\(\int (c+d x)^3 \Gamma (-3,a+b x) \, dx\) [153]

Optimal result

Integrand size = 15, antiderivative size = 139 \[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=-\frac {d^3 e^{-a-b x}}{4 b^4}-\frac {(b c-a d)^4 \Gamma (-3,a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \Gamma (-3,a+b x)}{4 d}-\frac {(b c-a d)^3 \Gamma (-2,a+b x)}{b^4}-\frac {3 d (b c-a d)^2 \Gamma (-1,a+b x)}{2 b^4}-\frac {d^2 (b c-a d) \Gamma (0,a+b x)}{b^4} \] Output:

-1/4*d^3*exp(-b*x-a)/b^4-1/4*(-a*d+b*c)^4/(b*x+a)^3*Ei(4,b*x+a)/b^4/d+1/4* 
(d*x+c)^4/(b*x+a)^3*Ei(4,b*x+a)/d-(-a*d+b*c)^3/(b*x+a)^2*Ei(3,b*x+a)/b^4-3 
/2*d*(-a*d+b*c)^2/(b*x+a)*Ei(2,b*x+a)/b^4-d^2*(-a*d+b*c)*Ei(1,b*x+a)/b^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(482\) vs. \(2(139)=278\).

Time = 0.32 (sec) , antiderivative size = 482, normalized size of antiderivative = 3.47 \[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=\frac {1}{24} \left (\frac {e^{-a-b x} \left (-6 d^3-\frac {2 a \left (-4 b^3 c^3+6 a b^2 c^2 d-4 a^2 b c d^2+a^3 d^3\right )}{(a+b x)^3}+\frac {-4 (3+a) b^3 c^3+6 a (6+a) b^2 c^2 d-4 a^2 (9+a) b c d^2+a^3 (12+a) d^3}{(a+b x)^2}+\frac {4 (3+a) b^3 c^3-6 \left (6+6 a+a^2\right ) b^2 c^2 d+4 a \left (18+9 a+a^2\right ) b c d^2-a^2 (6+a)^2 d^3}{a+b x}\right )}{b^4}+\frac {12 c^3 \operatorname {ExpIntegralEi}(-a-b x)}{b}+\frac {4 a c^3 \operatorname {ExpIntegralEi}(-a-b x)}{b}-\frac {36 c^2 d \operatorname {ExpIntegralEi}(-a-b x)}{b^2}-\frac {36 a c^2 d \operatorname {ExpIntegralEi}(-a-b x)}{b^2}-\frac {6 a^2 c^2 d \operatorname {ExpIntegralEi}(-a-b x)}{b^2}+\frac {24 c d^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^3}+\frac {72 a c d^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^3}+\frac {36 a^2 c d^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^3}+\frac {4 a^3 c d^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^3}-\frac {24 a d^3 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}-\frac {36 a^2 d^3 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}-\frac {12 a^3 d^3 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}-\frac {a^4 d^3 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}+6 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \Gamma (-3,a+b x)\right ) \] Input:

Integrate[(c + d*x)^3*Gamma[-3, a + b*x],x]
 

Output:

((E^(-a - b*x)*(-6*d^3 - (2*a*(-4*b^3*c^3 + 6*a*b^2*c^2*d - 4*a^2*b*c*d^2 
+ a^3*d^3))/(a + b*x)^3 + (-4*(3 + a)*b^3*c^3 + 6*a*(6 + a)*b^2*c^2*d - 4* 
a^2*(9 + a)*b*c*d^2 + a^3*(12 + a)*d^3)/(a + b*x)^2 + (4*(3 + a)*b^3*c^3 - 
 6*(6 + 6*a + a^2)*b^2*c^2*d + 4*a*(18 + 9*a + a^2)*b*c*d^2 - a^2*(6 + a)^ 
2*d^3)/(a + b*x)))/b^4 + (12*c^3*ExpIntegralEi[-a - b*x])/b + (4*a*c^3*Exp 
IntegralEi[-a - b*x])/b - (36*c^2*d*ExpIntegralEi[-a - b*x])/b^2 - (36*a*c 
^2*d*ExpIntegralEi[-a - b*x])/b^2 - (6*a^2*c^2*d*ExpIntegralEi[-a - b*x])/ 
b^2 + (24*c*d^2*ExpIntegralEi[-a - b*x])/b^3 + (72*a*c*d^2*ExpIntegralEi[- 
a - b*x])/b^3 + (36*a^2*c*d^2*ExpIntegralEi[-a - b*x])/b^3 + (4*a^3*c*d^2* 
ExpIntegralEi[-a - b*x])/b^3 - (24*a*d^3*ExpIntegralEi[-a - b*x])/b^4 - (3 
6*a^2*d^3*ExpIntegralEi[-a - b*x])/b^4 - (12*a^3*d^3*ExpIntegralEi[-a - b* 
x])/b^4 - (a^4*d^3*ExpIntegralEi[-a - b*x])/b^4 + 6*x*(4*c^3 + 6*c^2*d*x + 
 4*c*d^2*x^2 + d^3*x^3)*Gamma[-3, a + b*x])/24
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(355\) vs. \(2(139)=278\).

Time = 0.86 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7119, 2629, 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.

\(\displaystyle \int (c+d x)^3 \Gamma (-3,a+b x) \, dx\)

\(\Big \downarrow \) 7119

\(\displaystyle \frac {b \int \frac {e^{-a-b x} (c+d x)^4}{(a+b x)^4}dx}{4 d}+\frac {(c+d x)^4 \Gamma (-3,a+b x)}{4 d}\)

\(\Big \downarrow \) 2629

\(\displaystyle \frac {b \int \left (\frac {e^{-a-b x} d^4}{b^4}+\frac {4 (b c-a d) e^{-a-b x} d^3}{b^4 (a+b x)}+\frac {6 (b c-a d)^2 e^{-a-b x} d^2}{b^4 (a+b x)^2}+\frac {4 (b c-a d)^3 e^{-a-b x} d}{b^4 (a+b x)^3}+\frac {(b c-a d)^4 e^{-a-b x}}{b^4 (a+b x)^4}\right )dx}{4 d}+\frac {(c+d x)^4 \Gamma (-3,a+b x)}{4 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {4 d^3 (b c-a d) \operatorname {ExpIntegralEi}(-a-b x)}{b^5}-\frac {6 d^2 (b c-a d)^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^5}-\frac {6 d^2 e^{-a-b x} (b c-a d)^2}{b^5 (a+b x)}+\frac {2 d (b c-a d)^3 \operatorname {ExpIntegralEi}(-a-b x)}{b^5}-\frac {(b c-a d)^4 \operatorname {ExpIntegralEi}(-a-b x)}{6 b^5}+\frac {2 d e^{-a-b x} (b c-a d)^3}{b^5 (a+b x)}-\frac {2 d e^{-a-b x} (b c-a d)^3}{b^5 (a+b x)^2}-\frac {e^{-a-b x} (b c-a d)^4}{6 b^5 (a+b x)}+\frac {e^{-a-b x} (b c-a d)^4}{6 b^5 (a+b x)^2}-\frac {e^{-a-b x} (b c-a d)^4}{3 b^5 (a+b x)^3}-\frac {d^4 e^{-a-b x}}{b^5}\right )}{4 d}+\frac {(c+d x)^4 \Gamma (-3,a+b x)}{4 d}\)

Input:

Int[(c + d*x)^3*Gamma[-3, a + b*x],x]
 

Output:

(b*(-((d^4*E^(-a - b*x))/b^5) - ((b*c - a*d)^4*E^(-a - b*x))/(3*b^5*(a + b 
*x)^3) - (2*d*(b*c - a*d)^3*E^(-a - b*x))/(b^5*(a + b*x)^2) + ((b*c - a*d) 
^4*E^(-a - b*x))/(6*b^5*(a + b*x)^2) - (6*d^2*(b*c - a*d)^2*E^(-a - b*x))/ 
(b^5*(a + b*x)) + (2*d*(b*c - a*d)^3*E^(-a - b*x))/(b^5*(a + b*x)) - ((b*c 
 - a*d)^4*E^(-a - b*x))/(6*b^5*(a + b*x)) + (4*d^3*(b*c - a*d)*ExpIntegral 
Ei[-a - b*x])/b^5 - (6*d^2*(b*c - a*d)^2*ExpIntegralEi[-a - b*x])/b^5 + (2 
*d*(b*c - a*d)^3*ExpIntegralEi[-a - b*x])/b^5 - ((b*c - a*d)^4*ExpIntegral 
Ei[-a - b*x])/(6*b^5)))/(4*d) + ((c + d*x)^4*Gamma[-3, a + b*x])/(4*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2629

Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte 
grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ 
Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

rule 7119

Int[Gamma[n_, (a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> 
Block[{$UseGamma = True}, Simp[(c + d*x)^(m + 1)*(Gamma[n, a + b*x]/(d*(m + 
 1))), x] + Simp[b/(d*(m + 1))   Int[(c + d*x)^(m + 1)*((a + b*x)^(n - 1)/E 
^(a + b*x)), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ 
[n, 0] || IntegersQ[m, n]) && NeQ[m, -1]
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{3} \operatorname {expIntegral}_{4}\left (b x +a \right )}{\left (b x +a \right )^{3}}d x\]

Input:

int((d*x+c)^3/(b*x+a)^3*Ei(4,b*x+a),x)
 

Output:

int((d*x+c)^3/(b*x+a)^3*Ei(4,b*x+a),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (130) = 260\).

Time = 0.09 (sec) , antiderivative size = 662, normalized size of antiderivative = 4.76 \[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=-\frac {{\left (b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} - 6 \, {\left (a + 2\right )} b^{2} c^{2} d + 4 \, {\left (a^{2} + 5 \, a + 2\right )} b c d^{2} - {\left (a^{3} + 8 \, a^{2} + 8 \, a\right )} d^{3} + {\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (6 \, b^{3} c^{2} d - 4 \, {\left (a + 1\right )} b^{2} c d^{2} + {\left (a^{2} + 4 \, a\right )} b d^{3}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{7} d^{3} x^{7} + 4 \, {\left (a^{4} + 3 \, a^{3}\right )} b^{3} c^{3} + {\left (4 \, b^{7} c d^{2} + 3 \, a b^{6} d^{3}\right )} x^{6} - 6 \, {\left (a^{5} + 6 \, a^{4} + 6 \, a^{3}\right )} b^{2} c^{2} d + 3 \, {\left (2 \, b^{7} c^{2} d + 4 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{5} + 4 \, {\left (a^{6} + 9 \, a^{5} + 18 \, a^{4} + 6 \, a^{3}\right )} b c d^{2} + {\left (4 \, b^{7} c^{3} + 18 \, a b^{6} c^{2} d + 12 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x^{4} - {\left (a^{7} + 12 \, a^{6} + 36 \, a^{5} + 24 \, a^{4}\right )} d^{3} + {\left (4 \, {\left (4 \, a + 3\right )} b^{6} c^{3} + 12 \, {\left (a^{2} - 3 \, a - 3\right )} b^{5} c^{2} d + 4 \, {\left (2 \, a^{3} + 9 \, a^{2} + 18 \, a + 6\right )} b^{4} c d^{2} - {\left (a^{4} + 12 \, a^{3} + 36 \, a^{2} + 24 \, a\right )} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (4 \, {\left (2 \, a^{2} + 3 \, a\right )} b^{5} c^{3} - 4 \, {\left (a^{3} + 9 \, a^{2} + 9 \, a\right )} b^{4} c^{2} d + 4 \, {\left (a^{4} + 9 \, a^{3} + 18 \, a^{2} + 6 \, a\right )} b^{3} c d^{2} - {\left (a^{5} + 12 \, a^{4} + 36 \, a^{3} + 24 \, a^{2}\right )} b^{2} d^{3}\right )} x^{2} + {\left (4 \, {\left (4 \, a^{3} + 9 \, a^{2}\right )} b^{4} c^{3} - 18 \, {\left (a^{4} + 6 \, a^{3} + 6 \, a^{2}\right )} b^{3} c^{2} d + 12 \, {\left (a^{5} + 9 \, a^{4} + 18 \, a^{3} + 6 \, a^{2}\right )} b^{2} c d^{2} - 3 \, {\left (a^{6} + 12 \, a^{5} + 36 \, a^{4} + 24 \, a^{3}\right )} b d^{3}\right )} x\right )} \Gamma \left (-3, b x + a\right )}{4 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \] Input:

integrate((d*x+c)^3*gamma(-3,b*x+a),x, algorithm="fricas")
 

Output:

-1/4*((b^3*d^3*x^3 + 4*b^3*c^3 - 6*(a + 2)*b^2*c^2*d + 4*(a^2 + 5*a + 2)*b 
*c*d^2 - (a^3 + 8*a^2 + 8*a)*d^3 + (4*b^3*c*d^2 - a*b^2*d^3)*x^2 + (6*b^3* 
c^2*d - 4*(a + 1)*b^2*c*d^2 + (a^2 + 4*a)*b*d^3)*x)*e^(-b*x - a) - (b^7*d^ 
3*x^7 + 4*(a^4 + 3*a^3)*b^3*c^3 + (4*b^7*c*d^2 + 3*a*b^6*d^3)*x^6 - 6*(a^5 
 + 6*a^4 + 6*a^3)*b^2*c^2*d + 3*(2*b^7*c^2*d + 4*a*b^6*c*d^2 + a^2*b^5*d^3 
)*x^5 + 4*(a^6 + 9*a^5 + 18*a^4 + 6*a^3)*b*c*d^2 + (4*b^7*c^3 + 18*a*b^6*c 
^2*d + 12*a^2*b^5*c*d^2 + a^3*b^4*d^3)*x^4 - (a^7 + 12*a^6 + 36*a^5 + 24*a 
^4)*d^3 + (4*(4*a + 3)*b^6*c^3 + 12*(a^2 - 3*a - 3)*b^5*c^2*d + 4*(2*a^3 + 
 9*a^2 + 18*a + 6)*b^4*c*d^2 - (a^4 + 12*a^3 + 36*a^2 + 24*a)*b^3*d^3)*x^3 
 + 3*(4*(2*a^2 + 3*a)*b^5*c^3 - 4*(a^3 + 9*a^2 + 9*a)*b^4*c^2*d + 4*(a^4 + 
 9*a^3 + 18*a^2 + 6*a)*b^3*c*d^2 - (a^5 + 12*a^4 + 36*a^3 + 24*a^2)*b^2*d^ 
3)*x^2 + (4*(4*a^3 + 9*a^2)*b^4*c^3 - 18*(a^4 + 6*a^3 + 6*a^2)*b^3*c^2*d + 
 12*(a^5 + 9*a^4 + 18*a^3 + 6*a^2)*b^2*c*d^2 - 3*(a^6 + 12*a^5 + 36*a^4 + 
24*a^3)*b*d^3)*x)*gamma(-3, b*x + a))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x 
 + a^3*b^4)
 

Sympy [F]

\[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=\int \frac {\left (c + d x\right )^{3} \operatorname {E}_{4}\left (a + b x\right )}{\left (a + b x\right )^{3}}\, dx \] Input:

integrate((d*x+c)**3*uppergamma(-3,b*x+a),x)
 

Output:

Integral((c + d*x)**3*expint(4, a + b*x)/(a + b*x)**3, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \Gamma \left (-3, b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^3*gamma(-3,b*x+a),x, algorithm="maxima")
 

Output:

((b*x + a)*gamma(-3, b*x + a) - gamma(-2, b*x + a))*c^3/b + integrate(d^3* 
x^3*gamma(-3, b*x + a) + 3*c*d^2*x^2*gamma(-3, b*x + a) + 3*c^2*d*x*gamma( 
-3, b*x + a), x)
 

Giac [F]

\[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \Gamma \left (-3, b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^3*gamma(-3,b*x+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^3*gamma(-3, b*x + a), x)
 

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.17 \[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=\int \frac {\mathrm {expint}\left (4,a+b\,x\right )\,{\left (c+d\,x\right )}^3}{{\left (a+b\,x\right )}^3} \,d x \] Input:

int((expint(4, a + b*x)*(c + d*x)^3)/(a + b*x)^3,x)
 

Output:

int((expint(4, a + b*x)*(c + d*x)^3)/(a + b*x)^3, x)
 

Reduce [F]

\[ \int (c+d x)^3 \Gamma (-3,a+b x) \, dx=\left (\int \frac {\mathit {ei} \left (4, b x +a \right )}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) c^{3}+\left (\int \frac {\mathit {ei} \left (4, b x +a \right ) x^{3}}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) d^{3}+3 \left (\int \frac {\mathit {ei} \left (4, b x +a \right ) x^{2}}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) c \,d^{2}+3 \left (\int \frac {\mathit {ei} \left (4, b x +a \right ) x}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) c^{2} d \] Input:

int((d*x+c)^3/(b*x+a)^3*Ei(4,b*x+a),x)
 

Output:

int(ei(4,a + b*x)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*c**3 
+ int((ei(4,a + b*x)*x**3)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3) 
,x)*d**3 + 3*int((ei(4,a + b*x)*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + 
 b**3*x**3),x)*c*d**2 + 3*int((ei(4,a + b*x)*x)/(a**3 + 3*a**2*b*x + 3*a*b 
**2*x**2 + b**3*x**3),x)*c**2*d