3.449 \(\int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=207 \[ \frac {\log ^2(f) (2 c d-b e)^2 \text {Int}\left (\frac {f^{a+b x+c x^2}}{d+e x},x\right )}{2 e^4}+\frac {c \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{d+e x},x\right )}{e^2}-\frac {\sqrt {\pi } \sqrt {c} \log ^{\frac {3}{2}}(f) f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{2 e^4}+\frac {\log (f) (2 c d-b e) f^{a+b x+c x^2}}{2 e^3 (d+e x)}-\frac {f^{a+b x+c x^2}}{2 e (d+e x)^2} \]

[Out]

-1/2*f^(c*x^2+b*x+a)/e/(e*x+d)^2+1/2*(-b*e+2*c*d)*f^(c*x^2+b*x+a)*ln(f)/e^3/(e*x+d)-1/2*(-b*e+2*c*d)*f^(a-1/4/
c*b^2)*erfi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*ln(f)^(3/2)*c^(1/2)*Pi^(1/2)/e^4+c*ln(f)*Unintegrable(f^(c*x^2+
b*x+a)/(e*x+d),x)/e^2+1/2*(-b*e+2*c*d)^2*ln(f)^2*Unintegrable(f^(c*x^2+b*x+a)/(e*x+d),x)/e^4

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Rubi [A]  time = 0.21, 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 = {} \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[f^(a + b*x + c*x^2)/(d + e*x)^3,x]

[Out]

-f^(a + b*x + c*x^2)/(2*e*(d + e*x)^2) + ((2*c*d - b*e)*f^(a + b*x + c*x^2)*Log[f])/(2*e^3*(d + e*x)) - (Sqrt[
c]*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Log[f]^(3/2))/(2*e^4)
+ (c*Log[f]*Defer[Int][f^(a + b*x + c*x^2)/(d + e*x), x])/e^2 + ((2*c*d - b*e)^2*Log[f]^2*Defer[Int][f^(a + b
*x + c*x^2)/(d + e*x), x])/(2*e^4)

Rubi steps

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

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Mathematica [A]  time = 0.85, size = 0, normalized size = 0.00 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[f^(a + b*x + c*x^2)/(d + e*x)^3,x]

[Out]

Integrate[f^(a + b*x + c*x^2)/(d + e*x)^3, x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{c x^{2} + b x + a}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral(f^(c*x^2 + b*x + a)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3, x)

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {f^{c \,x^{2}+b x +a}}{\left (e x +d \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x+a)/(e*x+d)^3,x)

[Out]

int(f^(c*x^2+b*x+a)/(e*x+d)^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)/(e*x+d)^3,x, algorithm="maxima")

[Out]

integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f^{c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x + c*x^2)/(d + e*x)^3,x)

[Out]

int(f^(a + b*x + c*x^2)/(d + e*x)^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x + c x^{2}}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+b*x+a)/(e*x+d)**3,x)

[Out]

Integral(f**(a + b*x + c*x**2)/(d + e*x)**3, x)

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