3.19.67 \(\int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx\) [1867]

Optimal. Leaf size=88 \[ \frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {104 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \]

[Out]

1/63*(1-2*x)^(5/2)/(2+3*x)^3-52/189*(1-2*x)^(3/2)/(2+3*x)^2-104/3969*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1
/2)+52/189*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 43, 65, 212} \begin {gather*} \frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}-\frac {52 (1-2 x)^{3/2}}{189 (3 x+2)^2}+\frac {52 \sqrt {1-2 x}}{189 (3 x+2)}-\frac {104 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(1 - 2*x)^(5/2)/(63*(2 + 3*x)^3) - (52*(1 - 2*x)^(3/2))/(189*(2 + 3*x)^2) + (52*Sqrt[1 - 2*x])/(189*(2 + 3*x))
- (104*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqrt[21])

Rule 43

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

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 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)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}+\frac {104}{63} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}-\frac {52}{63} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}+\frac {52}{189} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {52}{189} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {104 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 60, normalized size = 0.68 \begin {gather*} \frac {8 \left (\frac {21 \sqrt {1-2 x} \left (107+664 x+792 x^2\right )}{8 (2+3 x)^3}-13 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{3969} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(8*((21*Sqrt[1 - 2*x]*(107 + 664*x + 792*x^2))/(8*(2 + 3*x)^3) - 13*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])
)/3969

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Maple [A]
time = 0.12, size = 57, normalized size = 0.65

method	result	size
risch	\(-\frac {1584 x^{3}+536 x^{2}-450 x -107}{189 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {104 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\)	\(51\)
derivativedivides	\(\frac {-\frac {176 \left (1-2 x \right )^{\frac {5}{2}}}{21}+\frac {832 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {728 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{3}}-\frac {104 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\)	\(57\)
default	\(\frac {-\frac {176 \left (1-2 x \right )^{\frac {5}{2}}}{21}+\frac {832 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {728 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{3}}-\frac {104 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\)	\(57\)
trager	\(\frac {\left (792 x^{2}+664 x +107\right ) \sqrt {1-2 x}}{189 \left (2+3 x \right )^{3}}+\frac {52 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3969}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

216*(-22/567*(1-2*x)^(5/2)+104/729*(1-2*x)^(3/2)-91/729*(1-2*x)^(1/2))/(-4-6*x)^3-104/3969*arctanh(1/7*21^(1/2
)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.55, size = 92, normalized size = 1.05 \begin {gather*} \frac {52}{3969} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8 \, {\left (198 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 728 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 637 \, \sqrt {-2 \, x + 1}\right )}}{189 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x, algorithm="maxima")

[Out]

52/3969*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 8/189*(198*(-2*x + 1)^(5/
2) - 728*(-2*x + 1)^(3/2) + 637*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]
time = 0.68, size = 84, normalized size = 0.95 \begin {gather*} \frac {52 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (792 \, x^{2} + 664 \, x + 107\right )} \sqrt {-2 \, x + 1}}{3969 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/3969*(52*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(792*
x^2 + 664*x + 107)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**(3/2)*(3+5*x)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [A]
time = 3.43, size = 84, normalized size = 0.95 \begin {gather*} \frac {52}{3969} \, \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 {198 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 728 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 637 \, \sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x, algorithm="giac")

[Out]

52/3969*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/189*(198*(2*x
- 1)^2*sqrt(-2*x + 1) - 728*(-2*x + 1)^(3/2) + 637*sqrt(-2*x + 1))/(3*x + 2)^3

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Mupad [B]
time = 0.07, size = 71, normalized size = 0.81 \begin {gather*} \frac {\frac {728\,\sqrt {1-2\,x}}{729}-\frac {832\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {176\,{\left (1-2\,x\right )}^{5/2}}{567}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {104\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3969} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3))/(3*x + 2)^4,x)

[Out]

((728*(1 - 2*x)^(1/2))/729 - (832*(1 - 2*x)^(3/2))/729 + (176*(1 - 2*x)^(5/2))/567)/((98*x)/3 + 7*(2*x - 1)^2
+ (2*x - 1)^3 - 98/27) - (104*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/3969

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