3.1855 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=153 \[ \frac {34655 \sqrt {1-2 x}}{77 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{14 (5 x+3)^2}+\frac {139 \sqrt {1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac {43467}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

43467/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-66325/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-
1045/14*(1-2*x)^(1/2)/(3+5*x)^2+1/2*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+139/14*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+3
4655/77*(1-2*x)^(1/2)/(3+5*x)

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Rubi [A]  time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {34655 \sqrt {1-2 x}}{77 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{14 (5 x+3)^2}+\frac {139 \sqrt {1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac {43467}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^2) + (139*Sqrt[1 - 2*x])/(14*(
2 + 3*x)*(3 + 5*x)^2) + (34655*Sqrt[1 - 2*x])/(77*(3 + 5*x)) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/7 - (66325*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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 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 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)^3 (3+5 x)^3} \, dx &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}-\frac {1}{2} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}-\frac {1}{14} \int \frac {-2513+3475 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {1}{308} \int \frac {-180818+206910 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}-\frac {\int \frac {-7469374+4574460 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{3388}\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}-\frac {130401}{14} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {331625}{22} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {130401}{14} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {331625}{22} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {43467}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \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 = 0.13, size = 119, normalized size = 0.78 \[ \frac {10519014 \sqrt {21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-6499850 \sqrt {55} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+77 \sqrt {1-2 x} \left (3118950 x^3+5926515 x^2+3748007 x+788875\right )}{11858 (3 x+2)^2 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(77*Sqrt[1 - 2*x]*(788875 + 3748007*x + 5926515*x^2 + 3118950*x^3) + 10519014*Sqrt[21]*(6 + 19*x + 15*x^2)^2*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 6499850*Sqrt[55]*(6 + 19*x + 15*x^2)^2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1
1858*(2 + 3*x)^2*(3 + 5*x)^2)

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fricas [A]  time = 0.79, size = 162, normalized size = 1.06 \[ \frac {3249925 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5259507 \, \sqrt {7} \sqrt {3} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (3118950 \, x^{3} + 5926515 \, x^{2} + 3748007 \, x + 788875\right )} \sqrt {-2 \, x + 1}}{11858 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11858*(3249925*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*x +
1) + 5*x - 8)/(5*x + 3)) + 5259507*sqrt(7)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(-(sqrt(7)*s
qrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(3118950*x^3 + 5926515*x^2 + 3748007*x + 788875)*sqrt(-2*x +
1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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giac [A]  time = 1.25, size = 148, normalized size = 0.97 \[ \frac {66325}{242} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (1559475 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10604940 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

66325/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 43467/98*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/77*(1559475*(2*x - 1)^3*sq
rt(-2*x + 1) + 10604940*(2*x - 1)^2*sqrt(-2*x + 1) - 24027469*(-2*x + 1)^(3/2) + 18137504*sqrt(-2*x + 1))/(15*
(2*x - 1)^2 + 136*x + 9)^2

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maple [A]  time = 0.02, size = 94, normalized size = 0.61 \[ \frac {43467 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{49}-\frac {66325 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {-\frac {24875 \left (-2 x +1\right )^{\frac {3}{2}}}{11}+4925 \sqrt {-2 x +1}}{\left (-10 x -6\right )^{2}}-\frac {972 \left (\frac {209 \left (-2 x +1\right )^{\frac {3}{2}}}{252}-\frac {211 \sqrt {-2 x +1}}{108}\right )}{\left (-6 x -4\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2500*(-199/220*(-2*x+1)^(3/2)+197/100*(-2*x+1)^(1/2))/(-10*x-6)^2-66325/121*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/
2))*55^(1/2)-972*(209/252*(-2*x+1)^(3/2)-211/108*(-2*x+1)^(1/2))/(-6*x-4)^2+43467/49*arctanh(1/7*21^(1/2)*(-2*
x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.13, size = 146, normalized size = 0.95 \[ \frac {66325}{242} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1559475 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10604940 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

66325/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 43467/98*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/77*(1559475*(-2*x + 1)^(7/2) - 10604940*(-2*x
+ 1)^(5/2) + 24027469*(-2*x + 1)^(3/2) - 18137504*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*
(2*x - 1)^2 + 20944*x - 4543)

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mupad [B]  time = 0.10, size = 107, normalized size = 0.70 \[ \frac {43467\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {66325\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {\frac {471104\,\sqrt {1-2\,x}}{225}-\frac {48054938\,{\left (1-2\,x\right )}^{3/2}}{17325}+\frac {1413992\,{\left (1-2\,x\right )}^{5/2}}{1155}-\frac {13862\,{\left (1-2\,x\right )}^{7/2}}{77}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(43467*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (66325*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11)
)/121 + ((471104*(1 - 2*x)^(1/2))/225 - (48054938*(1 - 2*x)^(3/2))/17325 + (1413992*(1 - 2*x)^(5/2))/1155 - (1
3862*(1 - 2*x)^(7/2))/77)/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2*x - 1)^3)/15 + (2*x - 1)^4 - 4543/
225)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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