3.1415 \(\int \frac{5-x}{(3+2 x)^3 (2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{41 x+26}{70 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{331 \sqrt{3 x^2+2}}{8575 (2 x+3)}+\frac{9 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (9*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2) - (331*Sqrt[2 + 3*x^2])/(
8575*(3 + 2*x)) - (1962*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

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Rubi [A]  time = 0.0566041, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {823, 835, 807, 725, 206} \[ \frac{41 x+26}{70 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{331 \sqrt{3 x^2+2}}{8575 (2 x+3)}+\frac{9 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (9*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2) - (331*Sqrt[2 + 3*x^2])/(
8575*(3 + 2*x)) - (1962*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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 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}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}-\frac{1}{210} \int \frac{-468-492 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}+\frac{\int \frac{12360+1620 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{14700}\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{331 \sqrt{2+3 x^2}}{8575 (3+2 x)}+\frac{1962 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{8575}\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{331 \sqrt{2+3 x^2}}{8575 (3+2 x)}-\frac{1962 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{8575}\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{331 \sqrt{2+3 x^2}}{8575 (3+2 x)}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{8575 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0817641, size = 70, normalized size = 0.67 \[ \frac{-\frac{35 \left (3972 x^3+4068 x^2-7397 x-3658\right )}{(2 x+3)^2 \sqrt{3 x^2+2}}-3924 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{600250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*(-3658 - 7397*x + 4068*x^2 + 3972*x^3))/((3 + 2*x)^2*Sqrt[2 + 3*x^2]) - 3924*Sqrt[35]*ArcTanh[(4 - 9*x)/
(Sqrt[35]*Sqrt[2 + 3*x^2])])/600250

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Maple [A]  time = 0.009, size = 107, normalized size = 1. \begin{align*} -{\frac{103}{980} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{981}{8575}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{993\,x}{17150}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{1962\,\sqrt{35}}{300125}{\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{13}{280} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-103/980/(x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(1/2)+981/8575/(3*(x+3/2)^2-9*x-19/4)^(1/2)-993/17150*x/(3*(x+3/2)^2-9
*x-19/4)^(1/2)-1962/300125*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-13/280/(x+3/2)
^2/(3*(x+3/2)^2-9*x-19/4)^(1/2)

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Maxima [A]  time = 1.50509, size = 173, normalized size = 1.66 \begin{align*} \frac{1962}{300125} \, \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{993 \, x}{17150 \, \sqrt{3 \, x^{2} + 2}} + \frac{981}{8575 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{70 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{103}{490 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1962/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 993/17150*x/sqrt(3*x^2 +
2) + 981/8575/sqrt(3*x^2 + 2) - 13/70/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 10
3/490/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 1.57375, size = 332, normalized size = 3.19 \begin{align*} \frac{1962 \, \sqrt{35}{\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\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 (3972 \, x^{3} + 4068 \, x^{2} - 7397 \, x - 3658\right )} \sqrt{3 \, x^{2} + 2}}{600250 \,{\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/600250*(1962*sqrt(35)*(12*x^4 + 36*x^3 + 35*x^2 + 24*x + 18)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x
^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(3972*x^3 + 4068*x^2 - 7397*x - 3658)*sqrt(3*x^2 + 2))/(12*x^4 + 36*x
^3 + 35*x^2 + 24*x + 18)

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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+2*x)**3/(3*x**2+2)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.27443, size = 269, normalized size = 2.59 \begin{align*} \frac{1962}{300125} \, \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 (157 \, x - 1478\right )}}{85750 \, \sqrt{3 \, x^{2} + 2}} - \frac{768 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 2461 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 6168 \, \sqrt{3} x + 856 \, \sqrt{3} + 6168 \, \sqrt{3 \, x^{2} + 2}}{6125 \,{\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1962/300125*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))) - 3/85750*(157*x - 1478)/sqrt(3*x^2 + 2) - 1/6125*(768*(sqrt(3)*x - sqrt(3*
x^2 + 2))^3 + 2461*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6168*sqrt(3)*x + 856*sqrt(3) + 6168*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)^2