Integrand size = 24, antiderivative size = 100 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}+\frac {7559 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]
7559/11907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-53/63*(3+5*x)^2*(1 -2*x)^(1/2)/(2+3*x)-1/6*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^2+5/1134*(323+2815 *x)*(1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (-15815-32833 x+7350 x^2+31500 x^3\right )}{1134 (2+3 x)^2}+\frac {7559 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]
(Sqrt[1 - 2*x]*(-15815 - 32833*x + 7350*x^2 + 31500*x^3))/(1134*(2 + 3*x)^ 2) + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {108, 166, 164, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{6} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{21} \int \frac {(643-2815 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{21} \left (\frac {5}{9} \sqrt {1-2 x} (2815 x+323)-\frac {7559}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{21} \left (\frac {7559}{9} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {5}{9} \sqrt {1-2 x} (2815 x+323)\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{21} \left (\frac {15118 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}+\frac {5}{9} \sqrt {1-2 x} (2815 x+323)\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}\) |
-1/6*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^2 + ((-106*Sqrt[1 - 2*x]*(3 + 5 *x)^2)/(21*(2 + 3*x)) + ((5*Sqrt[1 - 2*x]*(323 + 2815*x))/9 + (15118*ArcTa nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21]))/21)/6
3.19.22.3.1 Defintions of rubi rules used
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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 = 0.97 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {63000 x^{4}-16800 x^{3}-73016 x^{2}+1203 x +15815}{1134 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {7559 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\) | \(56\) |
| pseudoelliptic | \(\frac {15118 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+21 \sqrt {1-2 x}\, \left (31500 x^{3}+7350 x^{2}-32833 x -15815\right )}{23814 \left (2+3 x \right )^{2}}\) | \(60\) |
| derivativedivides | \(-\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {50 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {211 \left (1-2 x \right )^{\frac {3}{2}}}{126}+\frac {209 \sqrt {1-2 x}}{54}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {7559 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\) | \(66\) |
| default | \(-\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {50 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {211 \left (1-2 x \right )^{\frac {3}{2}}}{126}+\frac {209 \sqrt {1-2 x}}{54}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {7559 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\) | \(66\) |
| trager | \(\frac {\left (31500 x^{3}+7350 x^{2}-32833 x -15815\right ) \sqrt {1-2 x}}{1134 \left (2+3 x \right )^{2}}+\frac {7559 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{23814}\) | \(77\) |
-1/1134*(63000*x^4-16800*x^3-73016*x^2+1203*x+15815)/(2+3*x)^2/(1-2*x)^(1/ 2)+7559/11907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=\frac {7559 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (31500 \, x^{3} + 7350 \, x^{2} - 32833 \, x - 15815\right )} \sqrt {-2 \, x + 1}}{23814 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/23814*(7559*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(31500*x^3 + 7350*x^2 - 32833*x - 15815)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Time = 137.51 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.55 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=- \frac {125 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {50 \sqrt {1 - 2 x}}{27} - \frac {185 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{567} - \frac {428 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} - \frac {56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} \]
-125*(1 - 2*x)**(3/2)/81 - 50*sqrt(1 - 2*x)/27 - 185*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/567 - 428*Piecewis e((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) < s qrt(21)/3)))/81 - 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) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/81
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {125}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7559}{23814} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {50}{27} \, \sqrt {-2 \, x + 1} + \frac {633 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{567 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2* x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 50/27*sqrt(-2*x + 1) + 1/567*(633 *(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {125}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7559}{23814} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {50}{27} \, \sqrt {-2 \, x + 1} + \frac {633 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{2268 \, {\left (3 \, x + 2\right )}^{2}} \]
-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6 *sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 50/27*sqrt(-2*x + 1) + 1 /2268*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {50\,\sqrt {1-2\,x}}{27}-\frac {125\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {\frac {209\,\sqrt {1-2\,x}}{729}-\frac {211\,{\left (1-2\,x\right )}^{3/2}}{1701}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,7559{}\mathrm {i}}{11907} \]