3.443 \(\int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx\)

Optimal. Leaf size=128 \[ (a d+b c) \text {Int}\left (\frac {e^{x (a d+b c)+a c+b d x^2}}{x},x\right )+\sqrt {\pi } \sqrt {b} \sqrt {d} e^{-\frac {(b c-a d)^2}{4 b d}} \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )-\frac {e^{x (a d+b c)+a c+b d x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^((a + b*x)*(c + d*x))/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(b*d*x^2 + a*c + (b*c + a*d)*x)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{\left (a+b\,x\right )\,\left (c+d\,x\right )}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((a + b*x)*(c + d*x))/x^2,x)

[Out]

int(exp((a + b*x)*(c + d*x))/x^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int \frac {e^{a d x} e^{b c x} e^{b d x^{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(a*c)*Integral(exp(a*d*x)*exp(b*c*x)*exp(b*d*x**2)/x**2, x)

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