3.19.51 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=208 \[ \frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}+\frac {12068887 \sqrt {1-2 x}}{1323 (3 x+2) (5 x+3)}+\frac {924025 \sqrt {1-2 x}}{1512 (3 x+2)^2 (5 x+3)}+\frac {16549 \sqrt {1-2 x}}{270 (3 x+2)^3 (5 x+3)}+\frac {1379 \sqrt {1-2 x}}{180 (3 x+2)^4 (5 x+3)}-\frac {323422735 \sqrt {1-2 x}}{3528 (5 x+3)}-\frac {2231141147 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 208, 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 = {98, 149, 151, 156, 63, 206} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}+\frac {12068887 \sqrt {1-2 x}}{1323 (3 x+2) (5 x+3)}+\frac {924025 \sqrt {1-2 x}}{1512 (3 x+2)^2 (5 x+3)}+\frac {16549 \sqrt {1-2 x}}{270 (3 x+2)^3 (5 x+3)}+\frac {1379 \sqrt {1-2 x}}{180 (3 x+2)^4 (5 x+3)}-\frac {323422735 \sqrt {1-2 x}}{3528 (5 x+3)}-\frac {2231141147 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

(-323422735*Sqrt[1 - 2*x])/(3528*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*(3 + 5*x)) + (1379*Sqrt[1 -
2*x])/(180*(2 + 3*x)^4*(3 + 5*x)) + (16549*Sqrt[1 - 2*x])/(270*(2 + 3*x)^3*(3 + 5*x)) + (924025*Sqrt[1 - 2*x])
/(1512*(2 + 3*x)^2*(3 + 5*x)) + (12068887*Sqrt[1 - 2*x])/(1323*(2 + 3*x)*(3 + 5*x)) - (2231141147*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) + 111650*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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])

Rule 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[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] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 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 {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1}{15} \int \frac {(263-295 x) \sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}-\frac {1}{180} \int \frac {-37432+59695 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}-\frac {\int \frac {-5374285+8109010 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx}{3780}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}-\frac {\int \frac {-587414870+808521875 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx}{52920}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac {12068887 \sqrt {1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac {\int \frac {-44297056545+50689325400 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{370440}\\ &=-\frac {323422735 \sqrt {1-2 x}}{3528 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac {12068887 \sqrt {1-2 x}}{1323 (2+3 x) (3+5 x)}+\frac {\int \frac {-1829861506935+1120659776775 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{4074840}\\ &=-\frac {323422735 \sqrt {1-2 x}}{3528 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac {12068887 \sqrt {1-2 x}}{1323 (2+3 x) (3+5 x)}+\frac {2231141147 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1176}-3070375 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {323422735 \sqrt {1-2 x}}{3528 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac {12068887 \sqrt {1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac {2231141147 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1176}+3070375 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {323422735 \sqrt {1-2 x}}{3528 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac {12068887 \sqrt {1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac {2231141147 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 105, normalized size = 0.50 \begin {gather*} -\frac {\sqrt {1-2 x} \left (130986207675 x^5+432275892930 x^4+570477768855 x^3+376323861626 x^2+124085884254 x+16360698684\right )}{5880 (3 x+2)^5 (5 x+3)}-\frac {2231141147 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

-1/5880*(Sqrt[1 - 2*x]*(16360698684 + 124085884254*x + 376323861626*x^2 + 570477768855*x^3 + 432275892930*x^4
+ 130986207675*x^5))/((2 + 3*x)^5*(3 + 5*x)) - (2231141147*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) +
111650*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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IntegrateAlgebraic [A]  time = 0.55, size = 133, normalized size = 0.64 \begin {gather*} \frac {\sqrt {1-2 x} \left (130986207675 (1-2 x)^5-1519482824235 (1-2 x)^4+7049980295610 (1-2 x)^3-16353496911178 (1-2 x)^2+18965427342155 (1-2 x)-8796956467915\right )}{2940 (3 (1-2 x)-7)^5 (5 (1-2 x)-11)}-\frac {2231141147 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

((-8796956467915 + 18965427342155*(1 - 2*x) - 16353496911178*(1 - 2*x)^2 + 7049980295610*(1 - 2*x)^3 - 1519482
824235*(1 - 2*x)^4 + 130986207675*(1 - 2*x)^5)*Sqrt[1 - 2*x])/(2940*(-7 + 3*(1 - 2*x))^5*(-11 + 5*(1 - 2*x)))
- (2231141147*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) + 111650*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]]

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fricas [A]  time = 1.52, size = 190, normalized size = 0.91 \begin {gather*} \frac {6893271000 \, \sqrt {55} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11155705735 \, \sqrt {21} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (130986207675 \, x^{5} + 432275892930 \, x^{4} + 570477768855 \, x^{3} + 376323861626 \, x^{2} + 124085884254 \, x + 16360698684\right )} \sqrt {-2 \, x + 1}}{123480 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/123480*(6893271000*sqrt(55)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log((5*x - s
qrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 11155705735*sqrt(21)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 336
0*x^2 + 880*x + 96)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(130986207675*x^5 + 432275892930*x
^4 + 570477768855*x^3 + 376323861626*x^2 + 124085884254*x + 16360698684)*sqrt(-2*x + 1))/(1215*x^6 + 4779*x^5
+ 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)

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giac [A]  time = 0.96, size = 171, normalized size = 0.82 \begin {gather*} -55825 \, \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 {2231141147}{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 {15125 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {21875000535 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 205345418670 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 722914128048 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1131203610530 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 663838720265 \, \sqrt {-2 \, x + 1}}{94080 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

-55825*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2231141147/24696
*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 15125*sqrt(-2*x + 1)/(5
*x + 3) - 1/94080*(21875000535*(2*x - 1)^4*sqrt(-2*x + 1) + 205345418670*(2*x - 1)^3*sqrt(-2*x + 1) + 72291412
8048*(2*x - 1)^2*sqrt(-2*x + 1) - 1131203610530*(-2*x + 1)^(3/2) + 663838720265*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.02, size = 109, normalized size = 0.52 \begin {gather*} -\frac {2231141147 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{12348}+111650 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {6050 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {\frac {1458333369 \left (-2 x +1\right )^{\frac {9}{2}}}{196}-\frac {139690761 \left (-2 x +1\right )^{\frac {7}{2}}}{2}+\frac {1229445796 \left (-2 x +1\right )^{\frac {5}{2}}}{5}-\frac {2308578797 \left (-2 x +1\right )^{\frac {3}{2}}}{6}+\frac {2709545797 \sqrt {-2 x +1}}{12}}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)/(3*x+2)^6/(5*x+3)^2,x)

[Out]

6050*(-2*x+1)^(1/2)/(-2*x-6/5)+111650*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+972*(54012347/7056*(-2*x+
1)^(9/2)-46563587/648*(-2*x+1)^(7/2)[phone]/1215*(-2*x+1)^(5/2)-2308578797/5832*(-2*x+1)^(3/2)[phone]/1
1664*(-2*x+1)^(1/2))/(-6*x-4)^5-2231141147/12348*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.25, size = 182, normalized size = 0.88 \begin {gather*} -55825 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2231141147}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {130986207675 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 1519482824235 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 7049980295610 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 16353496911178 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 18965427342155 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 8796956467915 \, \sqrt {-2 \, x + 1}}{2940 \, {\left (1215 \, {\left (2 \, x - 1\right )}^{6} + 16848 \, {\left (2 \, x - 1\right )}^{5} + 97335 \, {\left (2 \, x - 1\right )}^{4} + 299880 \, {\left (2 \, x - 1\right )}^{3} + 519645 \, {\left (2 \, x - 1\right )}^{2} + 960400 \, x - 295323\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

-55825*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2231141147/24696*sqrt(21)*
log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/2940*(130986207675*(-2*x + 1)^(11/2) - 1
519482824235*(-2*x + 1)^(9/2) + 7049980295610*(-2*x + 1)^(7/2) - 16353496911178*(-2*x + 1)^(5/2) + 18965427342
155*(-2*x + 1)^(3/2) - 8796956467915*sqrt(-2*x + 1))/(1215*(2*x - 1)^6 + 16848*(2*x - 1)^5 + 97335*(2*x - 1)^4
+ 299880*(2*x - 1)^3 + 519645*(2*x - 1)^2 + 960400*x - 295323)

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mupad [B]  time = 0.10, size = 144, normalized size = 0.69 \begin {gather*} 111650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {2231141147\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-\frac {\frac {35905944767\,\sqrt {1-2\,x}}{14580}-\frac {77409907519\,{\left (1-2\,x\right )}^{3/2}}{14580}+\frac {166872417461\,{\left (1-2\,x\right )}^{5/2}}{36450}-\frac {4795904963\,{\left (1-2\,x\right )}^{7/2}}{2430}+\frac {33766284983\,{\left (1-2\,x\right )}^{9/2}}{79380}-\frac {64684547\,{\left (1-2\,x\right )}^{11/2}}{1764}}{\frac {192080\,x}{243}+\frac {34643\,{\left (2\,x-1\right )}^2}{81}+\frac {6664\,{\left (2\,x-1\right )}^3}{27}+\frac {721\,{\left (2\,x-1\right )}^4}{9}+\frac {208\,{\left (2\,x-1\right )}^5}{15}+{\left (2\,x-1\right )}^6-\frac {98441}{405}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)/((3*x + 2)^6*(5*x + 3)^2),x)

[Out]

111650*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (2231141147*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7
))/12348 - ((35905944767*(1 - 2*x)^(1/2))/14580 - (77409907519*(1 - 2*x)^(3/2))/14580 + (166872417461*(1 - 2*x
)^(5/2))/36450 - (4795904963*(1 - 2*x)^(7/2))/2430 + (33766284983*(1 - 2*x)^(9/2))/79380 - (64684547*(1 - 2*x)
^(11/2))/1764)/((192080*x)/243 + (34643*(2*x - 1)^2)/81 + (6664*(2*x - 1)^3)/27 + (721*(2*x - 1)^4)/9 + (208*(
2*x - 1)^5)/15 + (2*x - 1)^6 - 98441/405)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**2,x)

[Out]

Timed out

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