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

Optimal. Leaf size=162 \[ \frac{(c x)^{m+1} \left (a^2 g-a b f+b^2 e\right )}{b^3 c (m+1)}+\frac{(c x)^{m+1} \left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^3 c (m+1)}+\frac{x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac{g x^{2 n+1} (c x)^m}{b (m+2 n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.312267, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(c x)^{m+1} \left (a^2 g-a b f+b^2 e\right )}{b^3 c (m+1)}+\frac{(c x)^{m+1} \left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^3 c (m+1)}+\frac{x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac{g x^{2 n+1} (c x)^m}{b (m+2 n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.3022, size = 190, normalized size = 1.17 \[ \frac{f x^{- m} x^{m + 2 n + 1} \left (c x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 2 n + 1}{n} \\ \frac{m + 3 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (m + 2 n + 1\right )} + \frac{d \left (c x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (m + 1\right )} + \frac{e x^{n} \left (c x\right )^{- n} \left (c x\right )^{m + n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (m + n + 1\right )} + \frac{g x^{3 n} \left (c x\right )^{- 3 n} \left (c x\right )^{m + 3 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 3 n + 1}{n} \\ \frac{m + 4 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (m + 3 n + 1\right )} \]

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),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.491568, size = 150, normalized size = 0.93 \[ x (c x)^m \left (\frac{a^2 g}{b^3 (m+1)}+\frac{\left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^3 (m+1)}-\frac{a \left (\frac{f}{m+1}+\frac{g x^n}{m+n+1}\right )}{b^2}+\frac{e}{b m+b}+\frac{f x^n}{b (m+n+1)}+\frac{g x^{2 n}}{b m+2 b n+b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{m} \left ( d+e{x}^{n}+f{x}^{2\,n}+g{x}^{3\,n} \right ) }{a+b{x}^{n}}}\, dx \]

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),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (b^{3} c^{m} d - a b^{2} c^{m} e + a^{2} b c^{m} f - a^{3} c^{m} g\right )} \int \frac{x^{m}}{b^{4} x^{n} + a b^{3}}\,{d x} + \frac{{\left (m^{2} + m{\left (n + 2\right )} + n + 1\right )} b^{2} c^{m} g x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} +{\left ({\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} b^{2} c^{m} e -{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a b c^{m} f +{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{2} c^{m} g\right )} x x^{m} +{\left ({\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} b^{2} c^{m} f -{\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} a b c^{m} g\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{{\left (m^{3} + 3 \, m^{2}{\left (n + 1\right )} +{\left (2 \, n^{2} + 6 \, n + 3\right )} m + 2 \, n^{2} + 3 \, n + 1\right )} b^{3}} \]

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/(b*x^n + a),x, algorithm="maxima")

[Out]

(b^3*c^m*d - a*b^2*c^m*e + a^2*b*c^m*f - a^3*c^m*g)*integrate(x^m/(b^4*x^n + a*b
^3), x) + ((m^2 + m*(n + 2) + n + 1)*b^2*c^m*g*x*e^(m*log(x) + 2*n*log(x)) + ((m
^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c^m*e - (m^2 + m*(3*n + 2) + 2*n^2 + 3*n
+ 1)*a*b*c^m*f + (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a^2*c^m*g)*x*x^m + ((m^2
+ 2*m*(n + 1) + 2*n + 1)*b^2*c^m*f - (m^2 + 2*m*(n + 1) + 2*n + 1)*a*b*c^m*g)*x
*e^(m*log(x) + n*log(x)))/((m^3 + 3*m^2*(n + 1) + (2*n^2 + 6*n + 3)*m + 2*n^2 +
3*n + 1)*b^3)

_______________________________________________________________________________________

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

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/(b*x^n + a),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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),x)

[Out]

Timed out

_______________________________________________________________________________________

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

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/(b*x^n + a),x, algorithm="giac")

[Out]

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