3.1856 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=180 \[ \frac{4031135 \sqrt{1-2 x}}{1078 (5 x+3)}-\frac{182335 \sqrt{1-2 x}}{294 (5 x+3)^2}+\frac{4042 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)^2}+\frac{29 \sqrt{1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}+\frac{2528082}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{551075}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-182335*Sqrt[1 - 2*x])/(294*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)^2) + (29*Sqrt[1 - 2*x])/(7*
(2 + 3*x)^2*(3 + 5*x)^2) + (4042*Sqrt[1 - 2*x])/(49*(2 + 3*x)*(3 + 5*x)^2) + (4031135*Sqrt[1 - 2*x])/(1078*(3
+ 5*x)) + (2528082*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (551075*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt
[1 - 2*x]])/11

________________________________________________________________________________________

Rubi [A]  time = 0.0734513, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {99, 151, 156, 63, 206} \[ \frac{4031135 \sqrt{1-2 x}}{1078 (5 x+3)}-\frac{182335 \sqrt{1-2 x}}{294 (5 x+3)^2}+\frac{4042 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)^2}+\frac{29 \sqrt{1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}+\frac{2528082}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{551075}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(-182335*Sqrt[1 - 2*x])/(294*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)^2) + (29*Sqrt[1 - 2*x])/(7*
(2 + 3*x)^2*(3 + 5*x)^2) + (4042*Sqrt[1 - 2*x])/(49*(2 + 3*x)*(3 + 5*x)^2) + (4031135*Sqrt[1 - 2*x])/(1078*(3
+ 5*x)) + (2528082*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (551075*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt
[1 - 2*x]])/11

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(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]

Rule 63

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}-\frac{1}{3} \int \frac{-28+45 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}-\frac{1}{42} \int \frac{-4024+6090 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac{4042 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)^2}-\frac{1}{294} \int \frac{-438494+606300 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{182335 \sqrt{1-2 x}}{294 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac{4042 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)^2}+\frac{\int \frac{-31549584+36102330 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{6468}\\ &=-\frac{182335 \sqrt{1-2 x}}{294 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac{4042 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)^2}+\frac{4031135 \sqrt{1-2 x}}{1078 (3+5 x)}-\frac{\int \frac{-1303277712+798164730 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{71148}\\ &=-\frac{182335 \sqrt{1-2 x}}{294 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac{4042 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)^2}+\frac{4031135 \sqrt{1-2 x}}{1078 (3+5 x)}-\frac{3792123}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{2755375}{22} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{182335 \sqrt{1-2 x}}{294 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac{4042 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)^2}+\frac{4031135 \sqrt{1-2 x}}{1078 (3+5 x)}+\frac{3792123}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{2755375}{22} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{182335 \sqrt{1-2 x}}{294 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac{29 \sqrt{1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac{4042 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)^2}+\frac{4031135 \sqrt{1-2 x}}{1078 (3+5 x)}+\frac{2528082}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{551075}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.18952, size = 167, normalized size = 0.93 \[ \frac{-22497365 (1-2 x)^{3/2} (3 x+2)^3+2977568 (1-2 x)^{3/2} (3 x+2)^2+150766 (1-2 x)^{3/2} (3 x+2)+11858 (1-2 x)^{3/2}+(5 x+3) (3 x+2)^3 \left (301398449 \sqrt{1-2 x}[phone] \sqrt{21} (5 x+3) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-378037450 \sqrt{55} (5 x+3) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{83006 (3 x+2)^3 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(11858*(1 - 2*x)^(3/2) + 150766*(1 - 2*x)^(3/2)*(2 + 3*x) + 2977568*(1 - 2*x)^(3/2)*(2 + 3*x)^2 - 22497365*(1
- 2*x)^(3/2)*(2 + 3*x)^3 + (2 + 3*x)^3*(3 + 5*x)*(301398449*Sqrt[1 - 2*x] + 611795844*Sqrt[21]*(3 + 5*x)*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 378037450*Sqrt[55]*(3 + 5*x)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/(83006*(2 + 3*x
)^3*(3 + 5*x)^2)

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 103, normalized size = 0.6 \begin{align*} -972\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{7297\, \left ( 1-2\,x \right ) ^{5/2}}{294}}-{\frac{22048\, \left ( 1-2\,x \right ) ^{3/2}}{189}}+{\frac{7403\,\sqrt{1-2\,x}}{54}} \right ) }+{\frac{2528082\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+62500\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{53\, \left ( 1-2\,x \right ) ^{3/2}}{220}}+{\frac{263\,\sqrt{1-2\,x}}{500}} \right ) }-{\frac{551075\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^3,x)

[Out]

-972*(7297/294*(1-2*x)^(5/2)-22048/189*(1-2*x)^(3/2)+7403/54*(1-2*x)^(1/2))/(-6*x-4)^3+2528082/343*arctanh(1/7
*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+62500*(-53/220*(1-2*x)^(3/2)+263/500*(1-2*x)^(1/2))/(-10*x-6)^2-551075/121*a
rctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.91336, size = 221, normalized size = 1.23 \begin{align*} \frac{551075}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1264041}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{544203225 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4970567340 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 17019867294 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 25893807436 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 14768524001 \, \sqrt{-2 \, x + 1}}{539 \,{\left (675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

551075/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1264041/343*sqrt(21)*l
og(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/539*(544203225*(-2*x + 1)^(9/2) - 4970567
340*(-2*x + 1)^(7/2) + 17019867294*(-2*x + 1)^(5/2) - 25893807436*(-2*x + 1)^(3/2) + 14768524001*sqrt(-2*x + 1
))/(675*(2*x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x - 49588)

________________________________________________________________________________________

Fricas [A]  time = 1.49805, size = 603, normalized size = 3.35 \begin{align*} \frac{189018725 \, \sqrt{11} \sqrt{5}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 305897922 \, \sqrt{7} \sqrt{3}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (544203225 \, x^{4} + 1396877220 \, x^{3} + 1343346156 \, x^{2} + 573620246 \, x + 91763734\right )} \sqrt{-2 \, x + 1}}{83006 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/83006*(189018725*sqrt(11)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((sqrt(11)*sqrt
(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 305897922*sqrt(7)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2
+ 564*x + 72)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(544203225*x^4 + 1396877220*x^3
+ 1343346156*x^2 + 573620246*x + 91763734)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x
+ 72)

________________________________________________________________________________________

Sympy [A]  time = 174.104, size = 811, normalized size = 4.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x)**3,x)

[Out]

36720*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, (x <= 1/2) & (x > -2/3))) - 7416*P
iecewise((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*(s
qrt(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, (x <= 1/2) & (x > -2/3))) + 1008*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, (x <= 1/2) & (x > -2/3))) + 67000*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, (x <= 1/2) & (x > -3/5))) + 11000*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sq
rt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(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, (x <= 1/2)
& (x > -3/5))) - 152100*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*a
tanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3)) + 253500*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)
/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 > -11/5))

________________________________________________________________________________________

Giac [A]  time = 2.11701, size = 204, normalized size = 1.13 \begin{align*} \frac{551075}{242} \, \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{1264041}{343} \, \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{125 \,{\left (1325 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2893 \, \sqrt{-2 \, x + 1}\right )}}{44 \,{\left (5 \, x + 3\right )}^{2}} + \frac{9 \,{\left (65673 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 308672 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 362747 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

551075/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1264041/343*
sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/44*(1325*(-2*x + 1)^
(3/2) - 2893*sqrt(-2*x + 1))/(5*x + 3)^2 + 9/196*(65673*(2*x - 1)^2*sqrt(-2*x + 1) - 308672*(-2*x + 1)^(3/2) +
362747*sqrt(-2*x + 1))/(3*x + 2)^3