3.1.100 \(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^3} \, dx\) [100]

Optimal. Leaf size=351 \[ \frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

1/2*(-d*g+e*f)^2*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(a/b/n)/n^3/((c*(e*x+d)^n)^(1/n))+4*g*(-d*g+
e*f)*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(2*a/b/n)/n^3/((c*(e*x+d)^n)^(2/n))+9/2*g^2*(e*x+d)^
3*Ei(3*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(3*a/b/n)/n^3/((c*(e*x+d)^n)^(3/n))-1/2*(e*x+d)*(g*x+f)^2/b/e/n/(
a+b*ln(c*(e*x+d)^n))^2+(-d*g+e*f)*(e*x+d)*(g*x+f)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))-3/2*(e*x+d)*(g*x+f)^2/b^2/
e/n^2/(a+b*ln(c*(e*x+d)^n))

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Rubi [A]
time = 0.61, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \begin {gather*} \frac {4 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {9 g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {(d+e x) (f+g x) (e f-d g)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*f - d*g)^2*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(2*b^3*e^3*E^(a/(b*n))*n^3*(c*(d + e
*x)^n)^n^(-1)) + (4*g*(e*f - d*g)*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^3*e^3*E^
((2*a)/(b*n))*n^3*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*
n)])/(2*b^3*e^3*E^((3*a)/(b*n))*n^3*(c*(d + e*x)^n)^(3/n)) - ((d + e*x)*(f + g*x)^2)/(2*b*e*n*(a + b*Log[c*(d
+ e*x)^n])^2) + ((e*f - d*g)*(d + e*x)*(f + g*x))/(b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])) - (3*(d + e*x)*(f +
g*x)^2)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

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

Rule 2446

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

Rule 2447

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

Rubi steps

\begin {align*} \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {3 \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{2 b n}-\frac {(e f-d g) \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{b e n}\\ &=-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {9 \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 n^2}-\frac {(2 (e f-d g)) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}-\frac {(3 (e f-d g)) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}+\frac {(e f-d g)^2 \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}\\ &=-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {9 \int \left (\frac {(e f-d g)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 g (e f-d g) (d+e x)}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g^2 (d+e x)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{2 b^2 n^2}-\frac {(2 (e f-d g)) \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b^2 e n^2}-\frac {(3 (e f-d g)) \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b^2 e n^2}+\frac {(e f-d g)^2 \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}\\ &=-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (9 g^2\right ) \int \frac {(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 e^2 n^2}-\frac {(2 g (e f-d g)) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac {(3 g (e f-d g)) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}+\frac {(9 g (e f-d g)) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac {\left (2 (e f-d g)^2\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac {\left (3 (e f-d g)^2\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}+\frac {\left (9 (e f-d g)^2\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 e^2 n^2}+\frac {\left ((e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (9 g^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^3 n^2}-\frac {(2 g (e f-d g)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {(3 g (e f-d g)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {(9 g (e f-d g)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {\left (2 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {\left (3 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {\left (9 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^3 n^2}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (9 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^3 n^3}-\frac {\left (2 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac {\left (3 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac {\left (9 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac {\left (2 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac {\left (3 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac {\left (9 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^3 n^3}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 0.95, size = 351, normalized size = 1.00 \begin {gather*} \frac {e^{-\frac {3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (e^{\frac {2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-8 e^{\frac {a}{b n}} g (-e f+d g) (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+9 g^2 (d+e x)^2 \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-b e e^{\frac {3 a}{b n}} n \left (c (d+e x)^n\right )^{3/n} (f+g x) \left (b e n (f+g x)+a (e f+2 d g+3 e g x)+b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(E^((2*a)/(b*n))*(e*f - d*g)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)
]*(a + b*Log[c*(d + e*x)^n])^2 - 8*E^(a/(b*n))*g*(-(e*f) + d*g)*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi
[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2 + 9*g^2*(d + e*x)^2*ExpIntegralEi[(3*(a +
b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2 - b*e*E^((3*a)/(b*n))*n*(c*(d + e*x)^n)^(3/n)*(f +
g*x)*(b*e*n*(f + g*x) + a*(e*f + 2*d*g + 3*e*g*x) + b*(2*d*g + e*(f + 3*g*x))*Log[c*(d + e*x)^n])))/(2*b^3*e^3
*E^((3*a)/(b*n))*n^3*(c*(d + e*x)^n)^(3/n)*(a + b*Log[c*(d + e*x)^n])^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.03, size = 6545, normalized size = 18.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(6545\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2/(a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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

[Out]

-1/2*(2*b*d^2*f*g*log(c) + 2*a*d^2*f*g + (3*a*g^2 + (g^2*n + 3*g^2*log(c))*b)*x^3*e^2 + (2*(2*a*f*g + (f*g*n +
2*f*g*log(c))*b)*e^2 + (5*a*d*g^2 + (d*g^2*n + 5*d*g^2*log(c))*b)*e)*x^2 + (2*b*d^2*g^2*log(c) + 2*a*d^2*g^2
+ (a*f^2 + (f^2*n + f^2*log(c))*b)*e^2 + 2*(3*a*d*f*g + (d*f*g*n + 3*d*f*g*log(c))*b)*e)*x + (a*d*f^2 + (d*f^2
*n + d*f^2*log(c))*b)*e + (3*b*g^2*x^3*e^2 + 2*b*d^2*f*g + b*d*f^2*e + (5*b*d*g^2*e + 4*b*f*g*e^2)*x^2 + (2*b*
d^2*g^2 + 6*b*d*f*g*e + b*f^2*e^2)*x)*log((x*e + d)^n))/(b^4*n^2*e^2*log((x*e + d)^n)^2 + 2*(b^4*n^2*log(c) +
a*b^3*n^2)*e^2*log((x*e + d)^n) + (b^4*n^2*log(c)^2 + 2*a*b^3*n^2*log(c) + a^2*b^2*n^2)*e^2) + integrate(1/2*(
9*g^2*x^2*e^2 + 2*d^2*g^2 + 6*d*f*g*e + f^2*e^2 + 2*(5*d*g^2*e + 4*f*g*e^2)*x)/(b^3*n^2*e^2*log((x*e + d)^n) +
(b^3*n^2*log(c) + a*b^2*n^2)*e^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (357) = 714\).
time = 0.37, size = 1088, normalized size = 3.10 \begin {gather*} -\frac {{\left (8 \, {\left (a^{2} d g^{2} - a^{2} f g e + {\left (b^{2} d g^{2} n^{2} - b^{2} f g n^{2} e\right )} \log \left (x e + d\right )^{2} + {\left (b^{2} d g^{2} - b^{2} f g e\right )} \log \left (c\right )^{2} + 2 \, {\left (a b d g^{2} n - a b f g n e + {\left (b^{2} d g^{2} n - b^{2} f g n e\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 2 \, {\left (a b d g^{2} - a b f g e\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) - {\left (a^{2} d^{2} g^{2} - 2 \, a^{2} d f g e + a^{2} f^{2} e^{2} + {\left (b^{2} d^{2} g^{2} n^{2} - 2 \, b^{2} d f g n^{2} e + b^{2} f^{2} n^{2} e^{2}\right )} \log \left (x e + d\right )^{2} + {\left (b^{2} d^{2} g^{2} - 2 \, b^{2} d f g e + b^{2} f^{2} e^{2}\right )} \log \left (c\right )^{2} + 2 \, {\left (a b d^{2} g^{2} n - 2 \, a b d f g n e + a b f^{2} n e^{2} + {\left (b^{2} d^{2} g^{2} n - 2 \, b^{2} d f g n e + b^{2} f^{2} n e^{2}\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 2 \, {\left (a b d^{2} g^{2} - 2 \, a b d f g e + a b f^{2} e^{2}\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (x e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) + {\left ({\left ({\left (b^{2} g^{2} n^{2} + 3 \, a b g^{2} n\right )} x^{3} + 2 \, {\left (b^{2} f g n^{2} + 2 \, a b f g n\right )} x^{2} + {\left (b^{2} f^{2} n^{2} + a b f^{2} n\right )} x\right )} e^{3} + {\left (b^{2} d f^{2} n^{2} + a b d f^{2} n + {\left (b^{2} d g^{2} n^{2} + 5 \, a b d g^{2} n\right )} x^{2} + 2 \, {\left (b^{2} d f g n^{2} + 3 \, a b d f g n\right )} x\right )} e^{2} + 2 \, {\left (a b d^{2} g^{2} n x + a b d^{2} f g n\right )} e + {\left ({\left (3 \, b^{2} g^{2} n^{2} x^{3} + 4 \, b^{2} f g n^{2} x^{2} + b^{2} f^{2} n^{2} x\right )} e^{3} + {\left (5 \, b^{2} d g^{2} n^{2} x^{2} + 6 \, b^{2} d f g n^{2} x + b^{2} d f^{2} n^{2}\right )} e^{2} + 2 \, {\left (b^{2} d^{2} g^{2} n^{2} x + b^{2} d^{2} f g n^{2}\right )} e\right )} \log \left (x e + d\right ) + {\left ({\left (3 \, b^{2} g^{2} n x^{3} + 4 \, b^{2} f g n x^{2} + b^{2} f^{2} n x\right )} e^{3} + {\left (5 \, b^{2} d g^{2} n x^{2} + 6 \, b^{2} d f g n x + b^{2} d f^{2} n\right )} e^{2} + 2 \, {\left (b^{2} d^{2} g^{2} n x + b^{2} d^{2} f g n\right )} e\right )} \log \left (c\right )\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 9 \, {\left (b^{2} g^{2} n^{2} \log \left (x e + d\right )^{2} + b^{2} g^{2} \log \left (c\right )^{2} + 2 \, a b g^{2} \log \left (c\right ) + a^{2} g^{2} + 2 \, {\left (b^{2} g^{2} n \log \left (c\right ) + a b g^{2} n\right )} \log \left (x e + d\right )\right )} \operatorname {log\_integral}\left ({\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} n^{5} e^{3} \log \left (x e + d\right )^{2} + b^{5} n^{3} e^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} e^{3} + 2 \, {\left (b^{5} n^{4} e^{3} \log \left (c\right ) + a b^{4} n^{4} e^{3}\right )} \log \left (x e + d\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(8*(a^2*d*g^2 - a^2*f*g*e + (b^2*d*g^2*n^2 - b^2*f*g*n^2*e)*log(x*e + d)^2 + (b^2*d*g^2 - b^2*f*g*e)*log(
c)^2 + 2*(a*b*d*g^2*n - a*b*f*g*n*e + (b^2*d*g^2*n - b^2*f*g*n*e)*log(c))*log(x*e + d) + 2*(a*b*d*g^2 - a*b*f*
g*e)*log(c))*e^((b*log(c) + a)/(b*n))*log_integral((x^2*e^2 + 2*d*x*e + d^2)*e^(2*(b*log(c) + a)/(b*n))) - (a^
2*d^2*g^2 - 2*a^2*d*f*g*e + a^2*f^2*e^2 + (b^2*d^2*g^2*n^2 - 2*b^2*d*f*g*n^2*e + b^2*f^2*n^2*e^2)*log(x*e + d)
^2 + (b^2*d^2*g^2 - 2*b^2*d*f*g*e + b^2*f^2*e^2)*log(c)^2 + 2*(a*b*d^2*g^2*n - 2*a*b*d*f*g*n*e + a*b*f^2*n*e^2
+ (b^2*d^2*g^2*n - 2*b^2*d*f*g*n*e + b^2*f^2*n*e^2)*log(c))*log(x*e + d) + 2*(a*b*d^2*g^2 - 2*a*b*d*f*g*e + a
*b*f^2*e^2)*log(c))*e^(2*(b*log(c) + a)/(b*n))*log_integral((x*e + d)*e^((b*log(c) + a)/(b*n))) + (((b^2*g^2*n
^2 + 3*a*b*g^2*n)*x^3 + 2*(b^2*f*g*n^2 + 2*a*b*f*g*n)*x^2 + (b^2*f^2*n^2 + a*b*f^2*n)*x)*e^3 + (b^2*d*f^2*n^2
+ a*b*d*f^2*n + (b^2*d*g^2*n^2 + 5*a*b*d*g^2*n)*x^2 + 2*(b^2*d*f*g*n^2 + 3*a*b*d*f*g*n)*x)*e^2 + 2*(a*b*d^2*g^
2*n*x + a*b*d^2*f*g*n)*e + ((3*b^2*g^2*n^2*x^3 + 4*b^2*f*g*n^2*x^2 + b^2*f^2*n^2*x)*e^3 + (5*b^2*d*g^2*n^2*x^2
+ 6*b^2*d*f*g*n^2*x + b^2*d*f^2*n^2)*e^2 + 2*(b^2*d^2*g^2*n^2*x + b^2*d^2*f*g*n^2)*e)*log(x*e + d) + ((3*b^2*
g^2*n*x^3 + 4*b^2*f*g*n*x^2 + b^2*f^2*n*x)*e^3 + (5*b^2*d*g^2*n*x^2 + 6*b^2*d*f*g*n*x + b^2*d*f^2*n)*e^2 + 2*(
b^2*d^2*g^2*n*x + b^2*d^2*f*g*n)*e)*log(c))*e^(3*(b*log(c) + a)/(b*n)) - 9*(b^2*g^2*n^2*log(x*e + d)^2 + b^2*g
^2*log(c)^2 + 2*a*b*g^2*log(c) + a^2*g^2 + 2*(b^2*g^2*n*log(c) + a*b*g^2*n)*log(x*e + d))*log_integral((x^3*e^
3 + 3*d*x^2*e^2 + 3*d^2*x*e + d^3)*e^(3*(b*log(c) + a)/(b*n))))*e^(-3*(b*log(c) + a)/(b*n))/(b^5*n^5*e^3*log(x
*e + d)^2 + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3 + 2*(b^5*n^4*e^3*log(c) + a*b^4*n^
4*e^3)*log(x*e + d))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f + g*x)**2/(a + b*log(c*(d + e*x)**n))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8396 vs. \(2 (357) = 714\).
time = 5.70, size = 8396, normalized size = 23.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x*e + d)^3*b^2*g^2*n^2*e^3*log(x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c)
+ 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + 2*(x*e + d
)^2*b^2*d*g^2*n^2*e^3*log(x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n
^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 1/2*(x*e + d)*b^2*d^2
*g^2*n^2*e^3*log(x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*lo
g(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + 1/2*b^2*d^2*g^2*n^2*Ei(log(c)/
n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 3)*log(x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x
*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^
6)*c^(1/n)) - 1/2*(x*e + d)^3*b^2*g^2*n^2*e^3/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c)
+ 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + (x*e + d)^
2*b^2*d*g^2*n^2*e^3/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e
+ d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 1/2*(x*e + d)*b^2*d^2*g^2*n^2*e^3/(b
^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log
(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 2*(x*e + d)^2*b^2*f*g*n^2*e^4*log(x*e + d)/(b^5*n^5*e^6*lo
g(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b
^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + (x*e + d)*b^2*d*f*g*n^2*e^4*log(x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 +
2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(
c) + a^2*b^3*n^3*e^6) - b^2*d*f*g*n^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 4)*log(x*e + d)^2/((
b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*lo
g(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) - 4*b^2*d*g^2*n^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*lo
g(x*e + d))*e^(-2*a/(b*n) + 3)*log(x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c)
+ 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(2/n)) -
3/2*(x*e + d)^3*b^2*g^2*n*e^3*log(c)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4
*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + 2*(x*e + d)^2*b^2*d
*g^2*n*e^3*log(c)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e +
d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 1/2*(x*e + d)*b^2*d^2*g^2*n*e^3*log(c)
/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*
log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + b^2*d^2*g^2*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(
-a/(b*n) + 3)*log(x*e + d)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n
^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) - (x*e + d)^2*
b^2*f*g*n^2*e^4/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d)
+ b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + (x*e + d)*b^2*d*f*g*n^2*e^4/(b^5*n^5*e^6
*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*
a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 3/2*(x*e + d)^3*a*b*g^2*n*e^3/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^
4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2
*b^3*n^3*e^6) + 2*(x*e + d)^2*a*b*d*g^2*n*e^3/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c)
+ 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 1/2*(x*e +
d)*a*b*d^2*g^2*n*e^3/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*
e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) - 1/2*(x*e + d)*b^2*f^2*n^2*e^5*log(
x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*
n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + a*b*d^2*g^2*n*Ei(log(c)/n + a/(b*n) + log(x*e +
d))*e^(-a/(b*n) + 3)*log(x*e + d)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*
n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) + 1/2*b^2*f^2
*n^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 5)*log(x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^
5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) +
a^2*b^3*n^3*e^6)*c^(1/n)) + 4*b^2*f*g*n^2*Ei(2...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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