Integrand size = 22, antiderivative size = 68 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {\sqrt {1-2 x}}{110 (3+5 x)^2}-\frac {69 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {69 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \] Output:
-1/110*(1-2*x)^(1/2)/(3+5*x)^2-69*(1-2*x)^(1/2)/(3630+6050*x)-69/33275*55^ (1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {\sqrt {1-2 x} (218+345 x)}{1210 (3+5 x)^2}-\frac {69 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \] Input:
Integrate[(2 + 3*x)/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]
Output:
-1/1210*(Sqrt[1 - 2*x]*(218 + 345*x))/(3 + 5*x)^2 - (69*ArcTanh[Sqrt[5/11] *Sqrt[1 - 2*x]])/(605*Sqrt[55])
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 52, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {3 x+2}{\sqrt {1-2 x} (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {69}{110} \int \frac {1}{\sqrt {1-2 x} (5 x+3)^2}dx-\frac {\sqrt {1-2 x}}{110 (5 x+3)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {69}{110} \left (\frac {1}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {\sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {\sqrt {1-2 x}}{110 (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {69}{110} \left (-\frac {1}{11} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {\sqrt {1-2 x}}{110 (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {69}{110} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{11 \sqrt {55}}-\frac {\sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {\sqrt {1-2 x}}{110 (5 x+3)^2}\) |
Input:
Int[(2 + 3*x)/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]
Output:
-1/110*Sqrt[1 - 2*x]/(3 + 5*x)^2 + (69*(-1/11*Sqrt[1 - 2*x]/(3 + 5*x) - (2 *ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(11*Sqrt[55])))/110
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(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] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {690 x^{2}+91 x -218}{1210 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {69 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{33275}\) | \(46\) |
derivativedivides | \(-\frac {100 \left (-\frac {69 \left (1-2 x \right )^{\frac {3}{2}}}{12100}+\frac {71 \sqrt {1-2 x}}{5500}\right )}{\left (-6-10 x \right )^{2}}-\frac {69 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{33275}\) | \(48\) |
default | \(-\frac {100 \left (-\frac {69 \left (1-2 x \right )^{\frac {3}{2}}}{12100}+\frac {71 \sqrt {1-2 x}}{5500}\right )}{\left (-6-10 x \right )^{2}}-\frac {69 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{33275}\) | \(48\) |
pseudoelliptic | \(\frac {-138 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}-55 \sqrt {1-2 x}\, \left (345 x +218\right )}{66550 \left (3+5 x \right )^{2}}\) | \(50\) |
trager | \(-\frac {\left (345 x +218\right ) \sqrt {1-2 x}}{1210 \left (3+5 x \right )^{2}}-\frac {69 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{66550}\) | \(67\) |
Input:
int((2+3*x)/(1-2*x)^(1/2)/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
1/1210*(690*x^2+91*x-218)/(3+5*x)^2/(1-2*x)^(1/2)-69/33275*55^(1/2)*arctan h(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {69 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (345 \, x + 218\right )} \sqrt {-2 \, x + 1}}{66550 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((2+3*x)/(1-2*x)^(1/2)/(3+5*x)^3,x, algorithm="fricas")
Output:
1/66550*(69*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1 ) - 8)/(5*x + 3)) - 55*(345*x + 218)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (58) = 116\).
Time = 71.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 4.22 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=- \frac {12 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{5} + \frac {8 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{5} \] Input:
integrate((2+3*x)/(1-2*x)**(1/2)/(3+5*x)**3,x)
Output:
-12*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt( 55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4 *(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (s qrt(1 - 2*x) < sqrt(55)/5)))/5 + 8*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqr t(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(s qrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(1 - 2*x)/ 11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(5 5)/5)))/5
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {69}{66550} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 781 \, \sqrt {-2 \, x + 1}}{605 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \] Input:
integrate((2+3*x)/(1-2*x)^(1/2)/(3+5*x)^3,x, algorithm="maxima")
Output:
69/66550*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) + 1/605*(345*(-2*x + 1)^(3/2) - 781*sqrt(-2*x + 1))/(25*(2*x - 1 )^2 + 220*x + 11)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {69}{66550} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 781 \, \sqrt {-2 \, x + 1}}{2420 \, {\left (5 \, x + 3\right )}^{2}} \] Input:
integrate((2+3*x)/(1-2*x)^(1/2)/(3+5*x)^3,x, algorithm="giac")
Output:
69/66550*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1/2420*(345*(-2*x + 1)^(3/2) - 781*sqrt(-2*x + 1))/( 5*x + 3)^2
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {69\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{33275}-\frac {\frac {71\,\sqrt {1-2\,x}}{1375}-\frac {69\,{\left (1-2\,x\right )}^{3/2}}{3025}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \] Input:
int((3*x + 2)/((1 - 2*x)^(1/2)*(5*x + 3)^3),x)
Output:
- (69*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/33275 - ((71*(1 - 2*x )^(1/2))/1375 - (69*(1 - 2*x)^(3/2))/3025)/((44*x)/5 + (2*x - 1)^2 + 11/25 )
Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.07 \[ \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {-18975 \sqrt {-2 x +1}\, x -11990 \sqrt {-2 x +1}+1725 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+2070 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +621 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-1725 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-2070 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -621 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{1663750 x^{2}+1996500 x +598950} \] Input:
int((2+3*x)/(1-2*x)^(1/2)/(3+5*x)^3,x)
Output:
( - 18975*sqrt( - 2*x + 1)*x - 11990*sqrt( - 2*x + 1) + 1725*sqrt(55)*log( 5*sqrt( - 2*x + 1) - sqrt(55))*x**2 + 2070*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x + 621*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 1725*sq rt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 2070*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 621*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55) ))/(66550*(25*x**2 + 30*x + 9))