3.21.68 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3} \, dx\) [2068]

Optimal. Leaf size=180 \[ -\frac {2076675 \sqrt {1-2 x}}{7546 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac {137735775 \sqrt {1-2 x}}{83006 (3+5 x)}+\frac {7852680}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2689875}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

7852680/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2689875/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55
^(1/2)-2076675/7546*(1-2*x)^(1/2)/(3+5*x)^2+1/7*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^2+90/49*(1-2*x)^(1/2)/(2+3*x)^
2/(3+5*x)^2+12555/343*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+137735775/83006*(1-2*x)^(1/2)/(3+5*x)

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Rubi [A]
time = 0.05, antiderivative size = 180, normalized size of antiderivative = 1.00, 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 = {105, 156, 162, 65, 212} \begin {gather*} \frac {137735775 \sqrt {1-2 x}}{83006 (5 x+3)}-\frac {2076675 \sqrt {1-2 x}}{7546 (5 x+3)^2}+\frac {12555 \sqrt {1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac {90 \sqrt {1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac {7852680}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2689875}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2076675*Sqrt[1 - 2*x])/(7546*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)^2) + (90*Sqrt[1 - 2*x])/(
49*(2 + 3*x)^2*(3 + 5*x)^2) + (12555*Sqrt[1 - 2*x])/(343*(2 + 3*x)*(3 + 5*x)^2) + (137735775*Sqrt[1 - 2*x])/(8
3006*(3 + 5*x)) + (7852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/121

Rule 65

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

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] && ILtQ[m, -1]

Rule 162

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {1}{21} \int \frac {90-135 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {1}{294} \int \frac {12510-18900 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac {\int \frac {1362060-1883250 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx}{2058}\\ &=-\frac {2076675 \sqrt {1-2 x}}{7546 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}-\frac {\int \frac {97998660-112140450 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{45276}\\ &=-\frac {2076675 \sqrt {1-2 x}}{7546 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac {137735775 \sqrt {1-2 x}}{83006 (3+5 x)}+\frac {\int \frac {4048216380-2479243950 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{498036}\\ &=-\frac {2076675 \sqrt {1-2 x}}{7546 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac {137735775 \sqrt {1-2 x}}{83006 (3+5 x)}-\frac {11779020}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {13449375}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {2076675 \sqrt {1-2 x}}{7546 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac {137735775 \sqrt {1-2 x}}{83006 (3+5 x)}+\frac {11779020}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {13449375}{242} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {2076675 \sqrt {1-2 x}}{7546 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac {90 \sqrt {1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac {12555 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac {137735775 \sqrt {1-2 x}}{83006 (3+5 x)}+\frac {7852680}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2689875}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 106, normalized size = 0.59 \begin {gather*} \frac {\sqrt {1-2 x} \left (3135381218+19599448500 x+45899434890 x^2+47728484550 x^3+18594329625 x^4\right )}{83006 (2+3 x)^3 (3+5 x)^2}+\frac {7852680}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2689875}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(3135381218 + 19599448500*x + 45899434890*x^2 + 47728484550*x^3 + 18594329625*x^4))/(83006*(2 +
3*x)^3*(3 + 5*x)^2) + (7852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]
time = 0.22, size = 103, normalized size = 0.57

method	result	size
risch	\(-\frac {37188659250 x^{5}[phone] x^{4}[phone] x^{3}-6700537890 x^{2}-13328686064 x -3135381218}{83006 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}-\frac {2689875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {7852680 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(86\)
derivativedivides	\(\frac {-\frac {815625 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {161875 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {2689875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {2916 \left (\frac {3755 \left (1-2 x \right )^{\frac {5}{2}}}{1029}-\frac {22690 \left (1-2 x \right )^{\frac {3}{2}}}{1323}+\frac {3809 \sqrt {1-2 x}}{189}\right )}{\left (-4-6 x \right )^{3}}+\frac {7852680 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(103\)
default	\(\frac {-\frac {815625 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {161875 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {2689875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {2916 \left (\frac {3755 \left (1-2 x \right )^{\frac {5}{2}}}{1029}-\frac {22690 \left (1-2 x \right )^{\frac {3}{2}}}{1323}+\frac {3809 \sqrt {1-2 x}}{189}\right )}{\left (-4-6 x \right )^{3}}+\frac {7852680 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(103\)
trager	\(\frac {\left (18594329625 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}}{83006 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {3375 \RootOf \left (\textit {\_Z}^{2}-34936495\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-34936495\right ) x +43835 \sqrt {1-2 x}+8 \RootOf \left (\textit {\_Z}^{2}-34936495\right )}{3+5 x}\right )}{2662}-\frac {3926340 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

312500*(-261/12100*(1-2*x)^(3/2)+259/5500*(1-2*x)^(1/2))/(-6-10*x)^2-2689875/1331*arctanh(1/11*55^(1/2)*(1-2*x
)^(1/2))*55^(1/2)-2916*(3755/1029*(1-2*x)^(5/2)-22690/1323*(1-2*x)^(3/2)+3809/189*(1-2*x)^(1/2))/(-4-6*x)^3+78
52680/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.58, size = 164, normalized size = 0.91 \begin {gather*} \frac {2689875}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3926340}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {18594329625 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 169834287600 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 581534624610 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 884739292920 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 504610725773 \, \sqrt {-2 \, x + 1}}{41503 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2689875/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3926340/2401*sqrt(21
)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/41503*(18594329625*(-2*x + 1)^(9/2) -
169834287600*(-2*x + 1)^(7/2) + 581534624610*(-2*x + 1)^(5/2) - 884739292920*(-2*x + 1)^(3/2) + 504610725773*s
qrt(-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)

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Fricas [A]
time = 0.65, size = 182, normalized size = 1.01 \begin {gather*} \frac {6458389875 \, \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 ) + 10451917080 \, \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 (18594329625 \, x^{4} + 47728484550 \, x^{3} + 45899434890 \, x^{2} + 19599448500 \, x + 3135381218\right )} \sqrt {-2 \, x + 1}}{6391462 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6391462*(6458389875*sqrt(11)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((sqrt(11)*s
qrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 10451917080*sqrt(7)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 176
6*x^2 + 564*x + 72)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(18594329625*x^4 + 4772848
4550*x^3 + 45899434890*x^2 + 19599448500*x + 3135381218)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766
*x^2 + 564*x + 72)

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Sympy [C] Result contains complex when optimal does not.
time = 18.49, size = 18726, normalized size = 104.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

47001794676480000000*sqrt(55)*I*(x - 1/2)**(47/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(1932336330946560000*(x -
1/2)**(47/2) + 31046203717208064000*(x - 1/2)**(45/2) + 231577627021738905600*(x - 1/2)**(43/2) + 10629387101
10936637440*(x - 1/2)**(41/2) + 3354048542457364663296*(x - 1/2)**(39/2) + 7696698250986854968320*(x - 1/2)**(
37/2) + 13245776891590104449280*(x - 1/2)**(35/2) + 17367111590212442741760*(x - 1/2)**(33/2) + 17432924800629
278445120*(x - 1/2)**(31/2) + 13331834207123103908544*(x - 1/2)**(29/2) + 7646223940064532709040*(x - 1/2)**(2
7/2) + 3189168923449757654560*(x - 1/2)**(25/2) + 914443203145284725220*(x - 1/2)**(23/2) + 161346104635697207
380*(x - 1/2)**(21/2) + 13216648996753920179*(x - 1/2)**(19/2)) - 3858139593907200000000*sqrt(55)*I*(x - 1/2)*
*(47/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(1932336330946560000*(x - 1/2)**(47/2) + 31046203717208064000*(x - 1/
2)**(45/2) + 231577627021738905600*(x - 1/2)**(43/2) + 1062938710110936637440*(x - 1/2)**(41/2) + 335404854245
7364663296*(x - 1/2)**(39/2) + 7696698250986854968320*(x - 1/2)**(37/2) + 13245776891590104449280*(x - 1/2)**(
35/2) + 17367111590212442741760*(x - 1/2)**(33/2) + 17432924800629278445120*(x - 1/2)**(31/2) + 13331834207123
103908544*(x - 1/2)**(29/2) + 7646223940064532709040*(x - 1/2)**(27/2) + 3189168923449757654560*(x - 1/2)**(25
/2) + 914443203145284725220*(x - 1/2)**(23/2) + 161346104635697207380*(x - 1/2)**(21/2) + 13216648996753920179
*(x - 1/2)**(19/2)) + 110358053746237440000*sqrt(21)*I*(x - 1/2)**(47/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(193
2336330946560000*(x - 1/2)**(47/2) + 31046203717208064000*(x - 1/2)**(45/2) + 231577627021738905600*(x - 1/2)*
*(43/2) + 1062938710110936637440*(x - 1/2)**(41/2) + 3354048542457364663296*(x - 1/2)**(39/2) + 76966982509868
54968320*(x - 1/2)**(37/2) + 13245776891590104449280*(x - 1/2)**(35/2) + 17367111590212442741760*(x - 1/2)**(3
3/2) + 17432924800629278445120*(x - 1/2)**(31/2) + 13331834207123103908544*(x - 1/2)**(29/2) + 764622394006453
2709040*(x - 1/2)**(27/2) + 3189168923449757654560*(x - 1/2)**(25/2) + 914443203145284725220*(x - 1/2)**(23/2)
+ 161346104635697207380*(x - 1/2)**(21/2) + 13216648996753920179*(x - 1/2)**(19/2)) + 6430232630713098240000*
sqrt(21)*I*(x - 1/2)**(47/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(1932336330946560000*(x - 1/2)**(47/2) + 310462037
17208064000*(x - 1/2)**(45/2) + 231577627021738905600*(x - 1/2)**(43/2) + 1062938710110936637440*(x - 1/2)**(4
1/2) + 3354048542457364663296*(x - 1/2)**(39/2) + 7696698250986854968320*(x - 1/2)**(37/2) + 13245776891590104
449280*(x - 1/2)**(35/2) + 17367111590212442741760*(x - 1/2)**(33/2) + 17432924800629278445120*(x - 1/2)**(31/
2) + 13331834207123103908544*(x - 1/2)**(29/2) + 7646223940064532709040*(x - 1/2)**(27/2) + 318916892344975765
4560*(x - 1/2)**(25/2) + 914443203145284725220*(x - 1/2)**(23/2) + 161346104635697207380*(x - 1/2)**(21/2) + 1
3216648996753920179*(x - 1/2)**(19/2)) - 3215116315356549120000*sqrt(21)*I*pi*(x - 1/2)**(47/2)/(1932336330946
560000*(x - 1/2)**(47/2) + 31046203717208064000*(x - 1/2)**(45/2) + 231577627021738905600*(x - 1/2)**(43/2) +
1062938710110936637440*(x - 1/2)**(41/2) + 3354048542457364663296*(x - 1/2)**(39/2) + 7696698250986854968320*(
x - 1/2)**(37/2) + 13245776891590104449280*(x - 1/2)**(35/2) + 17367111590212442741760*(x - 1/2)**(33/2) + 174
32924800629278445120*(x - 1/2)**(31/2) + 13331834207123103908544*(x - 1/2)**(29/2) + 7646223940064532709040*(x
- 1/2)**(27/2) + 3189168923449757654560*(x - 1/2)**(25/2) + 914443203145284725220*(x - 1/2)**(23/2) + 1613461
04635697207380*(x - 1/2)**(21/2) + 13216648996753920179*(x - 1/2)**(19/2)) + 1929069796953600000000*sqrt(55)*I
*pi*(x - 1/2)**(47/2)/(1932336330946560000*(x - 1/2)**(47/2) + 31046203717208064000*(x - 1/2)**(45/2) + 231577
627021738905600*(x - 1/2)**(43/2) + 1062938710110936637440*(x - 1/2)**(41/2) + 3354048542457364663296*(x - 1/2
)**(39/2) + 7696698250986854968320*(x - 1/2)**(37/2) + 13245776891590104449280*(x - 1/2)**(35/2) + 17367111590
212442741760*(x - 1/2)**(33/2) + 17432924800629278445120*(x - 1/2)**(31/2) + 13331834207123103908544*(x - 1/2)
**(29/2) + 7646223940064532709040*(x - 1/2)**(27/2) + 3189168923449757654560*(x - 1/2)**(25/2) + 9144432031452
84725220*(x - 1/2)**(23/2) + 161346104635697207380*(x - 1/2)**(21/2) + 13216648996753920179*(x - 1/2)**(19/2))
+ 755162167802112000000*sqrt(55)*I*(x - 1/2)**(45/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(1932336330946560000*
(x - 1/2)**(47/2) + 31046203717208064000*(x - 1/2)**(45/2) + 231577627021738905600*(x - 1/2)**(43/2) + 1062938
710110936637440*(x - 1/2)**(41/2) + 3354048542457364663296*(x - 1/2)**(39/2) + 7696698250986854968320*(x - 1/2
)**(37/2) + 13245776891590104449280*(x - 1/2)**(35/2) + 17367111590212442741760*(x - 1/2)**(33/2) + 1743292480
0629278445120*(x - 1/2)**(31/2) + 13331834207123103908544*(x - 1/2)**(29/2) + 7646223940064532709040*(x - 1/2)
**(27/2) + 3189168923449757654560*(x - 1/2)**(25/2) + 914443203145284725220*(x - 1/2)**(23/2) + 16134610463569
7207380*(x - 1/2)**(21/2) + 1321664899675392017...

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Giac [A]
time = 1.82, size = 151, normalized size = 0.84 \begin {gather*} \frac {2689875}{2662} \, \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 {3926340}{2401} \, \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 {625 \, {\left (1305 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2849 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} + \frac {27 \, {\left (33795 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 158830 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 186641 \, \sqrt {-2 \, x + 1}\right )}}{686 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2689875/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3926340/24
01*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 625/484*(1305*(-2*x +
1)^(3/2) - 2849*sqrt(-2*x + 1))/(5*x + 3)^2 + 27/686*(33795*(2*x - 1)^2*sqrt(-2*x + 1) - 158830*(-2*x + 1)^(3
/2) + 186641*sqrt(-2*x + 1))/(3*x + 2)^3

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Mupad [B]
time = 1.29, size = 125, normalized size = 0.69 \begin {gather*} \frac {\frac {936198007\,\sqrt {1-2\,x}}{51975}-\frac {936232056\,{\left (1-2\,x\right )}^{3/2}}{29645}+\frac {4307663886\,{\left (1-2\,x\right )}^{5/2}}{207515}-\frac {251606352\,{\left (1-2\,x\right )}^{7/2}}{41503}+\frac {27547155\,{\left (1-2\,x\right )}^{9/2}}{41503}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}}+\frac {7852680\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {2689875\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((936198007*(1 - 2*x)^(1/2))/51975 - (936232056*(1 - 2*x)^(3/2))/29645 + (4307663886*(1 - 2*x)^(5/2))/207515 -
(251606352*(1 - 2*x)^(7/2))/41503 + (27547155*(1 - 2*x)^(9/2))/41503)/((182182*x)/675 + (79954*(2*x - 1)^2)/6
75 + (3898*(2*x - 1)^3)/75 + (57*(2*x - 1)^4)/5 + (2*x - 1)^5 - 49588/675) + (7852680*21^(1/2)*atanh((21^(1/2)
*(1 - 2*x)^(1/2))/7))/2401 - (2689875*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331

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