3.627 \(\int e^{a+b x+c x^2} (b+2 c x) (a+b x+c x^2)^{7/2} \, dx\)

Optimal. Leaf size=142 \[ \frac {105}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac {35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {105}{8} e^{a+b x+c x^2} \sqrt {a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.63, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6707, 2176, 2180, 2204} \[ \frac {105}{16} \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac {35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {105}{8} e^{a+b x+c x^2} \sqrt {a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(7/2),x]

[Out]

(-105*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2])/8 + (35*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(3/2))/4 - (7*E
^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(5/2))/2 + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(7/2) + (105*Sqrt[Pi]*Er
fi[Sqrt[a + b*x + c*x^2]])/16

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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] &&  !$UseGamma === True

Rule 2204

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 6707

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

Rubi steps

\begin {align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{7/2} \, dx &=\operatorname {Subst}\left (\int e^x x^{7/2} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac {7}{2} \operatorname {Subst}\left (\int e^x x^{5/2} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac {35}{4} \operatorname {Subst}\left (\int e^x x^{3/2} \, dx,x,a+b x+c x^2\right )\\ &=\frac {35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac {105}{8} \operatorname {Subst}\left (\int e^x \sqrt {x} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {105}{8} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}+\frac {35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac {105}{16} \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {105}{8} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}+\frac {35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac {105}{8} \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )\\ &=-\frac {105}{8} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}+\frac {35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac {105}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 47, normalized size = 0.33 \[ -\frac {\sqrt {a+x (b+c x)} \Gamma \left (\frac {9}{2},-a-x (b+c x)\right )}{\sqrt {-a-x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(7/2),x]

[Out]

-((Sqrt[a + x*(b + c*x)]*Gamma[9/2, -a - x*(b + c*x)])/Sqrt[-a - x*(b + c*x)])

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, c^{4} x^{7} + 7 \, b c^{3} x^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{5} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{4} + a^{3} b + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{3} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x\right )} \sqrt {c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

integral((2*c^4*x^7 + 7*b*c^3*x^6 + 3*(3*b^2*c^2 + 2*a*c^3)*x^5 + 5*(b^3*c + 3*a*b*c^2)*x^4 + a^3*b + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*x^3 + 3*(a*b^3 + 3*a^2*b*c)*x^2 + (3*a^2*b^2 + 2*a^3*c)*x)*sqrt(c*x^2 + b*x + a)*e^(c*x
^2 + b*x + a), x)

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giac [A]  time = 0.46, size = 93, normalized size = 0.65 \[ \frac {105}{16} \, \sqrt {\pi } i \operatorname {erf}\left (-\sqrt {c x^{2} + b x + a} i\right ) + \frac {1}{8} \, {\left (8 \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} - 28 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} + 70 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} - 105 \, \sqrt {c x^{2} + b x + a}\right )} e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

105/16*sqrt(pi)*i*erf(-sqrt(c*x^2 + b*x + a)*i) + 1/8*(8*(c*x^2 + b*x + a)^(7/2) - 28*(c*x^2 + b*x + a)^(5/2)
+ 70*(c*x^2 + b*x + a)^(3/2) - 105*sqrt(c*x^2 + b*x + a))*e^(c*x^2 + b*x + a)

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maple [A]  time = 0.04, size = 119, normalized size = 0.84 \[ \frac {105 \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )}{16}+\frac {35 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} {\mathrm e}^{c \,x^{2}+b x +a}}{4}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} {\mathrm e}^{c \,x^{2}+b x +a}}{2}+\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} {\mathrm e}^{c \,x^{2}+b x +a}-\frac {105 \sqrt {c \,x^{2}+b x +a}\, {\mathrm e}^{c \,x^{2}+b x +a}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(7/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(7/2)*(2*c*x + b)*e^(c*x^2 + b*x + a), x)

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mupad [B]  time = 4.22, size = 135, normalized size = 0.95 \[ \frac {\left ({\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (\frac {105\,\sqrt {-c\,x^2-b\,x-a}}{8}+\frac {35\,{\left (-c\,x^2-b\,x-a\right )}^{3/2}}{4}+\frac {7\,{\left (-c\,x^2-b\,x-a\right )}^{5/2}}{2}+{\left (-c\,x^2-b\,x-a\right )}^{7/2}\right )+\frac {105\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )}{16}\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{{\left (-c\,x^2-b\,x-a\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(7/2),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x**2+b*x+a)*(2*c*x+b)*(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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