3.572 \(\int \frac{(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=305 \[ \frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{(m+3 n+1) \sqrt{a+b x^n}} \]

[Out]

(d*(c*x)^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m +
n)/n, -((b*x^n)/a)])/(c*(1 + m)*Sqrt[a + b*x^n]) + (e*x^(1 + n)*(c*x)^m*Sqrt[1 +
(b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)]
)/((1 + m + n)*Sqrt[a + b*x^n]) + (f*x^(1 + 2*n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hyp
ergeometric2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/((1 + m + 2
*n)*Sqrt[a + b*x^n]) + (g*x^(1 + 3*n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric
2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n)/a)])/((1 + m + 3*n)*Sqrt[a
+ b*x^n])

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Rubi [A]  time = 0.494093, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{(m+3 n+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/Sqrt[a + b*x^n],x]

[Out]

(d*(c*x)^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m +
n)/n, -((b*x^n)/a)])/(c*(1 + m)*Sqrt[a + b*x^n]) + (e*x^(1 + n)*(c*x)^m*Sqrt[1 +
(b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)]
)/((1 + m + n)*Sqrt[a + b*x^n]) + (f*x^(1 + 2*n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hyp
ergeometric2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/((1 + m + 2
*n)*Sqrt[a + b*x^n]) + (g*x^(1 + 3*n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric
2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n)/a)])/((1 + m + 3*n)*Sqrt[a
+ b*x^n])

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Rubi in Sympy [A]  time = 57.7425, size = 292, normalized size = 0.96 \[ \frac{d \left (c x\right )^{m + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right )} + \frac{e x^{n} \left (c x\right )^{- n} \left (c x\right )^{m + n + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \sqrt{1 + \frac{b x^{n}}{a}} \left (m + n + 1\right )} + \frac{f x^{2 n} \left (c x\right )^{- 2 n} \left (c x\right )^{m + 2 n + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 2 n + 1}{n} \\ \frac{m + 3 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 2 n + 1\right )} + \frac{g x^{3 n} \left (c x\right )^{- 3 n} \left (c x\right )^{m + 3 n + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 3 n + 1}{n} \\ \frac{m + 4 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**m*(d+e*x**n+f*x**(2*n)+g*x**(3*n))/(a+b*x**n)**(1/2),x)

[Out]

d*(c*x)**(m + 1)*sqrt(a + b*x**n)*hyper((1/2, (m + 1)/n), ((m + n + 1)/n,), -b*x
**n/a)/(a*c*sqrt(1 + b*x**n/a)*(m + 1)) + e*x**n*(c*x)**(-n)*(c*x)**(m + n + 1)*
sqrt(a + b*x**n)*hyper((1/2, (m + n + 1)/n), ((m + 2*n + 1)/n,), -b*x**n/a)/(a*c
*sqrt(1 + b*x**n/a)*(m + n + 1)) + f*x**(2*n)*(c*x)**(-2*n)*(c*x)**(m + 2*n + 1)
*sqrt(a + b*x**n)*hyper((1/2, (m + 2*n + 1)/n), ((m + 3*n + 1)/n,), -b*x**n/a)/(
a*c*sqrt(1 + b*x**n/a)*(m + 2*n + 1)) + g*x**(3*n)*(c*x)**(-3*n)*(c*x)**(m + 3*n
+ 1)*sqrt(a + b*x**n)*hyper((1/2, (m + 3*n + 1)/n), ((m + 4*n + 1)/n,), -b*x**n
/a)/(a*c*sqrt(1 + b*x**n/a)*(m + 3*n + 1))

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Mathematica [A]  time = 2.75986, size = 399, normalized size = 1.31 \[ \frac{x (c x)^m \left (2 (m+1) \left (a+b x^n\right ) \left (4 a^2 g \left (m^2+m (3 n+2)+2 n^2+3 n+1\right )-2 a b \left (f \left (2 m^2+m (7 n+4)+5 n^2+7 n+2\right )+g \left (2 m^2+m (5 n+4)+2 n^2+5 n+2\right ) x^n\right )+b^2 \left (e \left (4 m^2+8 m (2 n+1)+15 n^2+16 n+4\right )+(2 m+n+2) x^n \left (f (2 m+5 n+2)+g (2 m+3 n+2) x^n\right )\right )\right )+\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (-8 a^3 g (m+1) \left (m^2+m (3 n+2)+2 n^2+3 n+1\right )+4 a^2 b f (m+1) \left (2 m^2+m (7 n+4)+5 n^2+7 n+2\right )-2 a b^2 e (m+1) \left (4 m^2+8 m (2 n+1)+15 n^2+16 n+4\right )+b^3 d \left (8 m^3+12 m^2 (3 n+2)+m \left (46 n^2+72 n+24\right )+15 n^3+46 n^2+36 n+8\right )\right )\right )}{b^3 (m+1) (2 m+n+2) (2 m+3 n+2) (2 m+5 n+2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/Sqrt[a + b*x^n],x]

[Out]

(x*(c*x)^m*(2*(1 + m)*(a + b*x^n)*(4*a^2*g*(1 + m^2 + 3*n + 2*n^2 + m*(2 + 3*n))
- 2*a*b*(f*(2 + 2*m^2 + 7*n + 5*n^2 + m*(4 + 7*n)) + g*(2 + 2*m^2 + 5*n + 2*n^2
+ m*(4 + 5*n))*x^n) + b^2*(e*(4 + 4*m^2 + 16*n + 15*n^2 + 8*m*(1 + 2*n)) + (2 +
2*m + n)*x^n*(f*(2 + 2*m + 5*n) + g*(2 + 2*m + 3*n)*x^n))) + (-2*a*b^2*e*(1 + m
)*(4 + 4*m^2 + 16*n + 15*n^2 + 8*m*(1 + 2*n)) - 8*a^3*g*(1 + m)*(1 + m^2 + 3*n +
2*n^2 + m*(2 + 3*n)) + 4*a^2*b*f*(1 + m)*(2 + 2*m^2 + 7*n + 5*n^2 + m*(4 + 7*n)
) + b^3*d*(8 + 8*m^3 + 36*n + 46*n^2 + 15*n^3 + 12*m^2*(2 + 3*n) + m*(24 + 72*n
+ 46*n^2)))*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m + n)/n,
-((b*x^n)/a)]))/(b^3*(1 + m)*(2 + 2*m + n)*(2 + 2*m + 3*n)*(2 + 2*m + 5*n)*Sqrt
[a + b*x^n])

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Maple [F]  time = 0.26, size = 0, normalized size = 0. \[ \int{ \left ( cx \right ) ^{m} \left ( d+e{x}^{n}+f{x}^{2\,n}+g{x}^{3\,n} \right ){\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**m*(d+e*x**n+f*x**(2*n)+g*x**(3*n))/(a+b*x**n)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a), x)