3.4.15 \(\int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^9 \, dx\) [315]

Optimal. Leaf size=31 \[ -\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^2}\right ) \log ^5(F)}{2 d} \]

[Out]

1/2*F^a*(d*x+c)^10*Ei(6,-b*ln(F)/(d*x+c)^2)/d

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Rubi [A]
time = 0.03, antiderivative size = 31, 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^5 F^a \log ^5(F) \text {Gamma}\left (-5,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)*(c + d*x)^9,x]

[Out]

-1/2*(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^2)]*Log[F]^5)/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)^2}} (c+d x)^9 \, dx &=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^2}\right ) \log ^5(F)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} -\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^2}\right ) \log ^5(F)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)*(c + d*x)^9,x]

[Out]

-1/2*(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^2)]*Log[F]^5)/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(960\) vs. \(2(29)=58\).
time = 0.10, size = 961, normalized size = 31.00

method	result	size
risch	\(\frac {F^{a} b^{5} \ln \left (F \right )^{5} \expIntegral \left (1, -\frac {b \ln \left (F \right )}{\left (d x +c \right )^{2}}\right )}{240 d}+\frac {F^{a} d^{6} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{7}}{5}+\frac {7 F^{a} d^{5} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{6}}{10}+\frac {7 F^{a} d^{4} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{5}}{5}+\frac {7 F^{a} d^{2} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x^{3}}{5}+\frac {7 F^{a} d b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6} x^{2}}{10}+\frac {7 F^{a} d^{3} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{4}}{4}+\frac {F^{a} d^{4} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{5}}{20}+\frac {F^{a} d^{3} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{4}}{8}+\frac {F^{a} d^{2} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{3}}{6}+\frac {F^{a} d \,b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{2}}{8}+\frac {F^{a} d^{2} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{3}}{60}+\frac {F^{a} d \,b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{2}}{40}+\frac {9 F^{a} d^{7} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{8}}{2}+12 F^{a} d^{6} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{7}+21 F^{a} d^{5} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{6}+\frac {126 F^{a} d^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x^{5}}{5}+21 F^{a} d^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6} x^{4}+12 F^{a} d^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{7} x^{3}+\frac {9 F^{a} d \,F^{\frac {b}{\left (d x +c \right )^{2}}} c^{8} x^{2}}{2}+F^{a} d^{8} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{9}+F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{9} x +\frac {F^{a} d^{9} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{10}}{10}+\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{10}}{10 d}+\frac {F^{a} d^{3} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{4}}{240}+\frac {F^{a} d \,b^{4} \ln \left (F \right )^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{2}}{240}+\frac {F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{8}}{40 d}+\frac {F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6}}{120 d}+\frac {F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4}}{240 d}+\frac {F^{a} b^{4} \ln \left (F \right )^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2}}{240 d}+\frac {F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x}{20}+\frac {F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x}{60}+\frac {F^{a} b^{4} \ln \left (F \right )^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c x}{120}+\frac {F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{7} x}{5}+\frac {F^{a} d^{7} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} x^{8}}{40}+\frac {F^{a} d^{5} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{6}}{120}\)	\(961\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^2)*(d*x+c)^9,x,method=_RETURNVERBOSE)

[Out]

1/5*F^a*d^6*b*ln(F)*F^(b/(d*x+c)^2)*c*x^7+7/10*F^a*d^5*b*ln(F)*F^(b/(d*x+c)^2)*c^2*x^6+7/5*F^a*d^4*b*ln(F)*F^(
b/(d*x+c)^2)*c^3*x^5+7/5*F^a*d^2*b*ln(F)*F^(b/(d*x+c)^2)*c^5*x^3+7/10*F^a*d*b*ln(F)*F^(b/(d*x+c)^2)*c^6*x^2+7/
4*F^a*d^3*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^4+1/20*F^a*d^4*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^5+1/8*F^a*d^3*b^2*ln(F)
^2*F^(b/(d*x+c)^2)*c^2*x^4+1/6*F^a*d^2*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^3*x^3+1/8*F^a*d*b^2*ln(F)^2*F^(b/(d*x+c)^
2)*c^4*x^2+1/60*F^a*d^2*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x^3+1/40*F^a*d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2*x^2+9/2*F
^a*d^7*F^(b/(d*x+c)^2)*c^2*x^8+12*F^a*d^6*F^(b/(d*x+c)^2)*c^3*x^7+21*F^a*d^5*F^(b/(d*x+c)^2)*c^4*x^6+126/5*F^a
*d^4*F^(b/(d*x+c)^2)*c^5*x^5+21*F^a*d^3*F^(b/(d*x+c)^2)*c^6*x^4+12*F^a*d^2*F^(b/(d*x+c)^2)*c^7*x^3+9/2*F^a*d*F
^(b/(d*x+c)^2)*c^8*x^2+1/240*F^a/d*b^5*ln(F)^5*Ei(1,-b*ln(F)/(d*x+c)^2)+F^a*d^8*F^(b/(d*x+c)^2)*c*x^9+F^a*F^(b
/(d*x+c)^2)*c^9*x+1/10*F^a*d^9*F^(b/(d*x+c)^2)*x^10+1/10*F^a/d*F^(b/(d*x+c)^2)*c^10+1/240*F^a*d^3*b^3*ln(F)^3*
F^(b/(d*x+c)^2)*x^4+1/240*F^a*d*b^4*ln(F)^4*F^(b/(d*x+c)^2)*x^2+1/40*F^a/d*b*ln(F)*F^(b/(d*x+c)^2)*c^8+1/120*F
^a/d*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^6+1/240*F^a/d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^4+1/240*F^a/d*b^4*ln(F)^4*F^(b/
(d*x+c)^2)*c^2+1/20*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^5*x+1/60*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^3*x+1/120*F^a
*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c*x+1/5*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^7*x+1/40*F^a*d^7*b*ln(F)*F^(b/(d*x+c)^2)*x^
8+1/120*F^a*d^5*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^6

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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)^2)*(d*x+c)^9,x, algorithm="maxima")

[Out]

1/240*(24*F^a*d^9*x^10 + 240*F^a*c*d^8*x^9 + 6*(180*F^a*c^2*d^7 + F^a*b*d^7*log(F))*x^8 + 48*(60*F^a*c^3*d^6 +
F^a*b*c*d^6*log(F))*x^7 + 2*(2520*F^a*c^4*d^5 + 84*F^a*b*c^2*d^5*log(F) + F^a*b^2*d^5*log(F)^2)*x^6 + 12*(504
*F^a*c^5*d^4 + 28*F^a*b*c^3*d^4*log(F) + F^a*b^2*c*d^4*log(F)^2)*x^5 + (5040*F^a*c^6*d^3 + 420*F^a*b*c^4*d^3*l
og(F) + 30*F^a*b^2*c^2*d^3*log(F)^2 + F^a*b^3*d^3*log(F)^3)*x^4 + 4*(720*F^a*c^7*d^2 + 84*F^a*b*c^5*d^2*log(F)
+ 10*F^a*b^2*c^3*d^2*log(F)^2 + F^a*b^3*c*d^2*log(F)^3)*x^3 + (1080*F^a*c^8*d + 168*F^a*b*c^6*d*log(F) + 30*F
^a*b^2*c^4*d*log(F)^2 + 6*F^a*b^3*c^2*d*log(F)^3 + F^a*b^4*d*log(F)^4)*x^2 + 2*(120*F^a*c^9 + 24*F^a*b*c^7*log
(F) + 6*F^a*b^2*c^5*log(F)^2 + 2*F^a*b^3*c^3*log(F)^3 + F^a*b^4*c*log(F)^4)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))
+ integrate(1/120*(F^a*b^5*d^2*x^2*log(F)^5 + 2*F^a*b^5*c*d*x*log(F)^5 - 24*F^a*b*c^10*log(F) - 6*F^a*b^2*c^8
*log(F)^2 - 2*F^a*b^3*c^6*log(F)^3 - F^a*b^4*c^4*log(F)^4)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*
x^2 + 3*c^2*d*x + c^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (29) = 58\).
time = 0.10, size = 465, normalized size = 15.00 \begin {gather*} -\frac {F^{a} b^{5} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \left (F\right )^{5} - {\left (24 \, d^{10} x^{10} + 240 \, c d^{9} x^{9} + 1080 \, c^{2} d^{8} x^{8} + 2880 \, c^{3} d^{7} x^{7} + 5040 \, c^{4} d^{6} x^{6} + 6048 \, c^{5} d^{5} x^{5} + 5040 \, c^{6} d^{4} x^{4} + 2880 \, c^{7} d^{3} x^{3} + 1080 \, c^{8} d^{2} x^{2} + 240 \, c^{9} d x + 24 \, c^{10} + {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \left (F\right )^{4} + {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \left (F\right )^{3} + 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)*(d*x+c)^9,x, algorithm="fricas")

[Out]

-1/240*(F^a*b^5*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))*log(F)^5 - (24*d^10*x^10 + 240*c*d^9*x^9 + 1080*c^2*d^8
*x^8 + 2880*c^3*d^7*x^7 + 5040*c^4*d^6*x^6 + 6048*c^5*d^5*x^5 + 5040*c^6*d^4*x^4 + 2880*c^7*d^3*x^3 + 1080*c^8
*d^2*x^2 + 240*c^9*d*x + 24*c^10 + (b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)*log(F)^4 + (b^3*d^4*x^4 + 4*b^3*c*d^3
*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*log(F)^3 + 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d
^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 + 6*(b*d^8*x^8 + 8*b*c*d^
7*x^7 + 28*b*c^2*d^6*x^6 + 56*b*c^3*d^5*x^5 + 70*b*c^4*d^4*x^4 + 56*b*c^5*d^3*x^3 + 28*b*c^6*d^2*x^2 + 8*b*c^7
*d*x + b*c^8)*log(F))*F^((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{9}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**2)*(d*x+c)**9,x)

[Out]

Integral(F**(a + b/(c + d*x)**2)*(c + d*x)**9, 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)^2)*(d*x+c)^9,x, algorithm="giac")

[Out]

integrate((d*x + c)^9*F^(a + b/(d*x + c)^2), x)

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Mupad [B]
time = 3.95, size = 136, normalized size = 4.39 \begin {gather*} \frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}{240\,d}+\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^5\,{\ln \left (F\right )}^5\,\left (\frac {{\left (c+d\,x\right )}^2}{120\,b\,\ln \left (F\right )}+\frac {{\left (c+d\,x\right )}^4}{120\,b^2\,{\ln \left (F\right )}^2}+\frac {{\left (c+d\,x\right )}^6}{60\,b^3\,{\ln \left (F\right )}^3}+\frac {{\left (c+d\,x\right )}^8}{20\,b^4\,{\ln \left (F\right )}^4}+\frac {{\left (c+d\,x\right )}^{10}}{5\,b^5\,{\ln \left (F\right )}^5}\right )}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^2)*(c + d*x)^9,x)

[Out]

(F^a*b^5*log(F)^5*expint(-(b*log(F))/(c + d*x)^2))/(240*d) + (F^a*F^(b/(c + d*x)^2)*b^5*log(F)^5*((c + d*x)^2/
(120*b*log(F)) + (c + d*x)^4/(120*b^2*log(F)^2) + (c + d*x)^6/(60*b^3*log(F)^3) + (c + d*x)^8/(20*b^4*log(F)^4
) + (c + d*x)^10/(5*b^5*log(F)^5)))/(2*d)

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