Integrand size = 14, antiderivative size = 289 \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}+\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}+\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}-\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}-\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d} \]
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Time = 0.23 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6496, 2258, 2236, 2240, 2243} \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\frac {d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}+\frac {d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}-\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}+\frac {e^{-(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}+\frac {3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}-\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}+\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d} \]
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Rule 2236
Rule 2240
Rule 2243
Rule 2258
Rule 6496
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \text {erf}(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 {erf}(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 {erf}(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 {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(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 {erf}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(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 {erf}(a+b x)}{16 b^4}-\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.86 \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\frac {e^{-(a+b x)^2} \left (-2 a \left (5+2 a^2\right ) d^3+2 b d^2 \left (8 \left (1+a^2\right ) c+\left (3+2 a^2\right ) d x\right )-4 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )+4 b^3 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-e^{(a+b x)^2} \sqrt {\pi } \left (12 b^2 c^2 d-16 a^3 b c d^2+3 d^3+4 a^4 d^3-8 a \left (2 b^3 c^3+3 b c d^2\right )+12 a^2 \left (2 b^2 c^2 d+d^3\right )-4 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \text {erf}(a+b x)\right )}{16 b^4 \sqrt {\pi }} \]
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Time = 1.19 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.62
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {erf}\left (b x +a \right ) \left (a d -b c -d \left (b x +a \right )\right )^{4}}{4 b^{3} d}-\frac {\frac {a^{4} d^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{3}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{8}\right )-4 a \,d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 a^{2} d^{4} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+2 a^{3} d^{4} {\mathrm e}^{-\left (b x +a \right )^{2}}-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+4 b c \,d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 b^{2} c^{2} d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )-2 b^{3} c^{3} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}-6 a^{2} b c \,d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2 \sqrt {\pi }\, b^{3} d}}{b}\) | \(467\) |
default | \(\frac {\frac {\operatorname {erf}\left (b x +a \right ) \left (a d -b c -d \left (b x +a \right )\right )^{4}}{4 b^{3} d}-\frac {\frac {a^{4} d^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{3}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{8}\right )-4 a \,d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 a^{2} d^{4} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+2 a^{3} d^{4} {\mathrm e}^{-\left (b x +a \right )^{2}}-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+4 b c \,d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 b^{2} c^{2} d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )-2 b^{3} c^{3} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}-6 a^{2} b c \,d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2 \sqrt {\pi }\, b^{3} d}}{b}\) | \(467\) |
parallelrisch | \(\frac {-16 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} a \,b^{2} c \,d^{2}+16 c \,d^{2} \operatorname {erf}\left (b x +a \right ) x^{3} \sqrt {\pi }\, b^{4}+24 c^{2} d \,\operatorname {erf}\left (b x +a \right ) x^{2} \sqrt {\pi }\, b^{4}+16 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a^{3} b c \,d^{2}-24 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a^{2} b^{2} c^{2} d +24 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a b c \,d^{2}-4 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{3} d^{3}+16 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b^{3} c^{3}-10 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a \,d^{3}-3 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) d^{3}+4 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{2} b \,d^{3}+24 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} b^{3} c^{2} d -4 x^{2} {\mathrm e}^{-\left (b x +a \right )^{2}} a \,b^{2} d^{3}+16 x^{2} {\mathrm e}^{-\left (b x +a \right )^{2}} b^{3} c \,d^{2}+16 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{2} b c \,d^{2}-24 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a \,b^{2} c^{2} d +4 d^{3} \operatorname {erf}\left (b x +a \right ) x^{4} \sqrt {\pi }\, b^{4}+16 c^{3} \operatorname {erf}\left (b x +a \right ) x \sqrt {\pi }\, b^{4}+16 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a \,b^{3} c^{3}-12 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) b^{2} c^{2} d +4 d^{3} x^{3} {\mathrm e}^{-\left (b x +a \right )^{2}} b^{3}+6 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} b \,d^{3}+16 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b c \,d^{2}-4 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a^{4} d^{3}-12 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a^{2} d^{3}}{16 \sqrt {\pi }\, b^{4}}\) | \(483\) |
parts | \(\frac {\operatorname {erf}\left (b x +a \right ) d^{3} x^{4}}{4}+\operatorname {erf}\left (b x +a \right ) d^{2} c \,x^{3}+\frac {3 \,\operatorname {erf}\left (b x +a \right ) d \,c^{2} x^{2}}{2}+\operatorname {erf}\left (b x +a \right ) c^{3} x +\frac {\operatorname {erf}\left (b x +a \right ) c^{4}}{4 d}-\frac {b \left (\frac {c^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2 b}+{\mathrm e}^{-a^{2}} d^{4} \left (-\frac {x^{3} {\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )}{b}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}}{b^{2}}\right )}{b}+\frac {-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{4 b^{2}}-\frac {3 a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{2 b}+\frac {3 \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{8 b^{3}}}{b^{2}}\right )+4 \,{\mathrm e}^{-a^{2}} c \,d^{3} \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )}{b}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}}{b^{2}}\right )+6 \,{\mathrm e}^{-a^{2}} c^{2} d^{2} \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )+4 \,{\mathrm e}^{-a^{2}} c^{3} d \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )\right )}{2 d \sqrt {\pi }}\) | \(712\) |
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Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.92 \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 8 \, {\left (a^{2} + 1\right )} b c d^{2} - {\left (2 \, a^{3} + 5 \, a\right )} d^{3} + 2 \, {\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (12 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + {\left (2 \, a^{2} + 3\right )} b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x + \pi {\left (16 \, a b^{3} c^{3} - 12 \, {\left (2 \, a^{2} + 1\right )} b^{2} c^{2} d + 8 \, {\left (2 \, a^{3} + 3 \, a\right )} b c d^{2} - {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{16 \, \pi b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (258) = 516\).
Time = 1.83 (sec) , antiderivative size = 746, normalized size of antiderivative = 2.58 \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\begin {cases} - \frac {a^{4} d^{3} \operatorname {erf}{\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} c d^{2} \operatorname {erf}{\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 {erf}{\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 {erf}{\left (a + b x \right )}}{4 b^{4}} + \frac {a c^{3} \operatorname {erf}{\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 {erf}{\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 {erf}{\left (a + b x \right )} + \frac {3 c^{2} d x^{2} \operatorname {erf}{\left (a + b x \right )}}{2} + c d^{2} x^{3} \operatorname {erf}{\left (a + b x \right )} + \frac {d^{3} x^{4} \operatorname {erf}{\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 {erf}{\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 {erf}{\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 {erf}{\left (a \right )} & \text {otherwise} \end {cases} \]
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\[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \operatorname {erf}\left (b x + a\right ) \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.35 \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\frac {{\left (d x + c\right )}^{4} \operatorname {erf}\left (b x + a\right )}{4 \, d} + \frac {4 \, \sqrt {\pi } c^{4} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right ) - 16 \, {\left (\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c^{3} d + \frac {12 \, {\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c^{2} d^{2}}{b} - \frac {8 \, {\left (\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c d^{3}}{b^{2}} + \frac {{\left (\frac {\sqrt {\pi } {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{3} - 8 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2} + 12 \, a^{2} b {\left (x + \frac {a}{b}\right )} - 8 \, a^{3} + 3 \, b {\left (x + \frac {a}{b}\right )} - 8 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} d^{4}}{b^{3}}}{16 \, \sqrt {\pi } d} \]
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Time = 6.32 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.17 \[ \int (c+d x)^3 \text {erf}(a+b x) \, dx=\mathrm {erf}\left (a+b\,x\right )\,\left (c^3\,x+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {d^3\,x^4}{4}\right )-\frac {\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {5\,a\,d^3}{4}+\frac {a^3\,d^3}{2}-2\,b^3\,c^3-b\,\left (2\,c\,a^2\,d^2+2\,c\,d^2\right )+3\,a\,b^2\,c^2\,d\right )}{b^4}-\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^2\,d^3-8\,a\,b\,c\,d^2+12\,b^2\,c^2\,d+3\,d^3\right )}{4\,b^3}-\frac {d^3\,x^3\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{2\,b}+\frac {x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^3-4\,b\,c\,d^2\right )}{2\,b^2}}{2\,\sqrt {\pi }}+\frac {\mathrm {erfi}\left (a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )\,\left (4\,a^4\,d^3-16\,a^3\,b\,c\,d^2+24\,a^2\,b^2\,c^2\,d+12\,a^2\,d^3-16\,a\,b^3\,c^3-24\,a\,b\,c\,d^2+12\,b^2\,c^2\,d+3\,d^3\right )\,1{}\mathrm {i}}{16\,b^4} \]
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