3.1.64 \(\int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx\) [64]

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

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 = {2230, 2212} \begin {gather*} \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 {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 {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)} \end {gather*}

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 2212

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

Rule 2230

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] &&  !TrueQ[$UseGamma]

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*}

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Mathematica [A]
time = 0.24, size = 86, normalized size = 0.62 \begin {gather*} \frac {F^{a+b c} x^m (-b d x \log (F))^{-m} \left (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))\right )}{b^3 d^3 \log ^3(F)} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(432\) vs. \(2(139)=278\).
time = 0.10, size = 433, normalized size = 3.12

method	result	size
meijerg	\(-\frac {\ln \left (F \right )^{-3-m} \left (-b d \right )^{-m} F^{c b +a} f^{2} \left (x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \left (m^{2}+3 m +2\right ) \Gamma \left (m \right ) \left (-b d x \ln \left (F \right )\right )^{-m}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-m b d x \ln \left (F \right )+m^{2}-2 b d x \ln \left (F \right )+3 m +2\right ) {\mathrm e}^{b d x \ln \left (F \right )}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \left (m^{2}+3 m +2\right ) \left (-b d x \ln \left (F \right )\right )^{-m} \Gamma \left (m , -b d x \ln \left (F \right )\right )\right )}{b^{3} d^{3}}+\frac {2 \ln \left (F \right )^{-2-m} \left (-b d \right )^{-m} F^{c b +a} f e \left (x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (1+m \right ) m \Gamma \left (m \right ) \left (-b d x \ln \left (F \right )\right )^{-m}+x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (b d x \ln \left (F \right )-1-m \right ) {\mathrm e}^{b d x \ln \left (F \right )}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (1+m \right ) m \left (-b d x \ln \left (F \right )\right )^{-m} \Gamma \left (m , -b d x \ln \left (F \right )\right )\right )}{b^{2} d^{2}}-\frac {F^{c b +a} \left (-b d \right )^{-m} \ln \left (F \right )^{-m -1} e^{2} \left (x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \Gamma \left (m \right ) \left (-b d x \ln \left (F \right )\right )^{-m}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} {\mathrm e}^{b d x \ln \left (F \right )}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \left (-b d x \ln \left (F \right )\right )^{-m} \Gamma \left (m , -b d x \ln \left (F \right )\right )\right )}{b d}\)	\(433\)

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,method=_RETURNVERBOSE)

[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 = 0.18, size = 123, normalized size = 0.88 \begin {gather*} -\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} f x^{m + 2} e \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} x^{m + 1} e^{2} \Gamma \left (m + 1, -b d x \log \left (F\right )\right ) \end {gather*}

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

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Fricas [A]
time = 0.11, size = 161, normalized size = 1.16 \begin {gather*} -\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} f x e\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 \, {\left (b d f m + b d f\right )} e \log \left (F\right ) + 2 \, f^{2}\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 {gather*}

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*f*x*e)*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*(b*d*f*m + b*d*f)*e*log(F) + 2*f^2)*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + b \left (c + d x\right )} x^{m} \left (e + f x\right )^{2}\, dx \end {gather*}

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 = 0.00, 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]

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{a+b\,\left (c+d\,x\right )}\,x^m\,{\left (e+f\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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