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

Optimal. Leaf size=133 \[ \frac {337955 \sqrt {1-2 x}}{2744 (3 x+2)}+\frac {14555 \sqrt {1-2 x}}{1176 (3 x+2)^2}+\frac {139 \sqrt {1-2 x}}{84 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 133, 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} \begin {gather*} \frac {337955 \sqrt {1-2 x}}{2744 (3 x+2)}+\frac {14555 \sqrt {1-2 x}}{1176 (3 x+2)^2}+\frac {139 \sqrt {1-2 x}}{84 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \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)^5*(3 + 5*x)),x]

[Out]

Sqrt[1 - 2*x]/(4*(2 + 3*x)^4) + (139*Sqrt[1 - 2*x])/(84*(2 + 3*x)^3) + (14555*Sqrt[1 - 2*x])/(1176*(2 + 3*x)^2
) + (337955*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (11656955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 25
0*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 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)^5 (3+5 x)} \, dx &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}-\frac {1}{4} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}-\frac {1}{84} \int \frac {-2535+3475 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}-\frac {\int \frac {-192405+218325 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}-\frac {\int \frac {-8277405+5069325 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}-\frac {11656955 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2744}+6875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {11656955 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2744}-6875 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 88, normalized size = 0.66 \begin {gather*} \frac {\sqrt {1-2 x} \left (9124785 x^3+18555225 x^2+12587542 x+2849254\right )}{2744 (3 x+2)^4}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \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)^5*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(2849254 + 12587542*x + 18555225*x^2 + 9124785*x^3))/(2744*(2 + 3*x)^4) + (11656955*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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IntegrateAlgebraic [A]  time = 0.54, size = 104, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-2 x} \left (9124785 (1-2 x)^3-64484805 (1-2 x)^2+151945423 (1-2 x)-119379435\right )}{1372 (3 (1-2 x)-7)^4}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1372*((-119379435 + 151945423*(1 - 2*x) - 64484805*(1 - 2*x)^2 + 9124785*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(-7 +
3*(1 - 2*x))^4 + (11656955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[55]*ArcTanh[Sqrt[5/11]
*Sqrt[1 - 2*x]]

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fricas [A]  time = 1.45, size = 150, normalized size = 1.13 \begin {gather*} \frac {7203000 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11656955 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (9124785 \, x^{3} + 18555225 \, x^{2} + 12587542 \, x + 2849254\right )} \sqrt {-2 \, x + 1}}{57624 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/57624*(7203000*sqrt(55)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*
x + 3)) + 11656955*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(
3*x + 2)) + 21*(9124785*x^3 + 18555225*x^2 + 12587542*x + 2849254)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)

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giac [A]  time = 1.26, size = 139, normalized size = 1.05 \begin {gather*} 125 \, \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 {11656955}{57624} \, \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 {9124785 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 64484805 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 151945423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 119379435 \, \sqrt {-2 \, x + 1}}{21952 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 11656955/57624*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21952*(9124785*(2*x - 1)^3
*sqrt(-2*x + 1) + 64484805*(2*x - 1)^2*sqrt(-2*x + 1) - 151945423*(-2*x + 1)^(3/2) + 119379435*sqrt(-2*x + 1))
/(3*x + 2)^4

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maple [A]  time = 0.01, size = 84, normalized size = 0.63 \begin {gather*} \frac {11656955 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{28812}-250 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )-\frac {162 \left (\frac {337955 \left (-2 x +1\right )^{\frac {7}{2}}}{8232}-\frac {3070705 \left (-2 x +1\right )^{\frac {5}{2}}}{10584}+\frac {3100927 \left (-2 x +1\right )^{\frac {3}{2}}}{4536}-\frac {116015 \sqrt {-2 x +1}}{216}\right )}{\left (-6 x -4\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-162*(337955/8232*(-2*x+1)^(7/2)-3070705/10584*(-2*x+1)^(5/
2)+3100927/4536*(-2*x+1)^(3/2)-116015/216*(-2*x+1)^(1/2))/(-6*x-4)^4+11656955/28812*arctanh(1/7*21^(1/2)*(-2*x
+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.30, size = 146, normalized size = 1.10 \begin {gather*} 125 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {11656955}{57624} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9124785 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 64484805 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 151945423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 119379435 \, \sqrt {-2 \, x + 1}}{1372 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 11656955/57624*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/1372*(9124785*(-2*x + 1)^(7/2) - 64484805*(-2
*x + 1)^(5/2) + 151945423*(-2*x + 1)^(3/2) - 119379435*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 264
6*(2*x - 1)^2 + 8232*x - 1715)

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mupad [B]  time = 1.21, size = 107, normalized size = 0.80 \begin {gather*} \frac {11656955\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{28812}-250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {116015\,\sqrt {1-2\,x}}{108}-\frac {3100927\,{\left (1-2\,x\right )}^{3/2}}{2268}+\frac {3070705\,{\left (1-2\,x\right )}^{5/2}}{5292}-\frac {337955\,{\left (1-2\,x\right )}^{7/2}}{4116}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11656955*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/28812 - 250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/
11) + ((116015*(1 - 2*x)^(1/2))/108 - (3100927*(1 - 2*x)^(3/2))/2268 + (3070705*(1 - 2*x)^(5/2))/5292 - (33795
5*(1 - 2*x)^(7/2))/4116)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

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

[Out]

Timed out

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