3.1367 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=148 \[ -\frac{1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac{111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac{281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac{1017 (4-9 x) \sqrt{3 x^2+2}}{7503125 (2 x+3)^2}-\frac{6102 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

[Out]

(-1017*(4 - 9*x)*Sqrt[2 + 3*x^2])/(7503125*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(210*(3 + 2*x)^6) - (281*(2 +
3*x^2)^(3/2))/(12250*(3 + 2*x)^5) - (111*(2 + 3*x^2)^(3/2))/(17500*(3 + 2*x)^4) - (1207*(2 + 3*x^2)^(3/2))/(8
57500*(3 + 2*x)^3) - (6102*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*Sqrt[35])

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Rubi [A]  time = 0.0919153, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {835, 807, 721, 725, 206} \[ -\frac{1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac{111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac{281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac{1017 (4-9 x) \sqrt{3 x^2+2}}{7503125 (2 x+3)^2}-\frac{6102 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^7,x]

[Out]

(-1017*(4 - 9*x)*Sqrt[2 + 3*x^2])/(7503125*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(210*(3 + 2*x)^6) - (281*(2 +
3*x^2)^(3/2))/(12250*(3 + 2*x)^5) - (111*(2 + 3*x^2)^(3/2))/(17500*(3 + 2*x)^4) - (1207*(2 + 3*x^2)^(3/2))/(8
57500*(3 + 2*x)^3) - (6102*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*Sqrt[35])

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^7} \, dx &=-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{1}{210} \int \frac{(-246+117 x) \sqrt{2+3 x^2}}{(3+2 x)^6} \, dx\\ &=-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}+\frac{\int \frac{(8730-5058 x) \sqrt{2+3 x^2}}{(3+2 x)^5} \, dx}{36750}\\ &=-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac{111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac{\int \frac{(-233352+97902 x) \sqrt{2+3 x^2}}{(3+2 x)^4} \, dx}{5145000}\\ &=-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac{111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac{1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}+\frac{2034 \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{214375}\\ &=-\frac{1017 (4-9 x) \sqrt{2+3 x^2}}{7503125 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac{111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac{1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}+\frac{6102 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{7503125}\\ &=-\frac{1017 (4-9 x) \sqrt{2+3 x^2}}{7503125 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac{111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac{1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}-\frac{6102 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{7503125}\\ &=-\frac{1017 (4-9 x) \sqrt{2+3 x^2}}{7503125 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac{281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac{111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac{1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}-\frac{6102 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{7503125 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.089216, size = 80, normalized size = 0.54 \[ \frac{-\frac{35 \sqrt{3 x^2+2} \left (642132 x^5+5388660 x^4+18236055 x^3+30753930 x^2+18651300 x+22308548\right )}{(2 x+3)^6}-36612 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1575656250} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^7,x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(22308548 + 18651300*x + 30753930*x^2 + 18236055*x^3 + 5388660*x^4 + 642132*x^5))/(3 + 2
*x)^6 - 36612*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/1575656250

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Maple [A]  time = 0.014, size = 191, normalized size = 1.3 \begin{align*} -{\frac{281}{392000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{111}{280000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1207}{6860000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1017}{15006250} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{9153}{262609375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{6102}{262609375}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{6102\,\sqrt{35}}{262609375}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{27459\,x}{262609375}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{13}{13440} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x)

[Out]

-281/392000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(3/2)-111/280000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(3/2)-1207/6860
000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(3/2)-1017/15006250/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(3/2)-9153/262609375
/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(3/2)+6102/262609375*(12*(x+3/2)^2-36*x-19)^(1/2)-6102/262609375*35^(1/2)*arct
anh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+27459/262609375*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-13/1344
0/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 1.54856, size = 309, normalized size = 2.09 \begin{align*} \frac{6102}{262609375} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{3051}{15006250} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{210 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{281 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{12250 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{111 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{17500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1207 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{857500 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2034 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{7503125 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{9153 \, \sqrt{3 \, x^{2} + 2}}{15006250 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x, algorithm="maxima")

[Out]

6102/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 3051/15006250*sqrt(3*
x^2 + 2) - 13/210*(3*x^2 + 2)^(3/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 281/1
2250*(3*x^2 + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 111/17500*(3*x^2 + 2)^(3/2)/(16
*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1207/857500*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2034/7503
125*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 9153/15006250*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 2.28851, size = 473, normalized size = 3.2 \begin{align*} \frac{18306 \, \sqrt{35}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \,{\left (642132 \, x^{5} + 5388660 \, x^{4} + 18236055 \, x^{3} + 30753930 \, x^{2} + 18651300 \, x + 22308548\right )} \sqrt{3 \, x^{2} + 2}}{1575656250 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x, algorithm="fricas")

[Out]

1/1575656250*(18306*sqrt(35)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-(sqrt(35)
*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(642132*x^5 + 5388660*x^4 + 18236055
*x^3 + 30753930*x^2 + 18651300*x + 22308548)*sqrt(3*x^2 + 2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x
^2 + 2916*x + 729)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.21101, size = 490, normalized size = 3.31 \begin{align*} \frac{6102}{262609375} \, \sqrt{35} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{3 \,{\left (65088 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 1073952 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} + 20936640 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 87678735 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 199001970 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 258582989 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1280293308 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 755892540 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 1065400320 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 207134880 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 59561856 \, \sqrt{3} x + 2283136 \, \sqrt{3} + 59561856 \, \sqrt{3 \, x^{2} + 2}\right )}}{240100000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x, algorithm="giac")

[Out]

6102/262609375*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(
35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3/240100000*(65088*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 1073952*sqrt(3)*
(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 20936640*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 87678735*sqrt(3)*(sqrt(3)*x - sq
rt(3*x^2 + 2))^8 - 199001970*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 258582989*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))
^6 - 1280293308*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 755892540*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 10654003
20*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 207134880*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 59561856*sqrt(3)*x +
2283136*sqrt(3) + 59561856*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2)) - 2)^6