3.21.8 \(\int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx\) [2008]

Optimal. Leaf size=48 \[ \frac {\sqrt {1-2 x}}{21 (2+3 x)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]

[Out]

-68/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/21*(1-2*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {79, 65, 212} \begin {gather*} \frac {\sqrt {1-2 x}}{21 (3 x+2)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(21*(2 + 3*x)) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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 {3+5 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{21 (2+3 x)}+\frac {34}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{21 (2+3 x)}-\frac {34}{21} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{21 (2+3 x)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 45, normalized size = 0.94 \begin {gather*} \frac {\sqrt {1-2 x}}{42+63 x}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(42 + 63*x) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 36, normalized size = 0.75

method	result	size
derivativedivides	\(-\frac {2 \sqrt {1-2 x}}{63 \left (-\frac {4}{3}-2 x \right )}-\frac {68 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\)	\(36\)
default	\(-\frac {2 \sqrt {1-2 x}}{63 \left (-\frac {4}{3}-2 x \right )}-\frac {68 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\)	\(36\)
risch	\(-\frac {-1+2 x}{21 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {68 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\)	\(41\)
trager	\(\frac {\sqrt {1-2 x}}{42+63 x}+\frac {34 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{441}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/63*(1-2*x)^(1/2)/(-4/3-2*x)-68/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 53, normalized size = 1.10 \begin {gather*} \frac {34}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {\sqrt {-2 \, x + 1}}{21 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21*sqrt(-2*x + 1)/(3*x +
2)

________________________________________________________________________________________

Fricas [A]
time = 1.07, size = 54, normalized size = 1.12 \begin {gather*} \frac {34 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, \sqrt {-2 \, x + 1}}{441 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/441*(34*sqrt(21)*(3*x + 2)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*sqrt(-2*x + 1))/(3*x + 2)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (37) = 74\).
time = 217.01, size = 190, normalized size = 3.96 \begin {gather*} - \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (-1 + \frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{4} + \frac {\log {\left (1 + \frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{4} - \frac {1}{4 \cdot \left (1 + \frac {\sqrt {21}}{3 \sqrt {1 - 2 x}}\right )} - \frac {1}{4 \left (-1 + \frac {\sqrt {21}}{3 \sqrt {1 - 2 x}}\right )}\right )}{63} & \text {for}\: \frac {1}{\sqrt {1 - 2 x}} > - \frac {\sqrt {21}}{7} \wedge \frac {1}{\sqrt {1 - 2 x}} < \frac {\sqrt {21}}{7} \end {cases}\right )}{7} + \frac {22 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} > \frac {3}{7} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} < \frac {3}{7} \end {cases}\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*Piecewise((sqrt(21)*(-log(-1 + sqrt(21)/(3*sqrt(1 - 2*x)))/4 + log(1 + sqrt(21)/(3*sqrt(1 - 2*x)))/4 - 1/(4
*(1 + sqrt(21)/(3*sqrt(1 - 2*x)))) - 1/(4*(-1 + sqrt(21)/(3*sqrt(1 - 2*x)))))/63, (1/sqrt(1 - 2*x) > -sqrt(21)
/7) & (1/sqrt(1 - 2*x) < sqrt(21)/7)))/7 + 22*Piecewise((-sqrt(21)*acoth(sqrt(21)/(3*sqrt(1 - 2*x)))/21, 1/(1
- 2*x) > 3/7), (-sqrt(21)*atanh(sqrt(21)/(3*sqrt(1 - 2*x)))/21, 1/(1 - 2*x) < 3/7))/7

________________________________________________________________________________________

Giac [A]
time = 0.55, size = 56, normalized size = 1.17 \begin {gather*} \frac {34}{441} \, \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 {\sqrt {-2 \, x + 1}}{21 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/441*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21*sqrt(-2*x +
1)/(3*x + 2)

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 35, normalized size = 0.73 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{63\,\left (2\,x+\frac {4}{3}\right )}-\frac {68\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(1 - 2*x)^(1/2))/(63*(2*x + 4/3)) - (68*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/441

________________________________________________________________________________________