3.24.25 \(\int \frac {(5-x) (3+2 x)^{9/2}}{(2+5 x+3 x^2)^3} \, dx\)

Optimal. Leaf size=115 \[ -\frac {(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {(12473 x+10832) (2 x+3)^{3/2}}{18 \left (3 x^2+5 x+2\right )}-\frac {3983}{9} \sqrt {2 x+3}+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {818, 824, 826, 1166, 207} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {(12473 x+10832) (2 x+3)^{3/2}}{18 \left (3 x^2+5 x+2\right )}-\frac {3983}{9} \sqrt {2 x+3}+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

(-3983*Sqrt[3 + 2*x])/9 - ((3 + 2*x)^(7/2)*(121 + 139*x))/(6*(2 + 5*x + 3*x^2)^2) + ((3 + 2*x)^(3/2)*(10832 +
12473*x))/(18*(2 + 5*x + 3*x^2)) + 1962*ArcTanh[Sqrt[3 + 2*x]] - (13675*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2
*x]])/9

Rule 207

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

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 824

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {1}{6} \int \frac {(3+2 x)^{5/2} (-416+131 x)}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac {1}{18} \int \frac {(5709-11949 x) \sqrt {3+2 x}}{2+5 x+3 x^2} \, dx\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac {1}{54} \int \frac {99177+46203 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {59745+46203 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}-5886 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {68375}{9} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+1962 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 86, normalized size = 0.75 \begin {gather*} \frac {1}{54} \left (-\frac {3 \sqrt {2 x+3} \left (192 x^4-45083 x^3-112467 x^2-90465 x-23327\right )}{\left (3 x^2+5 x+2\right )^2}-27350 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right )+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

1962*ArcTanh[Sqrt[3 + 2*x]] + ((-3*Sqrt[3 + 2*x]*(-23327 - 90465*x - 112467*x^2 - 45083*x^3 + 192*x^4))/(2 + 5
*x + 3*x^2)^2 - 27350*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/54

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IntegrateAlgebraic [A]  time = 0.29, size = 122, normalized size = 1.06 \begin {gather*} \frac {-96 (2 x+3)^{9/2}+46235 (2 x+3)^{7/2}-185997 (2 x+3)^{5/2}+239865 (2 x+3)^{3/2}-99575 \sqrt {2 x+3}}{9 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

(-99575*Sqrt[3 + 2*x] + 239865*(3 + 2*x)^(3/2) - 185997*(3 + 2*x)^(5/2) + 46235*(3 + 2*x)^(7/2) - 96*(3 + 2*x)
^(9/2))/(9*(5 - 8*(3 + 2*x) + 3*(3 + 2*x)^2)^2) + 1962*ArcTanh[Sqrt[3 + 2*x]] - (13675*Sqrt[5/3]*ArcTanh[Sqrt[
3/5]*Sqrt[3 + 2*x]])/9

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fricas [A]  time = 0.41, size = 175, normalized size = 1.52 \begin {gather*} \frac {13675 \, \sqrt {5} \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 52974 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 52974 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 3 \, {\left (192 \, x^{4} - 45083 \, x^{3} - 112467 \, x^{2} - 90465 \, x - 23327\right )} \sqrt {2 \, x + 3}}{54 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="fricas")

[Out]

1/54*(13675*sqrt(5)*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(-(sqrt(5)*sqrt(3)*sqrt(2*x + 3) - 3*x - 7
)/(3*x + 2)) + 52974*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3) + 1) - 52974*(9*x^4 + 30*x^3 + 37*
x^2 + 20*x + 4)*log(sqrt(2*x + 3) - 1) - 3*(192*x^4 - 45083*x^3 - 112467*x^2 - 90465*x - 23327)*sqrt(2*x + 3))
/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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giac [A]  time = 0.20, size = 129, normalized size = 1.12 \begin {gather*} \frac {13675}{54} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {32}{27} \, \sqrt {2 \, x + 3} + \frac {137169 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 554983 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 717035 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 297925 \, \sqrt {2 \, x + 3}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 981 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 981 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

13675/54*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 32/27*sqrt(2*x +
3) + 1/27*(137169*(2*x + 3)^(7/2) - 554983*(2*x + 3)^(5/2) + 717035*(2*x + 3)^(3/2) - 297925*sqrt(2*x + 3))/(3
*(2*x + 3)^2 - 16*x - 19)^2 + 981*log(sqrt(2*x + 3) + 1) - 981*log(abs(sqrt(2*x + 3) - 1))

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maple [A]  time = 0.02, size = 133, normalized size = 1.16 \begin {gather*} -\frac {13675 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{27}-981 \ln \left (-1+\sqrt {2 x +3}\right )+981 \ln \left (\sqrt {2 x +3}+1\right )-\frac {32 \sqrt {2 x +3}}{27}+\frac {\frac {9625 \left (2 x +3\right )^{\frac {3}{2}}}{3}-\frac {165625 \sqrt {2 x +3}}{27}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {104}{\sqrt {2 x +3}+1}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {104}{-1+\sqrt {2 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(2*x+3)^(9/2)/(3*x^2+5*x+2)^3,x)

[Out]

-32/27*(2*x+3)^(1/2)+1250/3*(77/10*(2*x+3)^(3/2)-265/18*(2*x+3)^(1/2))/(6*x+4)^2-13675/27*arctanh(1/5*15^(1/2)
*(2*x+3)^(1/2))*15^(1/2)-3/((2*x+3)^(1/2)+1)^2+104/((2*x+3)^(1/2)+1)+981*ln((2*x+3)^(1/2)+1)+3/(-1+(2*x+3)^(1/
2))^2+104/(-1+(2*x+3)^(1/2))-981*ln(-1+(2*x+3)^(1/2))

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maxima [A]  time = 1.38, size = 143, normalized size = 1.24 \begin {gather*} \frac {13675}{54} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {32}{27} \, \sqrt {2 \, x + 3} + \frac {137169 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 554983 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 717035 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 297925 \, \sqrt {2 \, x + 3}}{27 \, {\left (9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 981 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 981 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="maxima")

[Out]

13675/54*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 32/27*sqrt(2*x + 3) + 1/27
*(137169*(2*x + 3)^(7/2) - 554983*(2*x + 3)^(5/2) + 717035*(2*x + 3)^(3/2) - 297925*sqrt(2*x + 3))/(9*(2*x + 3
)^4 - 48*(2*x + 3)^3 + 94*(2*x + 3)^2 - 160*x - 215) + 981*log(sqrt(2*x + 3) + 1) - 981*log(sqrt(2*x + 3) - 1)

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mupad [B]  time = 2.41, size = 116, normalized size = 1.01 \begin {gather*} \frac {\frac {297925\,\sqrt {2\,x+3}}{243}-\frac {717035\,{\left (2\,x+3\right )}^{3/2}}{243}+\frac {554983\,{\left (2\,x+3\right )}^{5/2}}{243}-\frac {15241\,{\left (2\,x+3\right )}^{7/2}}{27}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {32\,\sqrt {2\,x+3}}{27}-\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,1962{}\mathrm {i}+\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,13675{}\mathrm {i}}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x + 3)^(9/2)*(x - 5))/(5*x + 3*x^2 + 2)^3,x)

[Out]

((297925*(2*x + 3)^(1/2))/243 - (717035*(2*x + 3)^(3/2))/243 + (554983*(2*x + 3)^(5/2))/243 - (15241*(2*x + 3)
^(7/2))/27)/((160*x)/9 - (94*(2*x + 3)^2)/9 + (16*(2*x + 3)^3)/3 - (2*x + 3)^4 + 215/9) - atan((2*x + 3)^(1/2)
*1i)*1962i + (15^(1/2)*atan((15^(1/2)*(2*x + 3)^(1/2)*1i)/5)*13675i)/27 - (32*(2*x + 3)^(1/2))/27

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**(9/2)/(3*x**2+5*x+2)**3,x)

[Out]

Timed out

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