Integrand size = 18, antiderivative size = 99 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\frac {6 e F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {6 e F^{c (a+b x)} x}{b^3 c^3 \log ^3(F)}+\frac {3 e F^{c (a+b x)} x^2}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} \left (d-e x^3\right )}{b c \log (F)} \] Output:
6*e*F^(c*(b*x+a))/b^4/c^4/ln(F)^4-6*e*F^(c*(b*x+a))*x/b^3/c^3/ln(F)^3+3*e* F^(c*(b*x+a))*x^2/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(-e*x^3+d)/b/c/ln(F)
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.68 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\frac {F^{c (a+b x)} \left (6 e-6 b c e x \log (F)+3 b^2 c^2 e x^2 \log ^2(F)+b^3 c^3 \left (d-e x^3\right ) \log ^3(F)\right )}{b^4 c^4 \log ^4(F)} \] Input:
Integrate[F^(c*(a + b*x))*(d - e*x^3),x]
Output:
(F^(c*(a + b*x))*(6*e - 6*b*c*e*x*Log[F] + 3*b^2*c^2*e*x^2*Log[F]^2 + b^3* c^3*(d - e*x^3)*Log[F]^3))/(b^4*c^4*Log[F]^4)
Time = 0.48 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2626, 2009}
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.
\(\displaystyle \int \left (d-e x^3\right ) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 2626 |
\(\displaystyle \int \left (d F^{c (a+b x)}-e x^3 F^{c (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6 e F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {6 e x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {3 e x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}-\frac {e x^3 F^{c (a+b x)}}{b c \log (F)}\) |
Input:
Int[F^(c*(a + b*x))*(d - e*x^3),x]
Output:
(6*e*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4) - (6*e*F^(c*(a + b*x))*x)/(b^3*c^ 3*Log[F]^3) + (3*e*F^(c*(a + b*x))*x^2)/(b^2*c^2*Log[F]^2) + (d*F^(c*(a + b*x)))/(b*c*Log[F]) - (e*F^(c*(a + b*x))*x^3)/(b*c*Log[F])
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(\frac {\left (-e \,x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+\ln \left (F \right )^{3} b^{3} c^{3} d +3 e \,x^{2} \ln \left (F \right )^{2} b^{2} c^{2}-6 e x \ln \left (F \right ) b c +6 e \right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(77\) |
risch | \(\frac {\left (-e \,x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+\ln \left (F \right )^{3} b^{3} c^{3} d +3 e \,x^{2} \ln \left (F \right )^{2} b^{2} c^{2}-6 e x \ln \left (F \right ) b c +6 e \right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(77\) |
orering | \(\frac {\left (-e \,x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+\ln \left (F \right )^{3} b^{3} c^{3} d +3 e \,x^{2} \ln \left (F \right )^{2} b^{2} c^{2}-6 e x \ln \left (F \right ) b c +6 e \right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(77\) |
meijerg | \(-\frac {F^{a c} e \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{\ln \left (F \right )^{4} b^{4} c^{4}}-\frac {F^{a c} d \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{b c \ln \left (F \right )}\) | \(99\) |
parallelrisch | \(\frac {-x^{3} F^{c \left (b x +a \right )} e \ln \left (F \right )^{3} b^{3} c^{3}+\ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d +3 x^{2} F^{c \left (b x +a \right )} e \ln \left (F \right )^{2} b^{2} c^{2}-6 e \,F^{c \left (b x +a \right )} x \ln \left (F \right ) b c +6 F^{c \left (b x +a \right )} e}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(113\) |
norman | \(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} d +6 e \right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}-\frac {6 e x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} c^{3}}+\frac {3 e \,x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}-\frac {e \,x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}\) | \(115\) |
Input:
int(F^(c*(b*x+a))*(-e*x^3+d),x,method=_RETURNVERBOSE)
Output:
(-e*x^3*ln(F)^3*b^3*c^3+ln(F)^3*b^3*c^3*d+3*e*x^2*ln(F)^2*b^2*c^2-6*e*x*ln (F)*b*c+6*e)*F^(c*(b*x+a))/ln(F)^4/b^4/c^4
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.77 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\frac {{\left (3 \, b^{2} c^{2} e x^{2} \log \left (F\right )^{2} - 6 \, b c e x \log \left (F\right ) - {\left (b^{3} c^{3} e x^{3} - b^{3} c^{3} d\right )} \log \left (F\right )^{3} + 6 \, e\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4}} \] Input:
integrate(F^((b*x+a)*c)*(-e*x^3+d),x, algorithm="fricas")
Output:
(3*b^2*c^2*e*x^2*log(F)^2 - 6*b*c*e*x*log(F) - (b^3*c^3*e*x^3 - b^3*c^3*d) *log(F)^3 + 6*e)*F^(b*c*x + a*c)/(b^4*c^4*log(F)^4)
Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{3} c^{3} d \log {\left (F \right )}^{3} - b^{3} c^{3} e x^{3} \log {\left (F \right )}^{3} + 3 b^{2} c^{2} e x^{2} \log {\left (F \right )}^{2} - 6 b c e x \log {\left (F \right )} + 6 e\right )}{b^{4} c^{4} \log {\left (F \right )}^{4}} & \text {for}\: b^{4} c^{4} \log {\left (F \right )}^{4} \neq 0 \\d x - \frac {e x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(F**((b*x+a)*c)*(-e*x**3+d),x)
Output:
Piecewise((F**(c*(a + b*x))*(b**3*c**3*d*log(F)**3 - b**3*c**3*e*x**3*log( F)**3 + 3*b**2*c**2*e*x**2*log(F)**2 - 6*b*c*e*x*log(F) + 6*e)/(b**4*c**4* log(F)**4), Ne(b**4*c**4*log(F)**4, 0)), (d*x - e*x**4/4, True))
Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\frac {F^{b c x + a c} d}{b c \log \left (F\right )} - \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} e}{b^{4} c^{4} \log \left (F\right )^{4}} \] Input:
integrate(F^((b*x+a)*c)*(-e*x^3+d),x, algorithm="maxima")
Output:
F^(b*c*x + a*c)*d/(b*c*log(F)) - (F^(a*c)*b^3*c^3*x^3*log(F)^3 - 3*F^(a*c) *b^2*c^2*x^2*log(F)^2 + 6*F^(a*c)*b*c*x*log(F) - 6*F^(a*c))*F^(b*c*x)*e/(b ^4*c^4*log(F)^4)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 2601, normalized size of antiderivative = 26.27 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\text {Too large to display} \] Input:
integrate(F^((b*x+a)*c)*(-e*x^3+d),x, algorithm="giac")
Output:
(((3*pi^2*b^3*c^3*e*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*e*x^3*log(abs( F)) + 2*b^3*c^3*e*x^3*log(abs(F))^3 - 3*pi^2*b^3*c^3*d*log(abs(F))*sgn(F) + 3*pi^2*b^3*c^3*d*log(abs(F)) - 2*b^3*c^3*d*log(abs(F))^3 - 3*pi^2*b^2*c^ 2*e*x^2*sgn(F) + 3*pi^2*b^2*c^2*e*x^2 - 6*b^2*c^2*e*x^2*log(abs(F))^2 + 12 *b*c*e*x*log(abs(F)) - 12*e)*(pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs (F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*lo g(abs(F))^4)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2) - 4*(pi^3*b^3*c^3* e*x^3*sgn(F) - 3*pi*b^3*c^3*e*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*e*x^ 3 + 3*pi*b^3*c^3*e*x^3*log(abs(F))^2 - pi^3*b^3*c^3*d*sgn(F) + 3*pi*b^3*c^ 3*d*log(abs(F))^2*sgn(F) + pi^3*b^3*c^3*d - 3*pi*b^3*c^3*d*log(abs(F))^2 + 6*pi*b^2*c^2*e*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*e*x^2*log(abs(F)) - 6*pi*b*c*e*x*sgn(F) + 6*pi*b*c*e*x)*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi* b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(a bs(F))^3)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi ^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 1 6*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^ 3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2))*cos(-1/2*pi*b*c*x...
Time = 22.86 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (-e\,b^3\,c^3\,x^3\,{\ln \left (F\right )}^3+d\,b^3\,c^3\,{\ln \left (F\right )}^3+3\,e\,b^2\,c^2\,x^2\,{\ln \left (F\right )}^2-6\,e\,b\,c\,x\,\ln \left (F\right )+6\,e\right )}{b^4\,c^4\,{\ln \left (F\right )}^4} \] Input:
int(F^(c*(a + b*x))*(d - e*x^3),x)
Output:
(F^(a*c + b*c*x)*(6*e + b^3*c^3*d*log(F)^3 + 3*b^2*c^2*e*x^2*log(F)^2 - b^ 3*c^3*e*x^3*log(F)^3 - 6*b*c*e*x*log(F)))/(b^4*c^4*log(F)^4)
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \left (d-e x^3\right ) \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{3} b^{3} c^{3} d -\mathrm {log}\left (f \right )^{3} b^{3} c^{3} e \,x^{3}+3 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e \,x^{2}-6 \,\mathrm {log}\left (f \right ) b c e x +6 e \right )}{\mathrm {log}\left (f \right )^{4} b^{4} c^{4}} \] Input:
int(F^((b*x+a)*c)*(-e*x^3+d),x)
Output:
(f**(a*c + b*c*x)*(log(f)**3*b**3*c**3*d - log(f)**3*b**3*c**3*e*x**3 + 3* log(f)**2*b**2*c**2*e*x**2 - 6*log(f)*b*c*e*x + 6*e))/(log(f)**4*b**4*c**4 )