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

Optimal. Leaf size=147 \[ \frac {1020 \sqrt {1-2 x}}{5 x+3}-\frac {1015 \sqrt {1-2 x}}{6 (5 x+3)^2}+\frac {45 \sqrt {1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 147, 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 = {98, 151, 156, 63, 206} \begin {gather*} \frac {1020 \sqrt {1-2 x}}{5 x+3}-\frac {1015 \sqrt {1-2 x}}{6 (5 x+3)^2}+\frac {45 \sqrt {1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \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 - 2*x)^(3/2)/((2 + 3*x)^3*(3 + 5*x)^3),x]

[Out]

(-1015*Sqrt[1 - 2*x])/(6*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)^2) + (45*Sqrt[1 - 2*x])/(2*
(2 + 3*x)*(3 + 5*x)^2) + (1020*Sqrt[1 - 2*x])/(3 + 5*x) + 14073*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 1
3665*Sqrt[5/11]*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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {1}{6} \int \frac {157-237 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1}{42} \int \frac {17087-23625 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}-\frac {1}{924} \int \frac {1229382-1406790 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+\frac {\int \frac {50784426-31101840 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{10164}\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}-\frac {42219}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {68325}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+\frac {42219}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {68325}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \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 = 0.12, size = 119, normalized size = 0.81 \begin {gather*} \frac {309606 \sqrt {21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-191310 \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 (91800 x^3+174435 x^2+110315 x+23219\right )}{154 (3 x+2)^2 (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(77*Sqrt[1 - 2*x]*(23219 + 110315*x + 174435*x^2 + 91800*x^3) + 309606*Sqrt[21]*(6 + 19*x + 15*x^2)^2*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]] - 191310*Sqrt[55]*(6 + 19*x + 15*x^2)^2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(154*(2 +
3*x)^2*(3 + 5*x)^2)

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IntegrateAlgebraic [A]  time = 0.39, size = 113, normalized size = 0.77 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (45900 (1-2 x)^3-312135 (1-2 x)^2+707200 (1-2 x)-533841\right )}{\left (15 (1-2 x)^2-68 (1-2 x)+77\right )^2}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \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 - 2*x)^(3/2)/((2 + 3*x)^3*(3 + 5*x)^3),x]

[Out]

(-2*(-533841 + 707200*(1 - 2*x) - 312135*(1 - 2*x)^2 + 45900*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(77 - 68*(1 - 2*x) +
15*(1 - 2*x)^2)^2 + 14073*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]]

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fricas [A]  time = 1.33, size = 162, normalized size = 1.10 \begin {gather*} \frac {95655 \, \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 ) + 154803 \, \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 (91800 \, x^{3} + 174435 \, x^{2} + 110315 \, x + 23219\right )} \sqrt {-2 \, x + 1}}{154 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/154*(95655*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)) + 154803*sqrt(7)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(-(sqrt(7)*sqrt(3
)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(91800*x^3 + 174435*x^2 + 110315*x + 23219)*sqrt(-2*x + 1))/(225*x
^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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giac [A]  time = 0.95, size = 148, normalized size = 1.01 \begin {gather*} \frac {13665}{22} \, \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 {14073}{14} \, \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 (45900 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 312135 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 707200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 533841 \, \sqrt {-2 \, x + 1}\right )}}{{\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13665/22*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 14073/14*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2*(45900*(2*x - 1)^3*sqrt(-2*
x + 1) + 312135*(2*x - 1)^2*sqrt(-2*x + 1) - 707200*(-2*x + 1)^(3/2) + 533841*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 = 0.64 \begin {gather*} \frac {14073 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{7}-\frac {13665 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{11}+\frac {-5075 \left (-2 x +1\right )^{\frac {3}{2}}+11055 \sqrt {-2 x +1}}{\left (-10 x -6\right )^{2}}-\frac {324 \left (\frac {205 \left (-2 x +1\right )^{\frac {3}{2}}}{36}-\frac {161 \sqrt {-2 x +1}}{12}\right )}{\left (-6 x -4\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500*(-203/20*(-2*x+1)^(3/2)+2211/100*(-2*x+1)^(1/2))/(-10*x-6)^2-13665/11*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2)
)*55^(1/2)-324*(205/36*(-2*x+1)^(3/2)-161/12*(-2*x+1)^(1/2))/(-6*x-4)^2+14073/7*arctanh(1/7*21^(1/2)*(-2*x+1)^
(1/2))*21^(1/2)

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maxima [A]  time = 1.35, size = 146, normalized size = 0.99 \begin {gather*} \frac {13665}{22} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14073}{14} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (45900 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 312135 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 707200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 533841 \, \sqrt {-2 \, x + 1}\right )}}{225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13665/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 14073/14*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2*(45900*(-2*x + 1)^(7/2) - 312135*(-2*x + 1)^(5
/2) + 707200*(-2*x + 1)^(3/2) - 533841*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 = 1.22, size = 107, normalized size = 0.73 \begin {gather*} \frac {14073\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {13665\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {355894\,\sqrt {1-2\,x}}{75}-\frac {56576\,{\left (1-2\,x\right )}^{3/2}}{9}+\frac {41618\,{\left (1-2\,x\right )}^{5/2}}{15}-408\,{\left (1-2\,x\right )}^{7/2}}{\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(14073*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7 - (13665*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))
/11 + ((355894*(1 - 2*x)^(1/2))/75 - (56576*(1 - 2*x)^(3/2))/9 + (41618*(1 - 2*x)^(5/2))/15 - 408*(1 - 2*x)^(7
/2))/((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 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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