3.19.45 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx\) [1845]

Optimal. Leaf size=97 \[ -\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-138 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-138/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+134/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-10*(1
-2*x)^(1/2)/(3+5*x)+(1-2*x)^(1/2)/(2+3*x)/(3+5*x)

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Rubi [A]
time = 0.02, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {101, 156, 162, 65, 212} \begin {gather*} -\frac {10 \sqrt {1-2 x}}{5 x+3}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)}-138 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x])/(3 + 5*x) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)) - 138*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]] + 134*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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 101

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 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 {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-\int \frac {-13+15 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}+\frac {1}{11} \int \frac {-539+330 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}+207 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-335 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-207 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+335 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-138 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \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 = 0.20, size = 83, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {1-2 x} (19+30 x)}{6+19 x+15 x^2}-138 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 - 2*x]*(19 + 30*x))/(6 + 19*x + 15*x^2)) - 138*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 134*Sqrt
[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]
time = 0.16, size = 70, normalized size = 0.72

method	result	size
risch	\(\frac {\left (-1+2 x \right ) \left (30 x +19\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}+\frac {134 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {138 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(68\)
derivativedivides	\(\frac {2 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {134 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {2 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}-\frac {138 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(70\)
default	\(\frac {2 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {134 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {2 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}-\frac {138 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(70\)
trager	\(-\frac {\left (30 x +19\right ) \sqrt {1-2 x}}{15 x^{2}+19 x +6}-\frac {69 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{7}+\frac {67 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{11}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1-2*x)^(1/2)/(-6/5-2*x)+134/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2*(1-2*x)^(1/2)/(-4/3-2*x)-138
/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.51, size = 110, normalized size = 1.13 \begin {gather*} -\frac {67}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {69}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 34 \, \sqrt {-2 \, x + 1}\right )}}{15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-67/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 69/7*sqrt(21)*log(-(sqrt(2
1) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4*(15*(-2*x + 1)^(3/2) - 34*sqrt(-2*x + 1))/(15*(2*x -
1)^2 + 136*x + 9)

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Fricas [A]
time = 0.86, size = 122, normalized size = 1.26 \begin {gather*} \frac {469 \, \sqrt {11} \sqrt {5} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 759 \, \sqrt {7} \sqrt {3} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (30 \, x + 19\right )} \sqrt {-2 \, x + 1}}{77 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/77*(469*sqrt(11)*sqrt(5)*(15*x^2 + 19*x + 6)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 7
59*sqrt(7)*sqrt(3)*(15*x^2 + 19*x + 6)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(30*x +
19)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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Sympy [A]
time = 46.92, size = 355, normalized size = 3.66 \begin {gather*} - 84 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 220 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) + 408 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right ) - 680 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-84*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*(sq
rt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqr
t(1 - 2*x) < sqrt(21)/3))) - 220*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqr
t(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, (sq
rt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5))) + 408*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 -
2*x)/7)/21, x < -2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, x > -2/3)) - 680*Piecewise((-sqrt(55)*a
coth(sqrt(55)*sqrt(1 - 2*x)/11)/55, x < -3/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, x > -3/5))

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Giac [A]
time = 1.30, size = 116, normalized size = 1.20 \begin {gather*} -\frac {67}{11} \, \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 {69}{7} \, \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 {4 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 34 \, \sqrt {-2 \, x + 1}\right )}}{15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-67/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 69/7*sqrt(21)*lo
g(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4*(15*(-2*x + 1)^(3/2) - 34*sqrt(-2
*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)

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Mupad [B]
time = 0.10, size = 72, normalized size = 0.74 \begin {gather*} \frac {134\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}-\frac {138\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {\frac {136\,\sqrt {1-2\,x}}{15}-4\,{\left (1-2\,x\right )}^{3/2}}{\frac {136\,x}{15}+{\left (2\,x-1\right )}^2+\frac {3}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(134*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 - (138*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7
- ((136*(1 - 2*x)^(1/2))/15 - 4*(1 - 2*x)^(3/2))/((136*x)/15 + (2*x - 1)^2 + 3/5)

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