3.18.99 \(\int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx\) [1799]

Optimal. Leaf size=68 \[ \frac {(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac {23 \sqrt {1-2 x}}{42 (2+3 x)}+\frac {23 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]

[Out]

1/42*(1-2*x)^(3/2)/(2+3*x)^2+23/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-23/42*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, 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)^{3/2}}{42 (3 x+2)^2}-\frac {23 \sqrt {1-2 x}}{42 (3 x+2)}+\frac {23 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(3/2)/(42*(2 + 3*x)^2) - (23*Sqrt[1 - 2*x])/(42*(2 + 3*x)) + (23*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
21*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 {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx &=\frac {(1-2 x)^{3/2}}{42 (2+3 x)^2}+\frac {23}{14} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac {23 \sqrt {1-2 x}}{42 (2+3 x)}-\frac {23}{42} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac {23 \sqrt {1-2 x}}{42 (2+3 x)}+\frac {23}{42} \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)^{3/2}}{42 (2+3 x)^2}-\frac {23 \sqrt {1-2 x}}{42 (2+3 x)}+\frac {23 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 53, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-2 x} (45+71 x)}{42 (2+3 x)^2}+\frac {23 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/42*(Sqrt[1 - 2*x]*(45 + 71*x))/(2 + 3*x)^2 + (23*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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Maple [A]
time = 0.11, size = 48, normalized size = 0.71

method	result	size
risch	\(\frac {142 x^{2}+19 x -45}{42 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {23 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\)	\(46\)
derivativedivides	\(-\frac {36 \left (-\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{756}+\frac {23 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {23 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\)	\(48\)
default	\(-\frac {36 \left (-\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{756}+\frac {23 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {23 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\)	\(48\)
trager	\(-\frac {\left (71 x +45\right ) \sqrt {1-2 x}}{42 \left (2+3 x \right )^{2}}+\frac {23 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{882}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-36*(-71/756*(1-2*x)^(3/2)+23/108*(1-2*x)^(1/2))/(-4-6*x)^2+23/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2
)

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Maxima [A]
time = 0.57, size = 74, normalized size = 1.09 \begin {gather*} -\frac {23}{882} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {71 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 161 \, \sqrt {-2 \, x + 1}}{21 \, {\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((3+5*x)*(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="maxima")

[Out]

-23/882*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21*(71*(-2*x + 1)^(3/2)
- 161*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]
time = 1.35, size = 70, normalized size = 1.03 \begin {gather*} \frac {23 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (71 \, x + 45\right )} \sqrt {-2 \, x + 1}}{882 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/882*(23*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(71*x + 45)*sqrt
(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (56) = 112\).
time = 133.51, size = 350, normalized size = 5.15 \begin {gather*} - \frac {148 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} - \frac {56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} - \frac {20 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-148*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(s
qrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sq
rt(1 - 2*x) < sqrt(21)/3)))/9 - 56*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21
)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) +
3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqr
t(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/9 - 20*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, x <
-2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, x > -2/3))/9

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Giac [A]
time = 1.05, size = 68, normalized size = 1.00 \begin {gather*} -\frac {23}{882} \, \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 {71 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 161 \, \sqrt {-2 \, x + 1}}{84 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23/882*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/84*(71*(-2*x +
1)^(3/2) - 161*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.06, size = 54, normalized size = 0.79 \begin {gather*} \frac {23\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-\frac {\frac {23\,\sqrt {1-2\,x}}{27}-\frac {71\,{\left (1-2\,x\right )}^{3/2}}{189}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(23*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/441 - ((23*(1 - 2*x)^(1/2))/27 - (71*(1 - 2*x)^(3/2))/189)/(
(28*x)/3 + (2*x - 1)^2 + 7/9)

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