Optimal. Leaf size=120 \[ -\frac {f^{a+b x+c x^2}}{e (d+e x)}+\frac {\sqrt {c} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}}{e^2}-\frac {(2 c d-b e) \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{d+e x},x\right )}{e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, 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 = {}
\begin {gather*} \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx &=-\frac {f^{a+b x+c x^2}}{e (d+e x)}+\frac {(2 c \log (f)) \int f^{a+b x+c x^2} \, dx}{e^2}-\frac {((2 c d-b e) \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}\\ &=-\frac {f^{a+b x+c x^2}}{e (d+e x)}-\frac {((2 c d-b e) \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}+\frac {\left (2 c f^{a-\frac {b^2}{4 c}} \log (f)\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{e^2}\\ &=-\frac {f^{a+b x+c x^2}}{e (d+e x)}+\frac {\sqrt {c} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}}{e^2}-\frac {((2 c d-b e) \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {f^{c \,x^{2}+b x +a}}{\left (e x +d \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a + b x + c x^{2}}}{\left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f^{c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________