3.64 \(\int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx\)

Optimal. Leaf size=139 \[ -\frac{2 e f x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+2,-b d x \log (F))}{b^2 d^2 \log ^2(F)}+\frac{f^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+3,-b d x \log (F))}{b^3 d^3 \log ^3(F)}+\frac{e^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+1,-b d x \log (F))}{b d \log (F)} \]

[Out]

(f^2*F^(a + b*c)*x^m*Gamma[3 + m, -(b*d*x*Log[F])])/(b^3*d^3*Log[F]^3*(-(b*d*x*Log[F]))^m) - (2*e*f*F^(a + b*c
)*x^m*Gamma[2 + m, -(b*d*x*Log[F])])/(b^2*d^2*Log[F]^2*(-(b*d*x*Log[F]))^m) + (e^2*F^(a + b*c)*x^m*Gamma[1 + m
, -(b*d*x*Log[F])])/(b*d*Log[F]*(-(b*d*x*Log[F]))^m)

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Rubi [A]  time = 0.309197, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2199, 2181} \[ -\frac{2 e f x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+2,-b d x \log (F))}{b^2 d^2 \log ^2(F)}+\frac{f^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+3,-b d x \log (F))}{b^3 d^3 \log ^3(F)}+\frac{e^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+1,-b d x \log (F))}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^2*F^(a + b*c)*x^m*Gamma[3 + m, -(b*d*x*Log[F])])/(b^3*d^3*Log[F]^3*(-(b*d*x*Log[F]))^m) - (2*e*f*F^(a + b*c
)*x^m*Gamma[2 + m, -(b*d*x*Log[F])])/(b^2*d^2*Log[F]^2*(-(b*d*x*Log[F]))^m) + (e^2*F^(a + b*c)*x^m*Gamma[1 + m
, -(b*d*x*Log[F])])/(b*d*Log[F]*(-(b*d*x*Log[F]))^m)

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx &=\int \left (e^2 F^{a+b c+b d x} x^m+2 e f F^{a+b c+b d x} x^{1+m}+f^2 F^{a+b c+b d x} x^{2+m}\right ) \, dx\\ &=e^2 \int F^{a+b c+b d x} x^m \, dx+(2 e f) \int F^{a+b c+b d x} x^{1+m} \, dx+f^2 \int F^{a+b c+b d x} x^{2+m} \, dx\\ &=\frac{f^2 F^{a+b c} x^m \Gamma (3+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b^3 d^3 \log ^3(F)}-\frac{2 e f F^{a+b c} x^m \Gamma (2+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c} x^m \Gamma (1+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b d \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.124451, size = 86, normalized size = 0.62 \[ \frac{x^m F^{a+b c} (-b d x \log (F))^{-m} \left (b d e \log (F) (b d e \log (F) \text{Gamma}(m+1,-b d x \log (F))-2 f \text{Gamma}(m+2,-b d x \log (F)))+f^2 \text{Gamma}(m+3,-b d x \log (F))\right )}{b^3 d^3 \log ^3(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a + b*c)*x^m*(f^2*Gamma[3 + m, -(b*d*x*Log[F])] + b*d*e*Log[F]*(-2*f*Gamma[2 + m, -(b*d*x*Log[F])] + b*d*e
*Gamma[1 + m, -(b*d*x*Log[F])]*Log[F])))/(b^3*d^3*Log[F]^3*(-(b*d*x*Log[F]))^m)

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Maple [B]  time = 0.097, size = 433, normalized size = 3.1 \begin{align*} -{\frac{ \left ( \ln \left ( F \right ) \right ) ^{-3-m} \left ( -bd \right ) ^{-m}{F}^{bc+a}{f}^{2} \left ({x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m \left ({m}^{2}+3\,m+2 \right ) \Gamma \left ( m \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ({b}^{2}{d}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-mbdx\ln \left ( F \right ) +{m}^{2}-2\,bdx\ln \left ( F \right ) +3\,m+2 \right ){{\rm e}^{bdx\ln \left ( F \right ) }}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m \left ({m}^{2}+3\,m+2 \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}\Gamma \left ( m,-bdx\ln \left ( F \right ) \right ) \right ) }{{b}^{3}{d}^{3}}}+2\,{\frac{ \left ( \ln \left ( F \right ) \right ) ^{-2-m} \left ( -bd \right ) ^{-m}{F}^{bc+a}fe \left ({x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ( 1+m \right ) m\Gamma \left ( m \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}+{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ( bdx\ln \left ( F \right ) -1-m \right ){{\rm e}^{bdx\ln \left ( F \right ) }}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ( 1+m \right ) m \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}\Gamma \left ( m,-bdx\ln \left ( F \right ) \right ) \right ) }{{b}^{2}{d}^{2}}}-{\frac{{F}^{bc+a} \left ( -bd \right ) ^{-m} \left ( \ln \left ( F \right ) \right ) ^{-m-1}{e}^{2} \left ({x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m\Gamma \left ( m \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}{{\rm e}^{bdx\ln \left ( F \right ) }}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}\Gamma \left ( m,-bdx\ln \left ( F \right ) \right ) \right ) }{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c))*x^m*(f*x+e)^2,x)

[Out]

-1/b^3/d^3*ln(F)^(-3-m)*(-b*d)^(-m)*F^(b*c+a)*f^2*(x^m*(-b*d)^m*ln(F)^m*m*(m^2+3*m+2)*GAMMA(m)*(-b*d*x*ln(F))^
(-m)-x^m*(-b*d)^m*ln(F)^m*(b^2*d^2*x^2*ln(F)^2-m*b*d*x*ln(F)+m^2-2*b*d*x*ln(F)+3*m+2)*exp(b*d*x*ln(F))-x^m*(-b
*d)^m*ln(F)^m*m*(m^2+3*m+2)*(-b*d*x*ln(F))^(-m)*GAMMA(m,-b*d*x*ln(F)))+2/b^2/d^2*ln(F)^(-2-m)*(-b*d)^(-m)*F^(b
*c+a)*f*e*(x^m*(-b*d)^m*ln(F)^m*(1+m)*m*GAMMA(m)*(-b*d*x*ln(F))^(-m)+x^m*(-b*d)^m*ln(F)^m*(b*d*x*ln(F)-1-m)*ex
p(b*d*x*ln(F))-x^m*(-b*d)^m*ln(F)^m*(1+m)*m*(-b*d*x*ln(F))^(-m)*GAMMA(m,-b*d*x*ln(F)))-F^(b*c+a)*(-b*d)^(-m)*l
n(F)^(-m-1)*e^2/b/d*(x^m*(-b*d)^m*ln(F)^m*m*GAMMA(m)*(-b*d*x*ln(F))^(-m)-x^m*(-b*d)^m*ln(F)^m*exp(b*d*x*ln(F))
-x^m*(-b*d)^m*ln(F)^m*m*(-b*d*x*ln(F))^(-m)*GAMMA(m,-b*d*x*ln(F)))

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Maxima [A]  time = 1.27436, size = 166, normalized size = 1.19 \begin{align*} -\left (-b d x \log \left (F\right )\right )^{-m - 3} F^{b c + a} f^{2} x^{m + 3} \Gamma \left (m + 3, -b d x \log \left (F\right )\right ) - 2 \, \left (-b d x \log \left (F\right )\right )^{-m - 2} F^{b c + a} e f x^{m + 2} \Gamma \left (m + 2, -b d x \log \left (F\right )\right ) - \left (-b d x \log \left (F\right )\right )^{-m - 1} F^{b c + a} e^{2} x^{m + 1} \Gamma \left (m + 1, -b d x \log \left (F\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b*d*x*log(F))^(-m - 3)*F^(b*c + a)*f^2*x^(m + 3)*gamma(m + 3, -b*d*x*log(F)) - 2*(-b*d*x*log(F))^(-m - 2)*F
^(b*c + a)*e*f*x^(m + 2)*gamma(m + 2, -b*d*x*log(F)) - (-b*d*x*log(F))^(-m - 1)*F^(b*c + a)*e^2*x^(m + 1)*gamm
a(m + 1, -b*d*x*log(F))

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Fricas [A]  time = 1.60851, size = 386, normalized size = 2.78 \begin{align*} -\frac{{\left ({\left (b d f^{2} m + 2 \, b d f^{2}\right )} x \log \left (F\right ) -{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} d^{2} e f x\right )} \log \left (F\right )^{2}\right )} F^{b d x + b c + a} x^{m} -{\left (b^{2} d^{2} e^{2} \log \left (F\right )^{2} + f^{2} m^{2} + 3 \, f^{2} m + 2 \, f^{2} - 2 \,{\left (b d e f m + b d e f\right )} \log \left (F\right )\right )} e^{\left (-m \log \left (-b d \log \left (F\right )\right ) +{\left (b c + a\right )} \log \left (F\right )\right )} \Gamma \left (m + 1, -b d x \log \left (F\right )\right )}{b^{3} d^{3} \log \left (F\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(((b*d*f^2*m + 2*b*d*f^2)*x*log(F) - (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x)*log(F)^2)*F^(b*d*x + b*c + a)*x^m -
(b^2*d^2*e^2*log(F)^2 + f^2*m^2 + 3*f^2*m + 2*f^2 - 2*(b*d*e*f*m + b*d*e*f)*log(F))*e^(-m*log(-b*d*log(F)) + (
b*c + a)*log(F))*gamma(m + 1, -b*d*x*log(F)))/(b^3*d^3*log(F)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + b \left (c + d x\right )} x^{m} \left (e + f x\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c))*x**m*(f*x+e)**2,x)

[Out]

Integral(F**(a + b*(c + d*x))*x**m*(e + f*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x + e)^2*F^((d*x + c)*b + a)*x^m, x)