3.221 \(\int (c+d x)^3 \text {erfi}(a+b x) \, dx\)

Optimal. Leaf size=279 \[ -\frac {d^2 e^{(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}+\frac {d^2 e^{(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}-\frac {(b c-a d)^4 \text {erfi}(a+b x)}{4 b^4 d}+\frac {3 d (b c-a d)^2 \text {erfi}(a+b x)}{4 b^4}-\frac {e^{(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}-\frac {3 d e^{(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}-\frac {3 d^3 \text {erfi}(a+b x)}{16 b^4}-\frac {d^3 e^{(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}+\frac {3 d^3 e^{(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6363, 2226, 2204, 2209, 2212} \[ -\frac {d^2 e^{(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}+\frac {d^2 e^{(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}-\frac {(b c-a d)^4 \text {Erfi}(a+b x)}{4 b^4 d}+\frac {3 d (b c-a d)^2 \text {Erfi}(a+b x)}{4 b^4}-\frac {e^{(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}-\frac {3 d e^{(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}-\frac {3 d^3 \text {Erfi}(a+b x)}{16 b^4}-\frac {d^3 e^{(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}+\frac {3 d^3 e^{(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {Erfi}(a+b x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Erfi[a + b*x],x]

[Out]

(d^2*(b*c - a*d)*E^(a + b*x)^2)/(b^4*Sqrt[Pi]) - ((b*c - a*d)^3*E^(a + b*x)^2)/(b^4*Sqrt[Pi]) + (3*d^3*E^(a +
b*x)^2*(a + b*x))/(8*b^4*Sqrt[Pi]) - (3*d*(b*c - a*d)^2*E^(a + b*x)^2*(a + b*x))/(2*b^4*Sqrt[Pi]) - (d^2*(b*c
- a*d)*E^(a + b*x)^2*(a + b*x)^2)/(b^4*Sqrt[Pi]) - (d^3*E^(a + b*x)^2*(a + b*x)^3)/(4*b^4*Sqrt[Pi]) - (3*d^3*E
rfi[a + b*x])/(16*b^4) + (3*d*(b*c - a*d)^2*Erfi[a + b*x])/(4*b^4) - ((b*c - a*d)^4*Erfi[a + b*x])/(4*b^4*d) +
((c + d*x)^4*Erfi[a + b*x])/(4*d)

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])

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 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]

Rubi steps

\begin {align*} \int (c+d x)^3 \text {erfi}(a+b x) \, dx &=\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d}-\frac {b \int e^{(a+b x)^2} (c+d x)^4 \, dx}{2 d \sqrt {\pi }}\\ &=\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d}-\frac {b \int \left (\frac {(b c-a d)^4 e^{(a+b x)^2}}{b^4}+\frac {4 d (b c-a d)^3 e^{(a+b x)^2} (a+b x)}{b^4}+\frac {6 d^2 (b c-a d)^2 e^{(a+b x)^2} (a+b x)^2}{b^4}+\frac {4 d^3 (b c-a d) e^{(a+b x)^2} (a+b x)^3}{b^4}+\frac {d^4 e^{(a+b x)^2} (a+b x)^4}{b^4}\right ) \, dx}{2 d \sqrt {\pi }}\\ &=\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d}-\frac {d^3 \int e^{(a+b x)^2} (a+b x)^4 \, dx}{2 b^3 \sqrt {\pi }}-\frac {\left (2 d^2 (b c-a d)\right ) \int e^{(a+b x)^2} (a+b x)^3 \, dx}{b^3 \sqrt {\pi }}-\frac {\left (3 d (b c-a d)^2\right ) \int e^{(a+b x)^2} (a+b x)^2 \, dx}{b^3 \sqrt {\pi }}-\frac {\left (2 (b c-a d)^3\right ) \int e^{(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt {\pi }}-\frac {(b c-a d)^4 \int e^{(a+b x)^2} \, dx}{2 b^3 d \sqrt {\pi }}\\ &=-\frac {(b c-a d)^3 e^{(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}-\frac {(b c-a d)^4 \text {erfi}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d}+\frac {\left (3 d^3\right ) \int e^{(a+b x)^2} (a+b x)^2 \, dx}{4 b^3 \sqrt {\pi }}+\frac {\left (2 d^2 (b c-a d)\right ) \int e^{(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt {\pi }}+\frac {\left (3 d (b c-a d)^2\right ) \int e^{(a+b x)^2} \, dx}{2 b^3 \sqrt {\pi }}\\ &=\frac {d^2 (b c-a d) e^{(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {(b c-a d)^3 e^{(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {3 d^3 e^{(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}+\frac {3 d (b c-a d)^2 \text {erfi}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erfi}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d}-\frac {\left (3 d^3\right ) \int e^{(a+b x)^2} \, dx}{8 b^3 \sqrt {\pi }}\\ &=\frac {d^2 (b c-a d) e^{(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {(b c-a d)^3 e^{(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {3 d^3 e^{(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}-\frac {3 d^3 \text {erfi}(a+b x)}{16 b^4}+\frac {3 d (b c-a d)^2 \text {erfi}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erfi}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfi}(a+b x)}{4 d}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 237, normalized size = 0.85 \[ \frac {\sqrt {\pi } \text {erfi}(a+b x) \left (-4 a^4 d^3+16 a^3 b c d^2+12 a^2 d \left (d^2-2 b^2 c^2\right )+8 a \left (2 b^3 c^3-3 b c d^2\right )+4 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+12 b^2 c^2 d-3 d^3\right )-2 e^{(a+b x)^2} \left (b d^2 \left (8 \left (a^2-1\right ) c+\left (2 a^2-3\right ) d x\right )+a \left (5-2 a^2\right ) d^3-2 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )+2 b^3 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right )}{16 \sqrt {\pi } b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*E^(a + b*x)^2*(a*(5 - 2*a^2)*d^3 + b*d^2*(8*(-1 + a^2)*c + (-3 + 2*a^2)*d*x) - 2*a*b^2*d*(6*c^2 + 4*c*d*x
+ d^2*x^2) + 2*b^3*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)) + Sqrt[Pi]*(12*b^2*c^2*d + 16*a^3*b*c*d^2 - 3*
d^3 - 4*a^4*d^3 + 12*a^2*d*(-2*b^2*c^2 + d^2) + 8*a*(2*b^3*c^3 - 3*b*c*d^2) + 4*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c
*d^2*x^2 + d^3*x^3))*Erfi[a + b*x])/(16*b^4*Sqrt[Pi])

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fricas [A]  time = 0.48, size = 263, normalized size = 0.94 \[ -\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 8 \, {\left (a^{2} - 1\right )} b c d^{2} - {\left (2 \, a^{3} - 5 \, a\right )} d^{3} + 2 \, {\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (12 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + {\left (2 \, a^{2} - 3\right )} b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x + \pi {\left (16 \, a b^{3} c^{3} - 12 \, {\left (2 \, a^{2} - 1\right )} b^{2} c^{2} d + 8 \, {\left (2 \, a^{3} - 3 \, a\right )} b c d^{2} - {\left (4 \, a^{4} - 12 \, a^{2} + 3\right )} d^{3}\right )}\right )} \operatorname {erfi}\left (b x + a\right )}{16 \, \pi b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*sqrt(pi)*(2*b^3*d^3*x^3 + 8*b^3*c^3 - 12*a*b^2*c^2*d + 8*(a^2 - 1)*b*c*d^2 - (2*a^3 - 5*a)*d^3 + 2*(4
*b^3*c*d^2 - a*b^2*d^3)*x^2 + (12*b^3*c^2*d - 8*a*b^2*c*d^2 + (2*a^2 - 3)*b*d^3)*x)*e^(b^2*x^2 + 2*a*b*x + a^2
) - (4*pi*b^4*d^3*x^4 + 16*pi*b^4*c*d^2*x^3 + 24*pi*b^4*c^2*d*x^2 + 16*pi*b^4*c^3*x + pi*(16*a*b^3*c^3 - 12*(2
*a^2 - 1)*b^2*c^2*d + 8*(2*a^3 - 3*a)*b*c*d^2 - (4*a^4 - 12*a^2 + 3)*d^3))*erfi(b*x + a))/(pi*b^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \operatorname {erfi}\left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 703, normalized size = 2.52 \[ \frac {\frac {d^{3} \erfi \left (b x +a \right ) \left (b x +a \right )^{4}}{4 b^{3}}-\frac {d^{3} \erfi \left (b x +a \right ) \left (b x +a \right )^{3} a}{b^{3}}+\frac {d^{2} \erfi \left (b x +a \right ) \left (b x +a \right )^{3} c}{b^{2}}+\frac {3 d^{3} \erfi \left (b x +a \right ) \left (b x +a \right )^{2} a^{2}}{2 b^{3}}-\frac {3 d^{2} \erfi \left (b x +a \right ) \left (b x +a \right )^{2} a c}{b^{2}}+\frac {3 d \erfi \left (b x +a \right ) \left (b x +a \right )^{2} c^{2}}{2 b}-\frac {d^{3} \erfi \left (b x +a \right ) \left (b x +a \right ) a^{3}}{b^{3}}+\frac {3 d^{2} \erfi \left (b x +a \right ) \left (b x +a \right ) a^{2} c}{b^{2}}-\frac {3 d \erfi \left (b x +a \right ) \left (b x +a \right ) a \,c^{2}}{b}+\erfi \left (b x +a \right ) \left (b x +a \right ) c^{3}+\frac {d^{3} \erfi \left (b x +a \right ) a^{4}}{4 b^{3}}-\frac {d^{2} \erfi \left (b x +a \right ) a^{3} c}{b^{2}}+\frac {3 d \erfi \left (b x +a \right ) a^{2} c^{2}}{2 b}-\erfi \left (b x +a \right ) a \,c^{3}+\frac {b \erfi \left (b x +a \right ) c^{4}}{4 d}-\frac {d^{4} \left (\frac {{\mathrm e}^{\left (b x +a \right )^{2}} \left (b x +a \right )^{3}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \erfi \left (b x +a \right )}{8}\right )+\frac {a^{4} d^{4} \sqrt {\pi }\, \erfi \left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \erfi \left (b x +a \right )}{2}-2 a^{3} d^{4} {\mathrm e}^{\left (b x +a \right )^{2}}+6 a^{2} d^{4} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x +a \right )}{4}\right )-4 a \,d^{4} \left (\frac {\left (b x +a \right )^{2} {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{2}\right )+2 b^{3} c^{3} d \,{\mathrm e}^{\left (b x +a \right )^{2}}+6 b^{2} c^{2} d^{2} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x +a \right )}{4}\right )+4 b c \,d^{3} \left (\frac {\left (b x +a \right )^{2} {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{2}\right )-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \erfi \left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \erfi \left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \erfi \left (b x +a \right )-6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{\left (b x +a \right )^{2}}+6 a^{2} b c \,d^{3} {\mathrm e}^{\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x +a \right )}{4}\right )}{2 b^{3} d \sqrt {\pi }}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*erfi(b*x+a),x)

[Out]

1/b*(1/4/b^3*d^3*erfi(b*x+a)*(b*x+a)^4-1/b^3*d^3*erfi(b*x+a)*(b*x+a)^3*a+1/b^2*d^2*erfi(b*x+a)*(b*x+a)^3*c+3/2
/b^3*d^3*erfi(b*x+a)*(b*x+a)^2*a^2-3/b^2*d^2*erfi(b*x+a)*(b*x+a)^2*a*c+3/2/b*d*erfi(b*x+a)*(b*x+a)^2*c^2-1/b^3
*d^3*erfi(b*x+a)*(b*x+a)*a^3+3/b^2*d^2*erfi(b*x+a)*(b*x+a)*a^2*c-3/b*d*erfi(b*x+a)*(b*x+a)*a*c^2+erfi(b*x+a)*(
b*x+a)*c^3+1/4/b^3*d^3*erfi(b*x+a)*a^4-1/b^2*d^2*erfi(b*x+a)*a^3*c+3/2/b*d*erfi(b*x+a)*a^2*c^2-erfi(b*x+a)*a*c
^3+1/4*b/d*erfi(b*x+a)*c^4-1/2/b^3/d/Pi^(1/2)*(d^4*(1/2*exp((b*x+a)^2)*(b*x+a)^3-3/4*(b*x+a)*exp((b*x+a)^2)+3/
8*Pi^(1/2)*erfi(b*x+a))+1/2*a^4*d^4*Pi^(1/2)*erfi(b*x+a)+1/2*b^4*c^4*Pi^(1/2)*erfi(b*x+a)-2*a^3*d^4*exp((b*x+a
)^2)+6*a^2*d^4*(1/2*(b*x+a)*exp((b*x+a)^2)-1/4*Pi^(1/2)*erfi(b*x+a))-4*a*d^4*(1/2*(b*x+a)^2*exp((b*x+a)^2)-1/2
*exp((b*x+a)^2))+2*b^3*c^3*d*exp((b*x+a)^2)+6*b^2*c^2*d^2*(1/2*(b*x+a)*exp((b*x+a)^2)-1/4*Pi^(1/2)*erfi(b*x+a)
)+4*b*c*d^3*(1/2*(b*x+a)^2*exp((b*x+a)^2)-1/2*exp((b*x+a)^2))-2*a*b^3*c^3*d*Pi^(1/2)*erfi(b*x+a)+3*a^2*b^2*c^2
*d^2*Pi^(1/2)*erfi(b*x+a)-2*a^3*b*c*d^3*Pi^(1/2)*erfi(b*x+a)-6*a*b^2*c^2*d^2*exp((b*x+a)^2)+6*a^2*b*c*d^3*exp(
(b*x+a)^2)-12*a*b*c*d^3*(1/2*(b*x+a)*exp((b*x+a)^2)-1/4*Pi^(1/2)*erfi(b*x+a))))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \operatorname {erfi}\left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.64, size = 357, normalized size = 1.28 \[ \mathrm {erfi}\left (a+b\,x\right )\,\left (c^3\,x+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {d^3\,x^4}{4}\right )-\frac {\frac {{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (-2\,a^3\,d^3+8\,a^2\,b\,c\,d^2-12\,a\,b^2\,c^2\,d+5\,a\,d^3+8\,b^3\,c^3-8\,b\,c\,d^2\right )}{4\,b^4}+\frac {d^3\,x^3\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b}-\frac {x^2\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a\,d^3-4\,b\,c\,d^2\right )}{2\,b^2}-\frac {x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (b^2\,\left (12\,c^2\,d-72\,a^2\,c^2\,d\right )+b\,\left (48\,a^3\,c\,d^2-8\,a\,c\,d^2\right )-3\,d^3+20\,a^2\,d^3-12\,a^4\,d^3\right )}{b^3\,\left (24\,a^2-4\right )}}{2\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (a+b\,x\right )\,\left (4\,a^4\,d^3-16\,a^3\,b\,c\,d^2+24\,a^2\,b^2\,c^2\,d-12\,a^2\,d^3-16\,a\,b^3\,c^3+24\,a\,b\,c\,d^2-12\,b^2\,c^2\,d+3\,d^3\right )}{16\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(a + b*x)*(c + d*x)^3,x)

[Out]

erfi(a + b*x)*(c^3*x + (d^3*x^4)/4 + (3*c^2*d*x^2)/2 + c*d^2*x^3) - ((exp(a^2 + b^2*x^2 + 2*a*b*x)*(5*a*d^3 -
2*a^3*d^3 + 8*b^3*c^3 - 8*b*c*d^2 - 12*a*b^2*c^2*d + 8*a^2*b*c*d^2))/(4*b^4) + (d^3*x^3*exp(a^2 + b^2*x^2 + 2*
a*b*x))/(2*b) - (x^2*exp(a^2 + b^2*x^2 + 2*a*b*x)*(a*d^3 - 4*b*c*d^2))/(2*b^2) - (x*exp(a^2 + b^2*x^2 + 2*a*b*
x)*(b^2*(12*c^2*d - 72*a^2*c^2*d) + b*(48*a^3*c*d^2 - 8*a*c*d^2) - 3*d^3 + 20*a^2*d^3 - 12*a^4*d^3))/(b^3*(24*
a^2 - 4)))/(2*pi^(1/2)) - (erfi(a + b*x)*(3*d^3 - 12*a^2*d^3 + 4*a^4*d^3 - 16*a*b^3*c^3 - 12*b^2*c^2*d + 24*a^
2*b^2*c^2*d + 24*a*b*c*d^2 - 16*a^3*b*c*d^2))/(16*b^4)

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sympy [A]  time = 8.01, size = 746, normalized size = 2.67 \[ \begin {cases} - \frac {a^{4} d^{3} \operatorname {erfi}{\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} c d^{2} \operatorname {erfi}{\left (a + b x \right )}}{b^{3}} + \frac {a^{3} d^{3} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{4 \sqrt {\pi } b^{4}} - \frac {3 a^{2} c^{2} d \operatorname {erfi}{\left (a + b x \right )}}{2 b^{2}} - \frac {a^{2} c d^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b^{3}} - \frac {a^{2} d^{3} x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{4 \sqrt {\pi } b^{3}} + \frac {3 a^{2} d^{3} \operatorname {erfi}{\left (a + b x \right )}}{4 b^{4}} + \frac {a c^{3} \operatorname {erfi}{\left (a + b x \right )}}{b} + \frac {3 a c^{2} d e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt {\pi } b^{2}} + \frac {a c d^{2} x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b^{2}} + \frac {a d^{3} x^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{4 \sqrt {\pi } b^{2}} - \frac {3 a c d^{2} \operatorname {erfi}{\left (a + b x \right )}}{2 b^{3}} - \frac {5 a d^{3} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{8 \sqrt {\pi } b^{4}} + c^{3} x \operatorname {erfi}{\left (a + b x \right )} + \frac {3 c^{2} d x^{2} \operatorname {erfi}{\left (a + b x \right )}}{2} + c d^{2} x^{3} \operatorname {erfi}{\left (a + b x \right )} + \frac {d^{3} x^{4} \operatorname {erfi}{\left (a + b x \right )}}{4} - \frac {c^{3} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {3 c^{2} d x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt {\pi } b} - \frac {c d^{2} x^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {d^{3} x^{3} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{4 \sqrt {\pi } b} + \frac {3 c^{2} d \operatorname {erfi}{\left (a + b x \right )}}{4 b^{2}} + \frac {c d^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b^{3}} + \frac {3 d^{3} x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{8 \sqrt {\pi } b^{3}} - \frac {3 d^{3} \operatorname {erfi}{\left (a + b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \operatorname {erfi}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**4*d**3*erfi(a + b*x)/(4*b**4) + a**3*c*d**2*erfi(a + b*x)/b**3 + a**3*d**3*exp(a**2)*exp(b**2*x
**2)*exp(2*a*b*x)/(4*sqrt(pi)*b**4) - 3*a**2*c**2*d*erfi(a + b*x)/(2*b**2) - a**2*c*d**2*exp(a**2)*exp(b**2*x*
*2)*exp(2*a*b*x)/(sqrt(pi)*b**3) - a**2*d**3*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(4*sqrt(pi)*b**3) + 3*a**
2*d**3*erfi(a + b*x)/(4*b**4) + a*c**3*erfi(a + b*x)/b + 3*a*c**2*d*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(2*s
qrt(pi)*b**2) + a*c*d**2*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi)*b**2) + a*d**3*x**2*exp(a**2)*exp(b
**2*x**2)*exp(2*a*b*x)/(4*sqrt(pi)*b**2) - 3*a*c*d**2*erfi(a + b*x)/(2*b**3) - 5*a*d**3*exp(a**2)*exp(b**2*x**
2)*exp(2*a*b*x)/(8*sqrt(pi)*b**4) + c**3*x*erfi(a + b*x) + 3*c**2*d*x**2*erfi(a + b*x)/2 + c*d**2*x**3*erfi(a
+ b*x) + d**3*x**4*erfi(a + b*x)/4 - c**3*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi)*b) - 3*c**2*d*x*exp(
a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(2*sqrt(pi)*b) - c*d**2*x**2*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi)
*b) - d**3*x**3*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(4*sqrt(pi)*b) + 3*c**2*d*erfi(a + b*x)/(4*b**2) + c*d**
2*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi)*b**3) + 3*d**3*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(8*sq
rt(pi)*b**3) - 3*d**3*erfi(a + b*x)/(16*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4
/4)*erfi(a), True))

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