3.20.49 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx\) [1949]

Optimal. Leaf size=109 \[ \frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {1055}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]

[Out]

1055/567*(1-2*x)^(3/2)+211/441*(1-2*x)^(5/2)-1/126*(1-2*x)^(7/2)/(2+3*x)^2+143/882*(1-2*x)^(7/2)/(2+3*x)-1055/
243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1055/81*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 = {91, 79, 52, 65, 212} \begin {gather*} \frac {143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac {211}{441} (1-2 x)^{5/2}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {1055}{81} \sqrt {1-2 x}-\frac {1055}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^3,x]

[Out]

(1055*Sqrt[1 - 2*x])/81 + (1055*(1 - 2*x)^(3/2))/567 + (211*(1 - 2*x)^(5/2))/441 - (1 - 2*x)^(7/2)/(126*(2 + 3
*x)^2) + (143*(1 - 2*x)^(7/2))/(882*(2 + 3*x)) - (1055*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 212

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

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {(1-2 x)^{5/2} (557+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{294} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{126} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{54} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {7385}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {7385}{162} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {1055}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 68, normalized size = 0.62 \begin {gather*} \frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (10007+25987 x+12828 x^2-3960 x^3+2160 x^4\right )}{(2+3 x)^2}-2110 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^3,x]

[Out]

((3*Sqrt[1 - 2*x]*(10007 + 25987*x + 12828*x^2 - 3960*x^3 + 2160*x^4))/(2 + 3*x)^2 - 2110*Sqrt[21]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/486

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Maple [A]
time = 0.14, size = 75, normalized size = 0.69

method	result	size
risch	\(-\frac {4320 x^{5}-10080 x^{4}+29616 x^{3}+39146 x^{2}-5973 x -10007}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {1055 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\)	\(61\)
derivativedivides	\(\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {130 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {1-2 x}}{81}+\frac {-\frac {1043 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {1055 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\)	\(75\)
default	\(\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {130 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {1-2 x}}{81}+\frac {-\frac {1043 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {1055 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\)	\(75\)
trager	\(\frac {\left (2160 x^{4}-3960 x^{3}+12828 x^{2}+25987 x +10007\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}-\frac {1055 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{486}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^3,x,method=_RETURNVERBOSE)

[Out]

10/27*(1-2*x)^(5/2)+130/81*(1-2*x)^(3/2)+1006/81*(1-2*x)^(1/2)+14/9*(-149/18*(1-2*x)^(3/2)+343/18*(1-2*x)^(1/2
))/(-4-6*x)^2-1055/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.49, size = 101, normalized size = 0.93 \begin {gather*} \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/27*(-2*x + 1)^(5/2) + 130/81*(-2*x + 1)^(3/2) + 1055/486*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) + 1006/81*sqrt(-2*x + 1) - 7/81*(149*(-2*x + 1)^(3/2) - 343*sqrt(-2*x + 1))/(9*(2*x -
1)^2 + 84*x + 7)

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Fricas [A]
time = 1.68, size = 90, normalized size = 0.83 \begin {gather*} \frac {1055 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \, {\left (2160 \, x^{4} - 3960 \, x^{3} + 12828 \, x^{2} + 25987 \, x + 10007\right )} \sqrt {-2 \, x + 1}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*(1055*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) + 3*(
2160*x^4 - 3960*x^3 + 12828*x^2 + 25987*x + 10007)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [F(-1)] Timed out
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((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.64, size = 102, normalized size = 0.94 \begin {gather*} \frac {10}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{324 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/27*(2*x - 1)^2*sqrt(-2*x + 1) + 130/81*(-2*x + 1)^(3/2) + 1055/486*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqr
t(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1006/81*sqrt(-2*x + 1) - 7/324*(149*(-2*x + 1)^(3/2) - 343*sqrt(
-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.06, size = 82, normalized size = 0.75 \begin {gather*} \frac {1006\,\sqrt {1-2\,x}}{81}+\frac {130\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {10\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {\frac {2401\,\sqrt {1-2\,x}}{729}-\frac {1043\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,1055{}\mathrm {i}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*1055i)/243 + (1006*(1 - 2*x)^(1/2))/81 + (130*(1 - 2*x)^(3/2))
/81 + (10*(1 - 2*x)^(5/2))/27 + ((2401*(1 - 2*x)^(1/2))/729 - (1043*(1 - 2*x)^(3/2))/729)/((28*x)/3 + (2*x - 1
)^2 + 7/9)

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