Integrand size = 24, antiderivative size = 133 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {13723 \sqrt {1-2 x}}{504 (2+3 x)^2}+\frac {318643 \sqrt {1-2 x}}{1176 (2+3 x)}+\frac {10990843 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}-550 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
7/12*(1-2*x)^(1/2)/(2+3*x)^4+131/36*(1-2*x)^(1/2)/(2+3*x)^3+13723/504*(1-2 *x)^(1/2)/(2+3*x)^2+318643*(1-2*x)^(1/2)/(2352+3528*x)+10990843/12348*21^( 1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-550*55^(1/2)*arctanh(1/11*55^(1/2 )*(1-2*x)^(1/2))
Time = 0.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (2686470+11868230 x+17494905 x^2+8603361 x^3\right )}{1176 (2+3 x)^4}+\frac {10990843 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}-550 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:
Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^5*(3 + 5*x)),x]
Output:
(Sqrt[1 - 2*x]*(2686470 + 11868230*x + 17494905*x^2 + 8603361*x^3))/(1176* (2 + 3*x)^4) + (10990843*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) - 550*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {109, 168, 27, 168, 27, 168, 174, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^5 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{12} \int \frac {153-229 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{21} \int \frac {7 (2391-3275 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \int \frac {2391-3275 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \int \frac {3 (60471-68615 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {13723 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {3}{14} \int \frac {60471-68615 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {13723 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {2601471-1593215 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {318643 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {13723 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {3}{14} \left (\frac {1}{7} \left (17787000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-10990843 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {318643 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {13723 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {3}{14} \left (\frac {1}{7} \left (10990843 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-17787000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {318643 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {13723 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {3}{14} \left (\frac {1}{7} \left (\frac {21981686 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-646800 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {318643 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {13723 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {131 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}\) |
Input:
Int[(1 - 2*x)^(3/2)/((2 + 3*x)^5*(3 + 5*x)),x]
Output:
(7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4) + ((131*Sqrt[1 - 2*x])/(3*(2 + 3*x)^3) + ((13723*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + (3*((318643*Sqrt[1 - 2*x])/(7* (2 + 3*x)) + ((21981686*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 64680 0*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/14)/3)/12
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {17206722 x^{4}+26386449 x^{3}+6241555 x^{2}-6495290 x -2686470}{1176 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {10990843 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{12348}-550 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )\) | \(74\) |
| derivativedivides | \(-550 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-\frac {162 \left (\frac {318643 \left (1-2 x \right )^{\frac {7}{2}}}{3528}-\frac {2895233 \left (1-2 x \right )^{\frac {5}{2}}}{4536}+\frac {2923727 \left (1-2 x \right )^{\frac {3}{2}}}{1944}-\frac {2297099 \sqrt {1-2 x}}{1944}\right )}{\left (-4-6 x \right )^{4}}+\frac {10990843 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{12348}\) | \(84\) |
| default | \(-550 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-\frac {162 \left (\frac {318643 \left (1-2 x \right )^{\frac {7}{2}}}{3528}-\frac {2895233 \left (1-2 x \right )^{\frac {5}{2}}}{4536}+\frac {2923727 \left (1-2 x \right )^{\frac {3}{2}}}{1944}-\frac {2297099 \sqrt {1-2 x}}{1944}\right )}{\left (-4-6 x \right )^{4}}+\frac {10990843 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{12348}\) | \(84\) |
| pseudoelliptic | \(\frac {21981686 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-13582800 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \sqrt {55}+21 \sqrt {1-2 x}\, \left (8603361 x^{3}+17494905 x^{2}+11868230 x +2686470\right )}{24696 \left (2+3 x \right )^{4}}\) | \(85\) |
| trager | \(\frac {\left (8603361 x^{3}+17494905 x^{2}+11868230 x +2686470\right ) \sqrt {1-2 x}}{1176 \left (2+3 x \right )^{4}}+275 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {10990843 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{24696}\) | \(121\) |
Input:
int((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-1/1176*(17206722*x^4+26386449*x^3+6241555*x^2-6495290*x-2686470)/(2+3*x)^ 4/(1-2*x)^(1/2)+10990843/12348*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2) )-550*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {6791400 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 10990843 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (8603361 \, x^{3} + 17494905 \, x^{2} + 11868230 \, x + 2686470\right )} \sqrt {-2 \, x + 1}}{24696 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x),x, algorithm="fricas")
Output:
1/24696*(6791400*sqrt(55)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((5* x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 10990843*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3* x + 2)) + 21*(8603361*x^3 + 17494905*x^2 + 11868230*x + 2686470)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Time = 136.70 (sec) , antiderivative size = 836, normalized size of antiderivative = 6.29 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=\text {Too large to display} \] Input:
integrate((1-2*x)**(3/2)/(2+3*x)**5/(3+5*x),x)
Output:
-3025*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 + 275*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) + 7260*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))) - 2904*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))) + 3472*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21) *sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48* (sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x )/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt( 21)/3)))/3 - 1568*Piecewise((sqrt(21)*(35*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1 )/256 - 35*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/256 + 35/(256*(sqrt(21)*sqrt( 1 - 2*x)/7 + 1)) + 15/(256*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 5/(192*(sq rt(21)*sqrt(1 - 2*x)/7 + 1)**3) + 1/(128*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)...
Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=275 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {10990843}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8603361 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 60799893 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 143262623 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 112557851 \, \sqrt {-2 \, x + 1}}{588 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x),x, algorithm="maxima")
Output:
275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 10990843/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) - 1/588*(8603361*(-2*x + 1)^(7/2) - 60799893*(-2*x + 1)^(5/2) + 143262623*(-2*x + 1)^(3/2) - 112557851*sqrt(-2*x + 1))/(81*(2 *x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=275 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {10990843}{24696} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8603361 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 60799893 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 143262623 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 112557851 \, \sqrt {-2 \, x + 1}}{9408 \, {\left (3 \, x + 2\right )}^{4}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x),x, algorithm="giac")
Output:
275*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) - 10990843/24696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/9408*(8603361*(2*x - 1)^3*sq rt(-2*x + 1) + 60799893*(2*x - 1)^2*sqrt(-2*x + 1) - 143262623*(-2*x + 1)^ (3/2) + 112557851*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {10990843\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-550\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {2297099\,\sqrt {1-2\,x}}{972}-\frac {2923727\,{\left (1-2\,x\right )}^{3/2}}{972}+\frac {2895233\,{\left (1-2\,x\right )}^{5/2}}{2268}-\frac {318643\,{\left (1-2\,x\right )}^{7/2}}{1764}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \] Input:
int((1 - 2*x)^(3/2)/((3*x + 2)^5*(5*x + 3)),x)
Output:
(10990843*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/12348 - 550*55^(1/ 2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((2297099*(1 - 2*x)^(1/2))/972 - (2923727*(1 - 2*x)^(3/2))/972 + (2895233*(1 - 2*x)^(5/2))/2268 - (318643* (1 - 2*x)^(7/2))/1764)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3 )/3 + (2*x - 1)^4 - 1715/81)
Time = 0.15 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.33 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {180670581 \sqrt {-2 x +1}\, x^{3}+367393005 \sqrt {-2 x +1}\, x^{2}+249232830 \sqrt {-2 x +1}\, x +56415870 \sqrt {-2 x +1}+550103400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{4}+1466942400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}+1466942400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+651974400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +108662400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-550103400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{4}-1466942400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-1466942400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-651974400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -108662400 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-890258283 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{4}-2374022088 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-2374022088 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-1055120928 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -175853488 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+890258283 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{4}+2374022088 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+2374022088 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+1055120928 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +175853488 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{2000376 x^{4}+5334336 x^{3}+5334336 x^{2}+2370816 x +395136} \] Input:
int((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x),x)
Output:
(180670581*sqrt( - 2*x + 1)*x**3 + 367393005*sqrt( - 2*x + 1)*x**2 + 24923 2830*sqrt( - 2*x + 1)*x + 56415870*sqrt( - 2*x + 1) + 550103400*sqrt(55)*l og(5*sqrt( - 2*x + 1) - sqrt(55))*x**4 + 1466942400*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**3 + 1466942400*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 + 651974400*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x + 108662400*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 550103400*sqrt(55 )*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**4 - 1466942400*sqrt(55)*log(5*sqrt ( - 2*x + 1) + sqrt(55))*x**3 - 1466942400*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 651974400*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))* x - 108662400*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) - 890258283*sqrt (21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**4 - 2374022088*sqrt(21)*log(3*s qrt( - 2*x + 1) - sqrt(21))*x**3 - 2374022088*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 1055120928*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(2 1))*x - 175853488*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 890258283* sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**4 + 2374022088*sqrt(21)*log (3*sqrt( - 2*x + 1) + sqrt(21))*x**3 + 2374022088*sqrt(21)*log(3*sqrt( - 2 *x + 1) + sqrt(21))*x**2 + 1055120928*sqrt(21)*log(3*sqrt( - 2*x + 1) + sq rt(21))*x + 175853488*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(24696* (81*x**4 + 216*x**3 + 216*x**2 + 96*x + 16))