3.6.76 \(\int \frac {(c x)^m (d+e x^n+f x^{2 n}+g x^{3 n})}{a+b x^n} \, dx\) [576]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1858, 20, 30, 371} \begin {gather*} \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)} \end {gather*}

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)*Hypergeometric2
F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*b^3*c*(1 + m))

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1858

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx &=\int \left (\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^m}{b^3}+\frac {(b f-a g) x^n (c x)^m}{b^2}+\frac {g x^{2 n} (c x)^m}{b}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^m}{b^3 \left (a+b x^n\right )}\right ) \, dx\\ &=\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {g \int x^{2 n} (c x)^m \, dx}{b}+\frac {(b f-a g) \int x^n (c x)^m \, dx}{b^2}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) \int \frac {(c x)^m}{a+b x^n} \, dx}{b^3}\\ &=\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^3 c (1+m)}+\frac {\left (g x^{-m} (c x)^m\right ) \int x^{m+2 n} \, dx}{b}+\frac {\left ((b f-a g) x^{-m} (c x)^m\right ) \int x^{m+n} \, dx}{b^2}\\ &=\frac {(b f-a g) x^{1+n} (c x)^m}{b^2 (1+m+n)}+\frac {g x^{1+2 n} (c x)^m}{b (1+m+2 n)}+\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^3 c (1+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.44, size = 150, normalized size = 0.93 \begin {gather*} x (c x)^m \left (\frac {a^2 g}{b^3 (1+m)}+\frac {e}{b+b m}+\frac {f x^n}{b (1+m+n)}+\frac {g x^{2 n}}{b+b m+2 b n}-\frac {a \left (\frac {f}{1+m}+\frac {g x^n}{1+m+n}\right )}{b^2}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^3 (1+m)}\right ) \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{m} \left (d +e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}\right )}{a +b \,x^{n}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

(b^3*c^m*d + a^2*b*c^m*f - a^3*c^m*g - a*b^2*c^m*e)*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)*a*b*c^m*f - (m^2 + m*(3*
n + 2) + 2*n^2 + 3*n + 1)*a^2*c^m*g - (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c^m*e)*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 = 0.37, size = 38, normalized size = 0.23 \begin {gather*} {\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 ) \end {gather*}

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, 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 [C] Result contains complex when optimal does not.
time = 23.93, size = 654, normalized size = 4.04 \begin {gather*} \frac {c^{m} d m x x^{m} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {c^{m} d x x^{m} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {c^{m} e m x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {c^{m} e x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {c^{m} e x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {c^{m} f m x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {2 c^{m} f x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {c^{m} f x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {c^{m} g m x x^{m} x^{3 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 3 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} + \frac {3 c^{m} g x x^{m} x^{3 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 3 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} + \frac {c^{m} g x x^{m} x^{3 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 3 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} \end {gather*}

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]

c**m*d*m*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**2*gamma(m/n + 1 + 1/n)
) + c**m*d*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**2*gamma(m/n + 1 + 1/
n)) + c**m*e*m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*n**2*g
amma(m/n + 2 + 1/n)) + c**m*e*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 +
1/n)/(a*n*gamma(m/n + 2 + 1/n)) + c**m*e*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gam
ma(m/n + 1 + 1/n)/(a*n**2*gamma(m/n + 2 + 1/n)) + c**m*f*m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*n**2*gamma(m/n + 3 + 1/n)) + 2*c**m*f*x*x**m*x**(2*n)*lerchphi(b*x**
n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*n*gamma(m/n + 3 + 1/n)) + c**m*f*x*x**m*x**(2*n
)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*n**2*gamma(m/n + 3 + 1/n)) + c*
*m*g*m*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 3 + 1/n)/(a*n**2*gamma
(m/n + 4 + 1/n)) + 3*c**m*g*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 3
+ 1/n)/(a*n*gamma(m/n + 4 + 1/n)) + c**m*g*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/
n)*gamma(m/n + 3 + 1/n)/(a*n**2*gamma(m/n + 4 + 1/n))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x\right )}^m\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right )}{a+b\,x^n} \,d x \end {gather*}

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)

________________________________________________________________________________________