\(\int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx\) [2020]

Optimal result

Integrand size = 24, antiderivative size = 68 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]

[Out]

-257/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1/126*(1-2*x)^(1/2)/(2+3*x)^2+137/882*(1-2*x)^(1/2)/(2+
3*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 79, 65, 212} \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {257 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}+\frac {137 \sqrt {1-2 x}}{882 (3 x+2)}-\frac {\sqrt {1-2 x}}{126 (3 x+2)^2} \]

[In]

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

[Out]

-1/126*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (137*Sqrt[1 - 2*x])/(882*(2 + 3*x)) - (257*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/(49*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 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*} \text {integral}& = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {563+1050 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}+\frac {257}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257}{98} \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}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\frac {7 \sqrt {1-2 x} (89+137 x)}{(2+3 x)^2}-514 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2058} \]

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*(89 + 137*x))/(2 + 3*x)^2 - 514*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2058

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {274 x^{2}+41 x -89}{294 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {257 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\)	\(46\)
derivativedivides	\(\frac {-\frac {137 \left (1-2 x \right )^{\frac {3}{2}}}{147}+\frac {15 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {257 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\)	\(48\)
default	\(\frac {-\frac {137 \left (1-2 x \right )^{\frac {3}{2}}}{147}+\frac {15 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {257 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\)	\(48\)
pseudoelliptic	\(\frac {-514 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+7 \sqrt {1-2 x}\, \left (137 x +89\right )}{2058 \left (2+3 x \right )^{2}}\)	\(50\)
trager	\(\frac {\left (137 x +89\right ) \sqrt {1-2 x}}{294 \left (2+3 x \right )^{2}}-\frac {257 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2058}\)	\(67\)

[In]

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

[Out]

-1/294*(274*x^2+41*x-89)/(2+3*x)^2/(1-2*x)^(1/2)-257/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {257 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (137 \, x + 89\right )} \sqrt {-2 \, x + 1}}{2058 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

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

[Out]

1/2058*(257*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 7*(137*x + 89)*sq
rt(-2*x + 1))/(9*x^2 + 12*x + 4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (56) = 112\).

Time = 95.64 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.85 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {25 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{189} + \frac {40 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} + \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \]

[In]

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

[Out]

25*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/189 + 40*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*(sqrt(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) & (sqrt(1 - 2*x) < sqrt(21)/3))
)/9 + 8*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) > -sqrt(21)/3) & (sqrt(1 - 2*x) <
sqrt(21)/3)))/9

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {257}{2058} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 315 \, \sqrt {-2 \, x + 1}}{147 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

[In]

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

[Out]

257/2058*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/147*(137*(-2*x + 1)^(3
/2) - 315*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {257}{2058} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 315 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \]

[In]

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

[Out]

257/2058*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/588*(137*(-2*
x + 1)^(3/2) - 315*sqrt(-2*x + 1))/(3*x + 2)^2

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\frac {5\,\sqrt {1-2\,x}}{21}-\frac {137\,{\left (1-2\,x\right )}^{3/2}}{1323}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {257\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029} \]

[In]

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

[Out]

((5*(1 - 2*x)^(1/2))/21 - (137*(1 - 2*x)^(3/2))/1323)/((28*x)/3 + (2*x - 1)^2 + 7/9) - (257*21^(1/2)*atanh((21
^(1/2)*(1 - 2*x)^(1/2))/7))/1029