x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b c^{2} x^{n}}{a c^{2} - d^{2}}} \right )}}{\left (m + 1\right ) \left (- a c^{2} + d^{2}\right )} + \frac{d x^{m + 1} \sqrt{a + b x^{n}} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},\frac{1}{2},1,\frac{m + n + 1}{n},- \frac{b x^{n}}{a},- \frac{b c^{2} x^{n}}{a c^{2} - d^{2}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right ) \left (- a c^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.27272, size = 373, normalized size = 2.23 \[ \frac{x^{m+1} \left (c \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right )-\frac{2 a d (m+n+1) \left (d^2-a c^2\right )^2 F_1\left (\frac{m+1}{n};\frac{1}{2},1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )}{\sqrt{a+b x^n} \left (a c^2+b c^2 x^n-d^2\right ) \left (2 a (m+n+1) \left (a c^2-d^2\right ) F_1\left (\frac{m+1}{n};\frac{1}{2},1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )-b n x^n \left (2 a c^2 F_1\left (\frac{m+n+1}{n};\frac{1}{2},2;\frac{m+1}{n}+2;-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )+\left (a c^2-d^2\right ) F_1\left (\frac{m+n+1}{n};\frac{3}{2},1;\frac{m+1}{n}+2;-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right )\right )\right )}\right )}{(m+1) \left (a c^2-d^2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^(1 + m)*((-2*a*d*(-(a*c^2) + d^2)^2*(1 + m + n)*AppellF1[(1 + m)/n, 1/2, 1, (
1 + m + n)/n, -((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*(a*c^2 - d^2)*(1 + m + n)*AppellF1[(1 + m)/n, 1/2, 1,
(1 + m + n)/n, -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))] - b*n*x^n*(2*a*c^2*A
ppellF1[(1 + m + n)/n, 1/2, 2, 2 + (1 + m)/n, -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2
- d^2))] + (a*c^2 - d^2)*AppellF1[(1 + m + n)/n, 3/2, 1, 2 + (1 + m)/n, -((b*x^
n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))]))) + c*Hypergeometric2F1[1, (1 + m)/n, (1 +
m + n)/n, -((b*c^2*x^n)/(a*c^2 - d^2))]))/((a*c^2 - d^2)*(1 + m))

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( ac+bc{x}^{n}+d\sqrt{a+b{x}^{n}} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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