\(\int \frac {e^{a+b x+c x^2} (b+2 c x)}{(a+b x+c x^2)^{9/2}} \, dx\) [558]

Optimal result

Integrand size = 33, antiderivative size = 145 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac {4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac {8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 e^{a+b x+c x^2}}{105 \sqrt {a+b x+c x^2}}+\frac {16}{105} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \] Output:

-2/7*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(7/2)-4/35*exp(c*x^2+b*x+a)/(c*x^2+b*x 
+a)^(5/2)-8/105*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(3/2)-16/105*exp(c*x^2+b*x+ 
a)/(c*x^2+b*x+a)^(1/2)+16/105*Pi^(1/2)*erfi((c*x^2+b*x+a)^(1/2))
 

Mathematica [A] (verified)

Time = 7.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.71 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \left (e^{a+x (b+c x)} \left (15+6 (a+x (b+c x))+4 (a+x (b+c x))^2+8 (a+x (b+c x))^3\right )+8 (-a-x (b+c x))^{7/2} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )\right )}{105 (a+x (b+c x))^{7/2}} \] Input:

Integrate[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(9/2),x]
 

Output:

(-2*(E^(a + x*(b + c*x))*(15 + 6*(a + x*(b + c*x)) + 4*(a + x*(b + c*x))^2 
 + 8*(a + x*(b + c*x))^3) + 8*(-a - x*(b + c*x))^(7/2)*Gamma[1/2, -a - x*( 
b + c*x)]))/(105*(a + x*(b + c*x))^(7/2))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {7258, 2608, 2608, 2608, 2608, 2611, 2633}

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.

\(\displaystyle \int \frac {(b+2 c x) e^{a+b x+c x^2}}{\left (a+b x+c x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 7258

\(\displaystyle \int \frac {e^{a+b x+c x^2}}{\left (a+b x+c x^2\right )^{9/2}}d\left (a+b x+c x^2\right )\)

\(\Big \downarrow \) 2608

\(\displaystyle \frac {2}{7} \int \frac {e^{c x^2+b x+a}}{\left (c x^2+b x+a\right )^{7/2}}d\left (c x^2+b x+a\right )-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2608

\(\displaystyle \frac {2}{7} \left (\frac {2}{5} \int \frac {e^{c x^2+b x+a}}{\left (c x^2+b x+a\right )^{5/2}}d\left (c x^2+b x+a\right )-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}}\right )-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2608

\(\displaystyle \frac {2}{7} \left (\frac {2}{5} \left (\frac {2}{3} \int \frac {e^{c x^2+b x+a}}{\left (c x^2+b x+a\right )^{3/2}}d\left (c x^2+b x+a\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\right )-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}}\right )-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2608

\(\displaystyle \frac {2}{7} \left (\frac {2}{5} \left (\frac {2}{3} \left (2 \int \frac {e^{c x^2+b x+a}}{\sqrt {c x^2+b x+a}}d\left (c x^2+b x+a\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\right )-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}}\right )-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {2}{7} \left (\frac {2}{5} \left (\frac {2}{3} \left (4 \int e^{c x^2+b x+a}d\sqrt {c x^2+b x+a}-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\right )-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}}\right )-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {2}{7} \left (\frac {2}{5} \left (\frac {2}{3} \left (2 \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\right )-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}}\right )-\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}\)

Input:

Int[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(9/2),x]
 

Output:

(-2*E^(a + b*x + c*x^2))/(7*(a + b*x + c*x^2)^(7/2)) + (2*((-2*E^(a + b*x 
+ c*x^2))/(5*(a + b*x + c*x^2)^(5/2)) + (2*((-2*E^(a + b*x + c*x^2))/(3*(a 
 + b*x + c*x^2)^(3/2)) + (2*((-2*E^(a + b*x + c*x^2))/Sqrt[a + b*x + c*x^2 
] + 2*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]))/3))/5))/7
 

Defintions of rubi rules used

rule 2608

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m 
_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))) 
, x] - Simp[f*g*n*(Log[F]/(d*(m + 1)))   Int[(c + d*x)^(m + 1)*(b*F^(g*(e + 
 f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && In 
tegerQ[2*m] &&  !TrueQ[$UseGamma]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 7258

Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, 
u, x]}, Simp[q   Subst[Int[x^m*F^x, x], x, v], x] /;  !FalseQ[q]] /; FreeQ[ 
{F, m}, x] && EqQ[w, v]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{7 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{35 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {16 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \sqrt {c \,x^{2}+b x +a}}+\frac {16 \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right )}{105}\)	\(120\)
default	\(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{7 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{35 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {16 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \sqrt {c \,x^{2}+b x +a}}+\frac {16 \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right )}{105}\)	\(120\)

Input:

int(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^(9/2),x,method=_RETURNVERBOSE 
)
 

Output:

-2/7*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(7/2)-4/35*exp(c*x^2+b*x+a)/(c*x^2+b*x 
+a)^(5/2)-8/105*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(3/2)-16/105*exp(c*x^2+b*x+ 
a)/(c*x^2+b*x+a)^(1/2)+16/105*Pi^(1/2)*erfi((c*x^2+b*x+a)^(1/2))
 

Fricas [F]

\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^(9/2),x, algorithm="fri 
cas")
 

Output:

integral(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*e^(c*x^2 + b*x + a)/(c^5*x^10 + 
 5*b*c^4*x^9 + 5*(2*b^2*c^3 + a*c^4)*x^8 + 10*(b^3*c^2 + 2*a*b*c^3)*x^7 + 
5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*x^6 + 5*a^4*b*x + (b^5 + 20*a*b^3*c + 
30*a^2*b*c^2)*x^5 + a^5 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*x^4 + 10*(a^ 
2*b^3 + 2*a^3*b*c)*x^3 + 5*(2*a^3*b^2 + a^4*c)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate(exp(c*x**2+b*x+a)*(2*c*x+b)/(c*x**2+b*x+a)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^(9/2),x, algorithm="max 
ima")
 

Output:

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(9/2), x)
 

Giac [F]

\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^(9/2),x, algorithm="gia 
c")
 

Output:

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(9/2), x)
 

Mupad [B] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.06 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (30\,c\,x^2+30\,b\,x+30\,a\right )+12\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^2+8\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^3+16\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^4-16\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{9/2}}{105\,{\left (c\,x^2+b\,x+a\right )}^{9/2}} \] Input:

int((exp(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(9/2),x)
 

Output:

-(exp(a + b*x + c*x^2)*(30*a + 30*b*x + 30*c*x^2) + 12*exp(a + b*x + c*x^2 
)*(a + b*x + c*x^2)^2 + 8*exp(a + b*x + c*x^2)*(a + b*x + c*x^2)^3 + 16*ex 
p(a + b*x + c*x^2)*(a + b*x + c*x^2)^4 - 16*pi^(1/2)*erfc((- a - b*x - c*x 
^2)^(1/2))*(- a - b*x - c*x^2)^(9/2))/(105*(a + b*x + c*x^2)^(9/2))
 

Reduce [F]

\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx=e^{a} \left (\left (\int \frac {e^{c \,x^{2}+b x}}{\sqrt {c \,x^{2}+b x +a}\, a^{4}+4 \sqrt {c \,x^{2}+b x +a}\, a^{3} b x +4 \sqrt {c \,x^{2}+b x +a}\, a^{3} c \,x^{2}+6 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x^{2}+12 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c \,x^{3}+6 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2} x^{4}+4 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{3}+12 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c \,x^{4}+12 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} x^{5}+4 \sqrt {c \,x^{2}+b x +a}\, a \,c^{3} x^{6}+\sqrt {c \,x^{2}+b x +a}\, b^{4} x^{4}+4 \sqrt {c \,x^{2}+b x +a}\, b^{3} c \,x^{5}+6 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} x^{6}+4 \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} x^{7}+\sqrt {c \,x^{2}+b x +a}\, c^{4} x^{8}}d x \right ) b +2 \left (\int \frac {e^{c \,x^{2}+b x} x}{\sqrt {c \,x^{2}+b x +a}\, a^{4}+4 \sqrt {c \,x^{2}+b x +a}\, a^{3} b x +4 \sqrt {c \,x^{2}+b x +a}\, a^{3} c \,x^{2}+6 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x^{2}+12 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c \,x^{3}+6 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2} x^{4}+4 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{3}+12 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c \,x^{4}+12 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} x^{5}+4 \sqrt {c \,x^{2}+b x +a}\, a \,c^{3} x^{6}+\sqrt {c \,x^{2}+b x +a}\, b^{4} x^{4}+4 \sqrt {c \,x^{2}+b x +a}\, b^{3} c \,x^{5}+6 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} x^{6}+4 \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} x^{7}+\sqrt {c \,x^{2}+b x +a}\, c^{4} x^{8}}d x \right ) c \right ) \] Input:

int(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^(9/2),x)
 

Output:

e**a*(int(e**(b*x + c*x**2)/(sqrt(a + b*x + c*x**2)*a**4 + 4*sqrt(a + b*x 
+ c*x**2)*a**3*b*x + 4*sqrt(a + b*x + c*x**2)*a**3*c*x**2 + 6*sqrt(a + b*x 
 + c*x**2)*a**2*b**2*x**2 + 12*sqrt(a + b*x + c*x**2)*a**2*b*c*x**3 + 6*sq 
rt(a + b*x + c*x**2)*a**2*c**2*x**4 + 4*sqrt(a + b*x + c*x**2)*a*b**3*x**3 
 + 12*sqrt(a + b*x + c*x**2)*a*b**2*c*x**4 + 12*sqrt(a + b*x + c*x**2)*a*b 
*c**2*x**5 + 4*sqrt(a + b*x + c*x**2)*a*c**3*x**6 + sqrt(a + b*x + c*x**2) 
*b**4*x**4 + 4*sqrt(a + b*x + c*x**2)*b**3*c*x**5 + 6*sqrt(a + b*x + c*x** 
2)*b**2*c**2*x**6 + 4*sqrt(a + b*x + c*x**2)*b*c**3*x**7 + sqrt(a + b*x + 
c*x**2)*c**4*x**8),x)*b + 2*int((e**(b*x + c*x**2)*x)/(sqrt(a + b*x + c*x* 
*2)*a**4 + 4*sqrt(a + b*x + c*x**2)*a**3*b*x + 4*sqrt(a + b*x + c*x**2)*a* 
*3*c*x**2 + 6*sqrt(a + b*x + c*x**2)*a**2*b**2*x**2 + 12*sqrt(a + b*x + c* 
x**2)*a**2*b*c*x**3 + 6*sqrt(a + b*x + c*x**2)*a**2*c**2*x**4 + 4*sqrt(a + 
 b*x + c*x**2)*a*b**3*x**3 + 12*sqrt(a + b*x + c*x**2)*a*b**2*c*x**4 + 12* 
sqrt(a + b*x + c*x**2)*a*b*c**2*x**5 + 4*sqrt(a + b*x + c*x**2)*a*c**3*x** 
6 + sqrt(a + b*x + c*x**2)*b**4*x**4 + 4*sqrt(a + b*x + c*x**2)*b**3*c*x** 
5 + 6*sqrt(a + b*x + c*x**2)*b**2*c**2*x**6 + 4*sqrt(a + b*x + c*x**2)*b*c 
**3*x**7 + sqrt(a + b*x + c*x**2)*c**4*x**8),x)*c)