3.4.3 \(\int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx\) [303]

Optimal. Leaf size=28 \[ \frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \]

[Out]

F^a*(d*x+c)^4*Ei(5,-b*ln(F)/(d*x+c))/d

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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250} \begin {gather*} \frac {b^4 F^a \log ^4(F) \text {Gamma}\left (-4,-\frac {b \log (F)}{c+d x}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x))]*Log[F]^4)/d

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx &=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x))]*Log[F]^4)/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(28)=56\).
time = 0.12, size = 368, normalized size = 13.14

method	result	size
risch	\(\frac {d^{3} F^{a} F^{\frac {b}{d x +c}} x^{4}}{4}+d^{2} F^{a} F^{\frac {b}{d x +c}} c \,x^{3}+\frac {3 d \,F^{a} F^{\frac {b}{d x +c}} c^{2} x^{2}}{2}+F^{a} F^{\frac {b}{d x +c}} c^{3} x +\frac {F^{a} F^{\frac {b}{d x +c}} c^{4}}{4 d}+\frac {d^{2} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} x^{3}}{12}+\frac {d b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c \,x^{2}}{4}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{2} x}{4}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{3}}{12 d}+\frac {d \,b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} x^{2}}{24}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c x}{12}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{2}}{24 d}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} x}{24}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c}{24 d}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} \expIntegral \left (1, -\frac {b \ln \left (F \right )}{d x +c}\right )}{24 d}\)	\(368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c))*(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*d^3*F^a*F^(b/(d*x+c))*x^4+d^2*F^a*F^(b/(d*x+c))*c*x^3+3/2*d*F^a*F^(b/(d*x+c))*c^2*x^2+F^a*F^(b/(d*x+c))*c^
3*x+1/4/d*F^a*F^(b/(d*x+c))*c^4+1/12*d^2*b*ln(F)*F^a*F^(b/(d*x+c))*x^3+1/4*d*b*ln(F)*F^a*F^(b/(d*x+c))*c*x^2+1
/4*b*ln(F)*F^a*F^(b/(d*x+c))*c^2*x+1/12/d*b*ln(F)*F^a*F^(b/(d*x+c))*c^3+1/24*d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*x
^2+1/12*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c*x+1/24/d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c^2+1/24*b^3*ln(F)^3*F^a*F^(b/(
d*x+c))*x+1/24/d*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*c+1/24/d*b^4*ln(F)^4*F^a*Ei(1,-b*ln(F)/(d*x+c))

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Maxima [F]
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]

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

[Out]

1/24*(6*F^a*d^3*x^4 + 2*(F^a*b*d^2*log(F) + 12*F^a*c*d^2)*x^3 + (F^a*b^2*d*log(F)^2 + 6*F^a*b*c*d*log(F) + 36*
F^a*c^2*d)*x^2 + (F^a*b^3*log(F)^3 + 2*F^a*b^2*c*log(F)^2 + 6*F^a*b*c^2*log(F) + 24*F^a*c^3)*x)*F^(b/(d*x + c)
) + integrate(1/24*(F^a*b^4*d*x*log(F)^4 - F^a*b^3*c^2*log(F)^3 - 2*F^a*b^2*c^3*log(F)^2 - 6*F^a*b*c^4*log(F))
*F^(b/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (28) = 56\).
time = 0.09, size = 175, normalized size = 6.25 \begin {gather*} -\frac {F^{a} b^{4} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{4} - {\left (6 \, d^{4} x^{4} + 24 \, c d^{3} x^{3} + 36 \, c^{2} d^{2} x^{2} + 24 \, c^{3} d x + 6 \, c^{4} + {\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(F^a*b^4*Ei(b*log(F)/(d*x + c))*log(F)^4 - (6*d^4*x^4 + 24*c*d^3*x^3 + 36*c^2*d^2*x^2 + 24*c^3*d*x + 6*c
^4 + (b^3*d*x + b^3*c)*log(F)^3 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(F)^2 + 2*(b*d^3*x^3 + 3*b*c*d^2*x^
2 + 3*b*c^2*d*x + b*c^3)*log(F))*F^((a*d*x + a*c + b)/(d*x + c)))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c))*(d*x+c)**3,x)

[Out]

Integral(F**(a + b/(c + d*x))*(c + d*x)**3, 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))*(d*x+c)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*F^(a + b/(d*x + c)), x)

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Mupad [B]
time = 3.70, size = 148, normalized size = 5.29 \begin {gather*} \frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^4}{4\,d}+\frac {F^a\,b^4\,{\ln \left (F\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{24\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^2}{24\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}{12\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \left (F\right )}^3\,\left (c+d\,x\right )}{24\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x))*(c + d*x)^3,x)

[Out]

(F^a*F^(b/(c + d*x))*(c + d*x)^4)/(4*d) + (F^a*b^4*log(F)^4*expint(-(b*log(F))/(c + d*x)))/(24*d) + (F^a*F^(b/
(c + d*x))*b^2*log(F)^2*(c + d*x)^2)/(24*d) + (F^a*F^(b/(c + d*x))*b*log(F)*(c + d*x)^3)/(12*d) + (F^a*F^(b/(c
+ d*x))*b^3*log(F)^3*(c + d*x))/(24*d)

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