3.223 \(\int (c+d x) \text{Erfi}(a+b x) \, dx\)

Optimal. Leaf size=115 \[ -\frac{(b c-a d)^2 \text{Erfi}(a+b x)}{2 b^2 d}-\frac{e^{(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^2}+\frac{d \text{Erfi}(a+b x)}{4 b^2}-\frac{d e^{(a+b x)^2} (a+b x)}{2 \sqrt{\pi } b^2}+\frac{(c+d x)^2 \text{Erfi}(a+b x)}{2 d} \]

[Out]

-(((b*c - a*d)*E^(a + b*x)^2)/(b^2*Sqrt[Pi])) - (d*E^(a + b*x)^2*(a + b*x))/(2*b^2*Sqrt[Pi]) + (d*Erfi[a + b*x
])/(4*b^2) - ((b*c - a*d)^2*Erfi[a + b*x])/(2*b^2*d) + ((c + d*x)^2*Erfi[a + b*x])/(2*d)

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Rubi [A]  time = 0.10527, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6363, 2226, 2204, 2209, 2212} \[ -\frac{(b c-a d)^2 \text{Erfi}(a+b x)}{2 b^2 d}-\frac{e^{(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^2}+\frac{d \text{Erfi}(a+b x)}{4 b^2}-\frac{d e^{(a+b x)^2} (a+b x)}{2 \sqrt{\pi } b^2}+\frac{(c+d x)^2 \text{Erfi}(a+b x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Erfi[a + b*x],x]

[Out]

-(((b*c - a*d)*E^(a + b*x)^2)/(b^2*Sqrt[Pi])) - (d*E^(a + b*x)^2*(a + b*x))/(2*b^2*Sqrt[Pi]) + (d*Erfi[a + b*x
])/(4*b^2) - ((b*c - a*d)^2*Erfi[a + b*x])/(2*b^2*d) + ((c + d*x)^2*Erfi[a + b*x])/(2*d)

Rule 6363

Int[Erfi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erfi[a + b*x])/(
d*(m + 1)), x] - Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)*E^(a + b*x)^2, x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), 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[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin{align*} \int (c+d x) \text{erfi}(a+b x) \, dx &=\frac{(c+d x)^2 \text{erfi}(a+b x)}{2 d}-\frac{b \int e^{(a+b x)^2} (c+d x)^2 \, dx}{d \sqrt{\pi }}\\ &=\frac{(c+d x)^2 \text{erfi}(a+b x)}{2 d}-\frac{b \int \left (\frac{(b c-a d)^2 e^{(a+b x)^2}}{b^2}+\frac{2 d (b c-a d) e^{(a+b x)^2} (a+b x)}{b^2}+\frac{d^2 e^{(a+b x)^2} (a+b x)^2}{b^2}\right ) \, dx}{d \sqrt{\pi }}\\ &=\frac{(c+d x)^2 \text{erfi}(a+b x)}{2 d}-\frac{d \int e^{(a+b x)^2} (a+b x)^2 \, dx}{b \sqrt{\pi }}-\frac{(2 (b c-a d)) \int e^{(a+b x)^2} (a+b x) \, dx}{b \sqrt{\pi }}-\frac{(b c-a d)^2 \int e^{(a+b x)^2} \, dx}{b d \sqrt{\pi }}\\ &=-\frac{(b c-a d) e^{(a+b x)^2}}{b^2 \sqrt{\pi }}-\frac{d e^{(a+b x)^2} (a+b x)}{2 b^2 \sqrt{\pi }}-\frac{(b c-a d)^2 \text{erfi}(a+b x)}{2 b^2 d}+\frac{(c+d x)^2 \text{erfi}(a+b x)}{2 d}+\frac{d \int e^{(a+b x)^2} \, dx}{2 b \sqrt{\pi }}\\ &=-\frac{(b c-a d) e^{(a+b x)^2}}{b^2 \sqrt{\pi }}-\frac{d e^{(a+b x)^2} (a+b x)}{2 b^2 \sqrt{\pi }}+\frac{d \text{erfi}(a+b x)}{4 b^2}-\frac{(b c-a d)^2 \text{erfi}(a+b x)}{2 b^2 d}+\frac{(c+d x)^2 \text{erfi}(a+b x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0706394, size = 78, normalized size = 0.68 \[ \frac{\sqrt{\pi } \text{Erfi}(a+b x) \left (-2 a^2 d+4 a b c+4 b^2 c x+2 b^2 d x^2+d\right )-2 e^{(a+b x)^2} (-a d+2 b c+b d x)}{4 \sqrt{\pi } b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*Erfi[a + b*x],x]

[Out]

(-2*E^(a + b*x)^2*(2*b*c - a*d + b*d*x) + Sqrt[Pi]*(4*a*b*c + d - 2*a^2*d + 4*b^2*c*x + 2*b^2*d*x^2)*Erfi[a +
b*x])/(4*b^2*Sqrt[Pi])

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Maple [A]  time = 0.047, size = 117, normalized size = 1. \begin{align*}{\frac{1}{b} \left ({\frac{d{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}}{2\,b}}-{\frac{{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ) ad}{b}}+{\it erfi} \left ( bx+a \right ) c \left ( bx+a \right ) -{\frac{1}{\sqrt{\pi }b} \left ( d \left ({\frac{ \left ( bx+a \right ){{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{2}}-{\frac{\sqrt{\pi }{\it erfi} \left ( bx+a \right ) }{4}} \right ) -ad{{\rm e}^{ \left ( bx+a \right ) ^{2}}}+{{\rm e}^{ \left ( bx+a \right ) ^{2}}}bc \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*erfi(b*x+a),x)

[Out]

1/b*(1/2/b*erfi(b*x+a)*d*(b*x+a)^2-1/b*erfi(b*x+a)*(b*x+a)*a*d+erfi(b*x+a)*c*(b*x+a)-1/Pi^(1/2)/b*(d*(1/2*(b*x
+a)*exp((b*x+a)^2)-1/4*Pi^(1/2)*erfi(b*x+a))-a*d*exp((b*x+a)^2)+exp((b*x+a)^2)*b*c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \operatorname{erfi}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*erfi(b*x+a),x, algorithm="maxima")

[Out]

integrate((d*x + c)*erfi(b*x + a), x)

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Fricas [A]  time = 2.32517, size = 212, normalized size = 1.84 \begin{align*} -\frac{2 \, \sqrt{\pi }{\left (b d x + 2 \, b c - a d\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} -{\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi{\left (4 \, a b c -{\left (2 \, a^{2} - 1\right )} d\right )}\right )} \operatorname{erfi}\left (b x + a\right )}{4 \, \pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*erfi(b*x+a),x, algorithm="fricas")

[Out]

-1/4*(2*sqrt(pi)*(b*d*x + 2*b*c - a*d)*e^(b^2*x^2 + 2*a*b*x + a^2) - (2*pi*b^2*d*x^2 + 4*pi*b^2*c*x + pi*(4*a*
b*c - (2*a^2 - 1)*d))*erfi(b*x + a))/(pi*b^2)

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Sympy [A]  time = 1.50053, size = 178, normalized size = 1.55 \begin{align*} \begin{cases} - \frac{a^{2} d \operatorname{erfi}{\left (a + b x \right )}}{2 b^{2}} + \frac{a c \operatorname{erfi}{\left (a + b x \right )}}{b} + \frac{a d e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt{\pi } b^{2}} + c x \operatorname{erfi}{\left (a + b x \right )} + \frac{d x^{2} \operatorname{erfi}{\left (a + b x \right )}}{2} - \frac{c e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt{\pi } b} - \frac{d x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt{\pi } b} + \frac{d \operatorname{erfi}{\left (a + b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \operatorname{erfi}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*erfi(b*x+a),x)

[Out]

Piecewise((-a**2*d*erfi(a + b*x)/(2*b**2) + a*c*erfi(a + b*x)/b + a*d*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(2
*sqrt(pi)*b**2) + c*x*erfi(a + b*x) + d*x**2*erfi(a + b*x)/2 - c*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(p
i)*b) - d*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(2*sqrt(pi)*b) + d*erfi(a + b*x)/(4*b**2), Ne(b, 0)), ((c*x
+ d*x**2/2)*erfi(a), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \operatorname{erfi}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*erfi(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)*erfi(b*x + a), x)