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]
________________________________________________________________________________________
Rubi [A] time = 0.249907, antiderivative size = 279, normalized size of antiderivative = 1., 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]
[Out]
Rule 6363
Rule 2226
Rule 2204
Rule 2209
Rule 2212
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*}
Mathematica [A] time = 0.26782, size = 237, normalized size = 0.85 \[ \frac{\sqrt{\pi } \text{Erfi}(a+b x) \left (12 a^2 d \left (d^2-2 b^2 c^2\right )+16 a^3 b c d^2-4 a^4 d^3+8 a \left (2 b^3 c^3-3 b c d^2\right )+4 b^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+12 b^2 c^2 d-3 d^3\right )-2 e^{(a+b x)^2} \left (b d^2 \left (8 \left (a^2-1\right ) c+\left (2 a^2-3\right ) d x\right )+a \left (5-2 a^2\right ) d^3-2 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )+2 b^3 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )}{16 \sqrt{\pi } b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 703, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3} \operatorname{erfi}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.461, size = 581, normalized size = 2.08 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 18.2334, size = 746, normalized size = 2.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3} \operatorname{erfi}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]