3.21.77 \(\int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)} \, dx\) [2077]

Optimal. Leaf size=41 \[ \frac {11}{7 \sqrt {1-2 x}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}} \]

[Out]

2/147*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7/(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 41, 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 {11}{7 \sqrt {1-2 x}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

11/(7*Sqrt[1 - 2*x]) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*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}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\frac {11}{7 \sqrt {1-2 x}}-\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {11}{7 \sqrt {1-2 x}}+\frac {1}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {11}{7 \sqrt {1-2 x}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 41, normalized size = 1.00 \begin {gather*} \frac {11}{7 \sqrt {1-2 x}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

11/(7*Sqrt[1 - 2*x]) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])

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Maple [A]
time = 0.12, size = 29, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{147}+\frac {11}{7 \sqrt {1-2 x}}\)	\(29\)
default	\(\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{147}+\frac {11}{7 \sqrt {1-2 x}}\)	\(29\)
risch	\(\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{147}+\frac {11}{7 \sqrt {1-2 x}}\)	\(29\)
trager	\(-\frac {11 \sqrt {1-2 x}}{7 \left (-1+2 x \right )}+\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{147}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/147*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7/(1-2*x)^(1/2)

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Maxima [A]
time = 0.50, size = 46, normalized size = 1.12 \begin {gather*} -\frac {1}{147} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {11}{7 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/147*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 11/7/sqrt(-2*x + 1)

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Fricas [A]
time = 0.78, size = 54, normalized size = 1.32 \begin {gather*} \frac {\sqrt {21} {\left (2 \, x - 1\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 231 \, \sqrt {-2 \, x + 1}}{147 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/147*(sqrt(21)*(2*x - 1)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 231*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [A]
time = 11.01, size = 71, normalized size = 1.73 \begin {gather*} - \frac {2 \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 )}{7} + \frac {11}{7 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*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))/7 + 11/(7*sqrt(1 - 2*x))

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Giac [A]
time = 1.28, size = 49, normalized size = 1.20 \begin {gather*} -\frac {1}{147} \, \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 {11}{7 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/147*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 11/7/sqrt(-2*x +
1)

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Mupad [B]
time = 0.06, size = 28, normalized size = 0.68 \begin {gather*} \frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{147}+\frac {11}{7\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/147 + 11/(7*(1 - 2*x)^(1/2))

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