3.21.60 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx\)
Optimal. Leaf size=179 \[ -\frac {77527480}{5021863 \sqrt {1-2 x}}+\frac {167960}{847 (1-2 x)^{3/2} (5 x+3)}-\frac {6845810}{195657 (1-2 x)^{3/2}}+\frac {9}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac {5165}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}+\frac {182655}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7570625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]
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Rubi [A] time = 0.08, antiderivative size = 179, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used
= {103, 151, 152, 156, 63, 206} \begin {gather*} -\frac {77527480}{5021863 \sqrt {1-2 x}}+\frac {167960}{847 (1-2 x)^{3/2} (5 x+3)}-\frac {6845810}{195657 (1-2 x)^{3/2}}+\frac {9}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac {5165}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}+\frac {182655}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7570625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x)^3),x]
[Out]
-6845810/(195657*(1 - 2*x)^(3/2)) - 77527480/(5021863*Sqrt[1 - 2*x]) - 5165/(154*(1 - 2*x)^(3/2)*(3 + 5*x)^2)
+ 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2) + 9/(2*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 167960/(847*(
1 - 2*x)^(3/2)*(3 + 5*x)) + (182655*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (7570625*Sqrt[5/11]*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641
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 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
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 152
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] && IntegersQ[2*m, 2*n, 2*p]
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 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 {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {1}{14} \int \frac {37-165 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {1}{98} \int \frac {2555-19845 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}-\frac {\int \frac {29470-1518510 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx}{2156}\\ &=-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}+\frac {\int \frac {-12649070-70543200 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx}{23716}\\ &=-\frac {6845810}{195657 (1-2 x)^{3/2}}-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {-264176535+2156430150 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{2739198}\\ &=-\frac {6845810}{195657 (1-2 x)^{3/2}}-\frac {77527480}{5021863 \sqrt {1-2 x}}-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}+\frac {\int \frac {\frac {39878518155}{2}-12210578100 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{105459123}\\ &=-\frac {6845810}{195657 (1-2 x)^{3/2}}-\frac {77527480}{5021863 \sqrt {1-2 x}}-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}-\frac {547965}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {37853125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=-\frac {6845810}{195657 (1-2 x)^{3/2}}-\frac {77527480}{5021863 \sqrt {1-2 x}}-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}+\frac {547965}{686} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {37853125 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=-\frac {6845810}{195657 (1-2 x)^{3/2}}-\frac {77527480}{5021863 \sqrt {1-2 x}}-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}+\frac {182655}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7570625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 83, normalized size = 0.46 \begin {gather*} \frac {-162075870 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+148384250 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+\frac {231 \left (15116400 x^3+28713705 x^2+18152609 x+3819389\right )}{(3 x+2)^2 (5 x+3)^2}}{391314 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x)^3),x]
[Out]
((231*(3819389 + 18152609*x + 28713705*x^2 + 15116400*x^3))/((2 + 3*x)^2*(3 + 5*x)^2) - 162075870*Hypergeometr
ic2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7] + 148384250*Hypergeometric2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11])/(391314*
(1 - 2*x)^(3/2))
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IntegrateAlgebraic [A] time = 0.35, size = 137, normalized size = 0.77 \begin {gather*} -\frac {2 \left (26165524500 (1-2 x)^5-177932259675 (1-2 x)^4+403131105480 (1-2 x)^3-304294845085 (1-2 x)^2-12901504 (1-2 x)-2087008\right )}{15065589 \left (15 (1-2 x)^2-68 (1-2 x)+77\right )^2 (1-2 x)^{3/2}}+\frac {182655}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7570625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x)^3),x]
[Out]
(-2*(-2087008 - 12901504*(1 - 2*x) - 304294845085*(1 - 2*x)^2 + 403131105480*(1 - 2*x)^3 - 177932259675*(1 - 2
*x)^4 + 26165524500*(1 - 2*x)^5))/(15065589*(77 - 68*(1 - 2*x) + 15*(1 - 2*x)^2)^2*(1 - 2*x)^(3/2)) + (182655*
Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (7570625*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641
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fricas [A] time = 1.31, size = 202, normalized size = 1.13 \begin {gather*} \frac {54531211875 \, \sqrt {11} \sqrt {5} {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 88250311215 \, \sqrt {7} \sqrt {3} {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (209324196000 \, x^{5} + 188418548700 \, x^{4} - 93885376440 \, x^{3} - 99160158305 \, x^{2} + 9944654283 \, x + 13236365823\right )} \sqrt {-2 \, x + 1}}{2320100706 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="fricas")
[Out]
1/2320100706*(54531211875*sqrt(11)*sqrt(5)*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(
(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 88250311215*sqrt(7)*sqrt(3)*(900*x^6 + 1380*x^5 + 109
*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(2093241
96000*x^5 + 188418548700*x^4 - 93885376440*x^3 - 99160158305*x^2 + 9944654283*x + 13236365823)*sqrt(-2*x + 1))
/(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)
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giac [A] time = 1.30, size = 169, normalized size = 0.94 \begin {gather*} \frac {7570625}{322102} \, \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 {182655}{4802} \, \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 {64 \, {\left (1224 \, x - 689\right )}}{105459123 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {2 \, {\left (5550396300 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 37744400445 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 85516621432 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 64553088299 \, \sqrt {-2 \, x + 1}\right )}}{3195731 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="giac")
[Out]
7570625/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 182655/4
802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/105459123*(1224*x
- 689)/((2*x - 1)*sqrt(-2*x + 1)) + 2/3195731*(5550396300*(2*x - 1)^3*sqrt(-2*x + 1) + 37744400445*(2*x - 1)^
2*sqrt(-2*x + 1) - 85516621432*(-2*x + 1)^(3/2) + 64553088299*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
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maple [A] time = 0.02, size = 112, normalized size = 0.63 \begin {gather*} \frac {182655 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}-\frac {7570625 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{161051}+\frac {64}{1369599 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {13056}{35153041 \sqrt {-2 x +1}}+\frac {-\frac {265625 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}+\frac {578125 \sqrt {-2 x +1}}{1331}}{\left (-10 x -6\right )^{2}}-\frac {26244 \left (\frac {221 \left (-2 x +1\right )^{\frac {3}{2}}}{36}-\frac {1561 \sqrt {-2 x +1}}{108}\right )}{2401 \left (-6 x -4\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-2*x+1)^(5/2)/(3*x+2)^3/(5*x+3)^3,x)
[Out]
64/1369599/(-2*x+1)^(3/2)+13056/35153041/(-2*x+1)^(1/2)+312500/14641*(-187/20*(-2*x+1)^(3/2)+407/20*(-2*x+1)^(
1/2))/(-10*x-6)^2-7570625/161051*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-26244/2401*(221/36*(-2*x+1)^(3
/2)-1561/108*(-2*x+1)^(1/2))/(-6*x-4)^2+182655/2401*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)
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maxima [A] time = 1.25, size = 164, normalized size = 0.92 \begin {gather*} \frac {7570625}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {182655}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (26165524500 \, {\left (2 \, x - 1\right )}^{5} + 177932259675 \, {\left (2 \, x - 1\right )}^{4} + 403131105480 \, {\left (2 \, x - 1\right )}^{3} + 304294845085 \, {\left (2 \, x - 1\right )}^{2} - 25803008 \, x + 14988512\right )}}{15065589 \, {\left (225 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2040 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 6934 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10472 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 5929 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="maxima")
[Out]
7570625/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 182655/4802*sqrt(2
1)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/15065589*(26165524500*(2*x - 1)^5 + 1
77932259675*(2*x - 1)^4 + 403131105480*(2*x - 1)^3 + 304294845085*(2*x - 1)^2 - 25803008*x + 14988512)/(225*(-
2*x + 1)^(11/2) - 2040*(-2*x + 1)^(9/2) + 6934*(-2*x + 1)^(7/2) - 10472*(-2*x + 1)^(5/2) + 5929*(-2*x + 1)^(3/
2))
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mupad [B] time = 0.10, size = 125, normalized size = 0.70 \begin {gather*} \frac {182655\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {7570625\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}+\frac {\frac {1580752442\,{\left (2\,x-1\right )}^2}{8804565}-\frac {8704\,x}{571725}+\frac {53750814064\,{\left (2\,x-1\right )}^3}{225983835}+\frac {1581620086\,{\left (2\,x-1\right )}^4}{15065589}+\frac {77527480\,{\left (2\,x-1\right )}^5}{5021863}+\frac {5056}{571725}}{\frac {5929\,{\left (1-2\,x\right )}^{3/2}}{225}-\frac {10472\,{\left (1-2\,x\right )}^{5/2}}{225}+\frac {6934\,{\left (1-2\,x\right )}^{7/2}}{225}-\frac {136\,{\left (1-2\,x\right )}^{9/2}}{15}+{\left (1-2\,x\right )}^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((1 - 2*x)^(5/2)*(3*x + 2)^3*(5*x + 3)^3),x)
[Out]
(182655*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (7570625*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2)
)/11))/161051 + ((1580752442*(2*x - 1)^2)/8804565 - (8704*x)/571725 + (53750814064*(2*x - 1)^3)/225983835 + (1
581620086*(2*x - 1)^4)/15065589 + (77527480*(2*x - 1)^5)/5021863 + 5056/571725)/((5929*(1 - 2*x)^(3/2))/225 -
(10472*(1 - 2*x)^(5/2))/225 + (6934*(1 - 2*x)^(7/2))/225 - (136*(1 - 2*x)^(9/2))/15 + (1 - 2*x)^(11/2))
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sympy [C] time = 30.11, size = 3028, normalized size = 16.92
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)**(5/2)/(2+3*x)**3/(3+5*x)**3,x)
[Out]
-3868311142080000*sqrt(2)*I*(x - 1/2)**(17/2)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 -
9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 56123978510
36592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 21920924557224000*sqrt(2)*I*(x - 1/2)**(15/2)/(-50114175
2496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x -
1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 49
676964263942400*sqrt(2)*I*(x - 1/2)**(13/2)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9
653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036
592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 56275446111672480*sqrt(2)*I*(x - 1/2)**(11/2)/(-5011417524
96000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/
2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 3186
7497856150880*sqrt(2)*I*(x - 1/2)**(9/2)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653
660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592
*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 7216395978913044*sqrt(2)*I*(x - 1/2)**(7/2)/(-501141752496000
*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6
- 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 131130886
656*sqrt(2)*I*(x - 1/2)**(5/2)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 96536606255812
80*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)
**4 - 1059202535612298*(x - 1/2)**3) - 12373870432*sqrt(2)*I*(x - 1/2)**(3/2)/(-501141752496000*(x - 1/2)**9 -
3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 123888644694
95976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 23557483530000000*sqrt(5
5)*I*(x - 1/2)**9*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)
**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 561239
7851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 38124134444880000*sqrt(21)*I*(x - 1/2)**9*atan(sqrt
(42)*sqrt(x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1
/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 105
9202535612298*(x - 1/2)**3) - 11778741765000000*sqrt(55)*I*pi*(x - 1/2)**9/(-501141752496000*(x - 1/2)**9 - 34
07763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 123888644694959
76*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 19062067222440000*sqrt(21)*
I*pi*(x - 1/2)**9/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)*
*7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202
535612298*(x - 1/2)**3) + 160190888004000000*sqrt(55)*I*(x - 1/2)**8*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-501141
752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x
- 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) -
259244114225184000*sqrt(21)*I*(x - 1/2)**8*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 340
7763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 1238886446949597
6*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 80095444002000000*sqrt(55)*I
*pi*(x - 1/2)**8/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**
7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 10592025
35612298*(x - 1/2)**3) + 129622057112592000*sqrt(21)*I*pi*(x - 1/2)**8/(-501141752496000*(x - 1/2)**9 - 340776
3916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(
x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 453795657732900000*sqrt(55)*I*(
x - 1/2)**7*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 -
9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 561239785103
6592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 734397909856538400*sqrt(21)*I*(x - 1/2)**7*atan(sqrt(42)*
sqrt(x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**
7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 10592025
35612298*(x - 1/2)**3) - 226897828866450000*sqrt(55)*I*pi*(x - 1/2)**7/(-501141752496000*(x - 1/2)**9 - 340776
3916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(
x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 367198954928269200*sqrt(21)*I*p
i*(x - 1/2)**7/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7
- 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535
612298*(x - 1/2)**3) + 685498340740080000*sqrt(55)*I*(x - 1/2)**6*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-501141752
496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1
/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 110
9372776206583680*sqrt(21)*I*(x - 1/2)**6*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 34077
63916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*
(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 342749170370040000*sqrt(55)*I*
pi*(x - 1/2)**6/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7
- 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 105920253
5612298*(x - 1/2)**3) + 554686388103291840*sqrt(21)*I*pi*(x - 1/2)**6/(-501141752496000*(x - 1/2)**9 - 3407763
916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x
- 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 582371094090555000*sqrt(55)*I*(x
- 1/2)**5*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9
653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036
592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 942477317649224280*sqrt(21)*I*(x - 1/2)**5*atan(sqrt(42)*s
qrt(x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7
- 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 105920253
5612298*(x - 1/2)**3) - 291185547045277500*sqrt(55)*I*pi*(x - 1/2)**5/(-501141752496000*(x - 1/2)**9 - 3407763
916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x
- 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 471238658824612140*sqrt(21)*I*pi
*(x - 1/2)**5/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 -
14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 10592025356
12298*(x - 1/2)**3) + 263825493048810000*sqrt(55)*I*(x - 1/2)**4*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-5011417524
96000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/
2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 4269
60653678087760*sqrt(21)*I*(x - 1/2)**4*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 3407763
916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x
- 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 131912746524405000*sqrt(55)*I*pi
*(x - 1/2)**4/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 -
14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 10592025356
12298*(x - 1/2)**3) + 213480326839043880*sqrt(21)*I*pi*(x - 1/2)**4/(-501141752496000*(x - 1/2)**9 - 340776391
6972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x -
1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 49790595501858750*sqrt(55)*I*(x -
1/2)**3*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653
660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592
*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) - 80578358659835190*sqrt(21)*I*(x - 1/2)**3*atan(sqrt(42)*sqrt(
x - 1/2)/7)/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 1
4582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612
298*(x - 1/2)**3) - 24895297750929375*sqrt(55)*I*pi*(x - 1/2)**3/(-501141752496000*(x - 1/2)**9 - 340776391697
2800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/
2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*(x - 1/2)**3) + 40289179329917595*sqrt(21)*I*pi*(x -
1/2)**3/(-501141752496000*(x - 1/2)**9 - 3407763916972800*(x - 1/2)**8 - 9653660625581280*(x - 1/2)**7 - 14582
705295075456*(x - 1/2)**6 - 12388864469495976*(x - 1/2)**5 - 5612397851036592*(x - 1/2)**4 - 1059202535612298*
(x - 1/2)**3)
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