Integrand size = 21, antiderivative size = 53 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3}{3 d}-\frac {b F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log (F)}{3 d} \] Output:
1/3*F^(a+b/(d*x+c)^3)*(d*x+c)^3/d-1/3*b*F^a*Ei(b*ln(F)/(d*x+c)^3)*ln(F)/d
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\frac {F^a \left (F^{\frac {b}{(c+d x)^3}} (c+d x)^3-b \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log (F)\right )}{3 d} \] Input:
Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^2,x]
Output:
(F^a*(F^(b/(c + d*x)^3)*(c + d*x)^3 - b*ExpIntegralEi[(b*Log[F])/(c + d*x) ^3]*Log[F]))/(3*d)
Time = 0.42 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2643, 2639}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c+d x)^2 F^{a+\frac {b}{(c+d x)^3}} \, dx\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle b \log (F) \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x}dx+\frac {(c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{3 d}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {(c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{3 d}-\frac {b F^a \log (F) \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d}\) |
Input:
Int[F^(a + b/(c + d*x)^3)*(c + d*x)^2,x]
Output:
(F^(a + b/(c + d*x)^3)*(c + d*x)^3)/(3*d) - (b*F^a*ExpIntegralEi[(b*Log[F] )/(c + d*x)^3]*Log[F])/(3*d)
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_ Symbol] :> Simp[F^a*(ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)), x] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ .), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) , x] - Simp[b*n*(Log[F]/(m + 1)) Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x) ^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[ -4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))
\[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}} \left (d x +c \right )^{2}d x\]
Input:
int(F^(a+b/(d*x+c)^3)*(d*x+c)^2,x)
Output:
int(F^(a+b/(d*x+c)^3)*(d*x+c)^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (49) = 98\).
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.66 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=-\frac {F^{a} b {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right ) - {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, d} \] Input:
integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^2,x, algorithm="fricas")
Output:
-1/3*(F^a*b*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F) - (d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*F^((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{2}\, dx \] Input:
integrate(F**(a+b/(d*x+c)**3)*(d*x+c)**2,x)
Output:
Integral(F**(a + b/(c + d*x)**3)*(c + d*x)**2, x)
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\int { {\left (d x + c\right )}^{2} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}} \,d x } \] Input:
integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^2,x, algorithm="maxima")
Output:
1/3*(F^a*d^2*x^3 + 3*F^a*c*d*x^2 + 3*F^a*c^2*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^ 2 + 3*c^2*d*x + c^3)) + integrate((F^a*b*d^3*x^3*log(F) + 3*F^a*b*c*d^2*x^ 2*log(F) + 3*F^a*b*c^2*d*x*log(F))*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), x)
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\int { {\left (d x + c\right )}^{2} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}} \,d x } \] Input:
integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^2,x, algorithm="giac")
Output:
integrate((d*x + c)^2*F^(a + b/(d*x + c)^3), x)
Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,{\left (c+d\,x\right )}^3}{3\,d}+\frac {F^a\,b\,\ln \left (F\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}{3\,d} \] Input:
int(F^(a + b/(c + d*x)^3)*(c + d*x)^2,x)
Output:
(F^a*F^(b/(c + d*x)^3)*(c + d*x)^3)/(3*d) + (F^a*b*log(F)*expint(-(b*log(F ))/(c + d*x)^3))/(3*d)
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx=\text {too large to display} \] Input:
int(F^(a+b/(d*x+c)^3)*(d*x+c)^2,x)
Output:
(270*f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*log(f)**3*b**3*c**2*d*x + 270* f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3* c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*log(f)**3*b**3*c*d**2*x**2 + 90*f** ((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c** 2*d*x + 3*c*d**2*x**2 + d**3*x**3))*log(f)**2*b**2*c**6 + 378*f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3 *c*d**2*x**2 + d**3*x**3))*log(f)**2*b**2*c**5*d*x - 675*f**((a*c**3 + 3*a *c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d* *2*x**2 + d**3*x**3))*log(f)**2*b**2*c**4*d**2*x**2 - 1800*f**((a*c**3 + 3 *a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c* d**2*x**2 + d**3*x**3))*log(f)**2*b**2*c**3*d**3*x**3 - 1350*f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3* c*d**2*x**2 + d**3*x**3))*log(f)**2*b**2*c**2*d**4*x**4 - 360*f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3 *c*d**2*x**2 + d**3*x**3))*log(f)**2*b**2*c*d**5*x**5 - 54*f**((a*c**3 + 3 *a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c* d**2*x**2 + d**3*x**3))*log(f)*b*c**9 - 1242*f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d* *3*x**3))*log(f)*b*c**8*d*x - 4302*f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d...