3.213 \(\int (f+g x)^2 \log (c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=181 \[ \frac {(f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right )}{3 g}-\frac {f^3 p \log \left (d+e x^n\right )}{3 g}-\frac {e f^2 n p x^{n+1} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {e x^n}{d}\right )}{d (n+1)}-\frac {e f g n p x^{n+2} \, _2F_1\left (1,\frac {n+2}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{d (n+2)}-\frac {e g^2 n p x^{n+3} \, _2F_1\left (1,\frac {n+3}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{3 d (n+3)} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2463, 1844, 260, 364} \[ \frac {(f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right )}{3 g}-\frac {f^3 p \log \left (d+e x^n\right )}{3 g}-\frac {e f^2 n p x^{n+1} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {e x^n}{d}\right )}{d (n+1)}-\frac {e f g n p x^{n+2} \, _2F_1\left (1,\frac {n+2}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{d (n+2)}-\frac {e g^2 n p x^{n+3} \, _2F_1\left (1,\frac {n+3}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{3 d (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^2*Log[c*(d + e*x^n)^p],x]

[Out]

-((e*f^2*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((e*x^n)/d)])/(d*(1 + n))) - (e*f*g*n*p*x
^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), -((e*x^n)/d)])/(d*(2 + n)) - (e*g^2*n*p*x^(3 + n)*Hyp
ergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(3*d*(3 + n)) - (f^3*p*Log[d + e*x^n])/(3*g) + ((f + g*x)
^3*Log[c*(d + e*x^n)^p])/(3*g)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1844

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]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[((
f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f
+ g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rubi steps

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

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Mathematica [A]  time = 0.25, size = 178, normalized size = 0.98 \[ \frac {(f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right )-e n p \left (\frac {f^3 \log \left (d+e x^n\right )}{e n}+\frac {3 f^2 g x^{n+1} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {e x^n}{d}\right )}{d (n+1)}+\frac {3 f g^2 x^{n+2} \, _2F_1\left (1,\frac {n+2}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{d (n+2)}+\frac {g^3 x^{n+3} \, _2F_1\left (1,\frac {n+3}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{d (n+3)}\right )}{3 g} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^2*Log[c*(d + e*x^n)^p],x]

[Out]

(-(e*n*p*((3*f^2*g*x^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((e*x^n)/d)])/(d*(1 + n)) + (3*f*g^
2*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), -((e*x^n)/d)])/(d*(2 + n)) + (g^3*x^(3 + n)*Hyperg
eometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(d*(3 + n)) + (f^3*Log[d + e*x^n])/(e*n))) + (f + g*x)^3*Log
[c*(d + e*x^n)^p])/(3*g)

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g^{2} x^{2} + 2 \, f g x + f^{2}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

integral((g^2*x^2 + 2*f*g*x + f^2)*log((e*x^n + d)^p*c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

integrate((g*x + f)^2*log((e*x^n + d)^p*c), x)

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maple [F]  time = 1.88, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right )^{2} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*ln(c*(e*x^n+d)^p),x)

[Out]

int((g*x+f)^2*ln(c*(e*x^n+d)^p),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{9} \, {\left (g^{2} n p - 3 \, g^{2} \log \relax (c)\right )} x^{3} - \frac {1}{2} \, {\left (f g n p - 2 \, f g \log \relax (c)\right )} x^{2} - {\left (f^{2} n p - f^{2} \log \relax (c)\right )} x + \frac {1}{3} \, {\left (g^{2} x^{3} + 3 \, f g x^{2} + 3 \, f^{2} x\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + \int \frac {d g^{2} n p x^{2} + 3 \, d f g n p x + 3 \, d f^{2} n p}{3 \, {\left (e x^{n} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*log(c*(d+e*x^n)^p),x, algorithm="maxima")

[Out]

-1/9*(g^2*n*p - 3*g^2*log(c))*x^3 - 1/2*(f*g*n*p - 2*f*g*log(c))*x^2 - (f^2*n*p - f^2*log(c))*x + 1/3*(g^2*x^3
+ 3*f*g*x^2 + 3*f^2*x)*log((e*x^n + d)^p) + integrate(1/3*(d*g^2*n*p*x^2 + 3*d*f*g*n*p*x + 3*d*f^2*n*p)/(e*x^
n + d), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^n)^p)*(f + g*x)^2,x)

[Out]

int(log(c*(d + e*x^n)^p)*(f + g*x)^2, x)

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sympy [C]  time = 20.06, size = 284, normalized size = 1.57 \[ f^{2} x \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {f^{2} p x \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n \Gamma \left (1 + \frac {1}{n}\right )} + f g x^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {g^{2} x^{3} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{3} - \frac {e f g p x^{2} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{d \Gamma \left (2 + \frac {2}{n}\right )} - \frac {2 e f g p x^{2} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{d n \Gamma \left (2 + \frac {2}{n}\right )} - \frac {e g^{2} p x^{3} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{3 d \Gamma \left (2 + \frac {3}{n}\right )} - \frac {e g^{2} p x^{3} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{d n \Gamma \left (2 + \frac {3}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*ln(c*(d+e*x**n)**p),x)

[Out]

f**2*x*log(c*(d + e*x**n)**p) + f**2*p*x*lerchphi(d*x**(-n)*exp_polar(I*pi)/e, 1, exp_polar(I*pi)/n)*gamma(1/n
)/(n*gamma(1 + 1/n)) + f*g*x**2*log(c*(d + e*x**n)**p) + g**2*x**3*log(c*(d + e*x**n)**p)/3 - e*f*g*p*x**2*x**
n*lerchphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 2/n)*gamma(1 + 2/n)/(d*gamma(2 + 2/n)) - 2*e*f*g*p*x**2*x**n*lerch
phi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 2/n)*gamma(1 + 2/n)/(d*n*gamma(2 + 2/n)) - e*g**2*p*x**3*x**n*lerchphi(e*
x**n*exp_polar(I*pi)/d, 1, 1 + 3/n)*gamma(1 + 3/n)/(3*d*gamma(2 + 3/n)) - e*g**2*p*x**3*x**n*lerchphi(e*x**n*e
xp_polar(I*pi)/d, 1, 1 + 3/n)*gamma(1 + 3/n)/(d*n*gamma(2 + 3/n))

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