3.20.16 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx\) [1916]

Optimal. Leaf size=127 \[ -\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-2311/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+204*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-335/2*(
1-2*x)^(1/2)/(3+5*x)+7/6*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+50/3*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)

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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 156, 162, 65, 212} \begin {gather*} -\frac {335 \sqrt {1-2 x}}{2 (5 x+3)}+\frac {50 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \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)^2),x]

[Out]

(-335*Sqrt[1 - 2*x])/(2*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)) + (50*Sqrt[1 - 2*x])/(3*(2 +
3*x)*(3 + 5*x)) - 2311*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 204*Sqrt[55]*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 100

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 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 {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {1}{6} \int \frac {122-167 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}+\frac {1}{42} \int \frac {9177-10500 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-\frac {1}{462} \int \frac {379071-232155 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}+\frac {6933}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-5610 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-\frac {6933}{2} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+5610 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 90, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {1-2 x} \left (1271+3920 x+3015 x^2\right )}{2 (2+3 x)^2 (3+5 x)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[1 - 2*x]*(1271 + 3920*x + 3015*x^2))/((2 + 3*x)^2*(3 + 5*x)) - 2311*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]] + 204*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]
time = 0.16, size = 82, normalized size = 0.65

method	result	size
risch	\(\frac {6030 x^{3}+4825 x^{2}-1378 x -1271}{2 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}+204 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {2311 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(76\)
derivativedivides	\(\frac {22 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+204 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {405 \left (1-2 x \right )^{\frac {3}{2}}-959 \sqrt {1-2 x}}{\left (-4-6 x \right )^{2}}-\frac {2311 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(82\)
default	\(\frac {22 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+204 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {405 \left (1-2 x \right )^{\frac {3}{2}}-959 \sqrt {1-2 x}}{\left (-4-6 x \right )^{2}}-\frac {2311 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(82\)
trager	\(-\frac {\left (3015 x^{2}+3920 x +1271\right ) \sqrt {1-2 x}}{2 \left (2+3 x \right )^{2} \left (3+5 x \right )}-102 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {2311 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{14}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22*(1-2*x)^(1/2)/(-6/5-2*x)+204*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+18*(45/2*(1-2*x)^(3/2)-959/18*(1
-2*x)^(1/2))/(-4-6*x)^2-2311/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.52, size = 128, normalized size = 1.01 \begin {gather*} -102 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2311}{14} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3015 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 13870 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15939 \, \sqrt {-2 \, x + 1}}{45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168} \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)^2,x, algorithm="maxima")

[Out]

-102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2311/14*sqrt(21)*log(-(sqrt(
21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - (3015*(-2*x + 1)^(5/2) - 13870*(-2*x + 1)^(3/2) + 159
39*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)

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Fricas [A]
time = 0.82, size = 136, normalized size = 1.07 \begin {gather*} \frac {2311 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 1428 \, \sqrt {55} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 7 \, {\left (3015 \, x^{2} + 3920 \, x + 1271\right )} \sqrt {-2 \, x + 1}}{14 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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)^2,x, algorithm="fricas")

[Out]

1/14*(2311*sqrt(7)*sqrt(3)*(45*x^3 + 87*x^2 + 56*x + 12)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x +
2)) + 1428*sqrt(55)*(45*x^3 + 87*x^2 + 56*x + 12)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 7*(301
5*x^2 + 3920*x + 1271)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)

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Sympy [F(-1)] Timed out
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)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.57, size = 123, normalized size = 0.97 \begin {gather*} -102 \, \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 {2311}{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 {55 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} + \frac {405 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 959 \, \sqrt {-2 \, x + 1}}{4 \, {\left (3 \, x + 2\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)^2,x, algorithm="giac")

[Out]

-102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2311/14*sqrt(21)*l
og(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 55*sqrt(-2*x + 1)/(5*x + 3) + 1/4*
(405*(-2*x + 1)^(3/2) - 959*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 1.21, size = 90, normalized size = 0.71 \begin {gather*} 204\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {2311\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {\frac {1771\,\sqrt {1-2\,x}}{5}-\frac {2774\,{\left (1-2\,x\right )}^{3/2}}{9}+67\,{\left (1-2\,x\right )}^{5/2}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \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)^2),x)

[Out]

204*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (2311*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7 - ((
1771*(1 - 2*x)^(1/2))/5 - (2774*(1 - 2*x)^(3/2))/9 + 67*(1 - 2*x)^(5/2))/((1414*x)/45 + (103*(2*x - 1)^2)/15 +
(2*x - 1)^3 - 56/15)

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