3.20.18 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=160 \[ -\frac {1676975 \sqrt {1-2 x}}{7546 (5 x+3)}+\frac {7585 \sqrt {1-2 x}}{343 (3 x+2) (5 x+3)}+\frac {145 \sqrt {1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 160, 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 = {103, 151, 156, 63, 206} \begin {gather*} -\frac {1676975 \sqrt {1-2 x}}{7546 (5 x+3)}+\frac {7585 \sqrt {1-2 x}}{343 (3 x+2) (5 x+3)}+\frac {145 \sqrt {1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \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)^2),x]

[Out]

(-1676975*Sqrt[1 - 2*x])/(7546*(3 + 5*x)) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)) + (145*Sqrt[1 - 2*x])/(98*
(2 + 3*x)^2*(3 + 5*x)) + (7585*Sqrt[1 - 2*x])/(343*(2 + 3*x)*(3 + 5*x)) - (1051695*Sqrt[3/7]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/343 + (32750*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 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {1}{21} \int \frac {75-105 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {1}{294} \int \frac {7920-10875 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}+\frac {\int \frac {596595-682650 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{2058}\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {\int \frac {24644085-15092775 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{22638}\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}+\frac {3155085}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {81875}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {3155085}{686} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {81875}{11} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{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 = 101, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {1-2 x} \left (45278325 x^3+89054820 x^2+58335165 x+12724912\right )}{7546 (3 x+2)^3 (5 x+3)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \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)^2),x]

[Out]

-1/7546*(Sqrt[1 - 2*x]*(12724912 + 58335165*x + 89054820*x^2 + 45278325*x^3))/((2 + 3*x)^3*(3 + 5*x)) - (10516
95*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (32750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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IntegrateAlgebraic [A]  time = 0.33, size = 121, normalized size = 0.76 \begin {gather*} \frac {\sqrt {1-2 x} \left (45278325 (1-2 x)^3-313944615 (1-2 x)^2+725394915 (1-2 x)-558527921\right )}{3773 (3 (1-2 x)-7)^3 (5 (1-2 x)-11)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-558527921 + 725394915*(1 - 2*x) - 313944615*(1 - 2*x)^2 + 45278325*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(3773*(-7 +
3*(1 - 2*x))^3*(-11 + 5*(1 - 2*x))) - (1051695*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (32750*Sqrt[5
/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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fricas [A]  time = 1.11, size = 162, normalized size = 1.01 \begin {gather*} \frac {78632750 \, \sqrt {11} \sqrt {5} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 127255095 \, \sqrt {7} \sqrt {3} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (45278325 \, x^{3} + 89054820 \, x^{2} + 58335165 \, x + 12724912\right )} \sqrt {-2 \, x + 1}}{581042 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/581042*(78632750*sqrt(11)*sqrt(5)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*
x + 1) - 5*x + 8)/(5*x + 3)) + 127255095*sqrt(7)*sqrt(3)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((sqrt(
7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(45278325*x^3 + 89054820*x^2 + 58335165*x + 12724912)*sqr
t(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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giac [A]  time = 1.26, size = 139, normalized size = 0.87 \begin {gather*} -\frac {16375}{121} \, \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 {1051695}{4802} \, \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 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} - \frac {9 \, {\left (68085 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 320740 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 377839 \, \sqrt {-2 \, x + 1}\right )}}{2744 \, {\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)^2/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-16375/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1051695/4802
*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 625/11*sqrt(-2*x + 1)/(
5*x + 3) - 9/2744*(68085*(2*x - 1)^2*sqrt(-2*x + 1) - 320740*(-2*x + 1)^(3/2) + 377839*sqrt(-2*x + 1))/(3*x +
2)^3

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maple [A]  time = 0.02, size = 91, normalized size = 0.57 \begin {gather*} -\frac {1051695 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}+\frac {32750 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {250 \sqrt {-2 x +1}}{11 \left (-2 x -\frac {6}{5}\right )}+\frac {\frac {612765 \left (-2 x +1\right )^{\frac {5}{2}}}{343}-\frac {412380 \left (-2 x +1\right )^{\frac {3}{2}}}{49}+\frac {69399 \sqrt {-2 x +1}}{7}}{\left (-6 x -4\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/11*(-2*x+1)^(1/2)/(-2*x-6/5)+32750/121*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+972*(7565/4116*(-2*x
+1)^(5/2)-11455/1323*(-2*x+1)^(3/2)+7711/756*(-2*x+1)^(1/2))/(-6*x-4)^3-1051695/2401*arctanh(1/7*21^(1/2)*(-2*
x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.23, size = 146, normalized size = 0.91 \begin {gather*} -\frac {16375}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1051695}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {45278325 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 313944615 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 725394915 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 558527921 \, \sqrt {-2 \, x + 1}}{3773 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16375/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1051695/4802*sqrt(21)*
log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/3773*(45278325*(-2*x + 1)^(7/2) - 313944
615*(-2*x + 1)^(5/2) + 725394915*(-2*x + 1)^(3/2) - 558527921*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1
)^3 + 4284*(2*x - 1)^2 + 13132*x - 2793)

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mupad [B]  time = 0.10, size = 108, normalized size = 0.68 \begin {gather*} \frac {32750\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {1051695\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {11398529\,\sqrt {1-2\,x}}{10395}-\frac {2302841\,{\left (1-2\,x\right )}^{3/2}}{1617}+\frac {6976547\,{\left (1-2\,x\right )}^{5/2}}{11319}-\frac {335395\,{\left (1-2\,x\right )}^{7/2}}{3773}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \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)^2),x)

[Out]

(32750*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - (1051695*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))
/7))/2401 - ((11398529*(1 - 2*x)^(1/2))/10395 - (2302841*(1 - 2*x)^(3/2))/1617 + (6976547*(1 - 2*x)^(5/2))/113
19 - (335395*(1 - 2*x)^(7/2))/3773)/((13132*x)/135 + (476*(2*x - 1)^2)/15 + (46*(2*x - 1)^3)/5 + (2*x - 1)^4 -
931/45)

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

[Out]

Timed out

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