3.21.44 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\) [2044]

Optimal. Leaf size=97 \[ \frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {219 \sqrt {1-2 x}}{98 (2+3 x)}+\frac {2523}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-50 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

2523/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+3/14
*(1-2*x)^(1/2)/(2+3*x)^2+219/98*(1-2*x)^(1/2)/(2+3*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 = {105, 156, 162, 65, 212} \begin {gather*} \frac {219 \sqrt {1-2 x}}{98 (3 x+2)}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2}+\frac {2523}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-50 \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)^3*(3 + 5*x)),x]

[Out]

(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + (219*Sqrt[1 - 2*x])/(98*(2 + 3*x)) + (2523*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/49 - 50*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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

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}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {1}{14} \int \frac {43-45 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {219 \sqrt {1-2 x}}{98 (2+3 x)}+\frac {1}{98} \int \frac {1793-1095 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {219 \sqrt {1-2 x}}{98 (2+3 x)}-\frac {7569}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {219 \sqrt {1-2 x}}{98 (2+3 x)}+\frac {7569}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-125 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {219 \sqrt {1-2 x}}{98 (2+3 x)}+\frac {2523}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-50 \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 = 0.23, size = 82, normalized size = 0.85 \begin {gather*} \frac {9 \sqrt {1-2 x} (51+73 x)}{98 (2+3 x)^2}+\frac {2523}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-50 \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)^3*(3 + 5*x)),x]

[Out]

(9*Sqrt[1 - 2*x]*(51 + 73*x))/(98*(2 + 3*x)^2) + (2523*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - 50*Sqr
t[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]
time = 0.17, size = 66, normalized size = 0.68

method	result	size
risch	\(-\frac {9 \left (146 x^{2}+29 x -51\right )}{98 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {2523 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(64\)
derivativedivides	\(-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {162 \left (\frac {73 \left (1-2 x \right )^{\frac {3}{2}}}{882}-\frac {25 \sqrt {1-2 x}}{126}\right )}{\left (-4-6 x \right )^{2}}+\frac {2523 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(66\)
default	\(-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {162 \left (\frac {73 \left (1-2 x \right )^{\frac {3}{2}}}{882}-\frac {25 \sqrt {1-2 x}}{126}\right )}{\left (-4-6 x \right )^{2}}+\frac {2523 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(66\)
trager	\(\frac {9 \left (73 x +51\right ) \sqrt {1-2 x}}{98 \left (2+3 x \right )^{2}}+\frac {2523 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{686}+\frac {25 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-162*(73/882*(1-2*x)^(3/2)-25/126*(1-2*x)^(1/2))/(-4-6*x)^
2+2523/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.52, size = 110, normalized size = 1.13 \begin {gather*} \frac {25}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2523}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9 \, {\left (73 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 175 \, \sqrt {-2 \, x + 1}\right )}}{49 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2523/686*sqrt(21)*log(-(sqr
t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 9/49*(73*(-2*x + 1)^(3/2) - 175*sqrt(-2*x + 1))/(9*
(2*x - 1)^2 + 84*x + 7)

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Fricas [A]
time = 1.11, size = 122, normalized size = 1.26 \begin {gather*} \frac {17150 \, \sqrt {11} \sqrt {5} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 27753 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 693 \, {\left (73 \, x + 51\right )} \sqrt {-2 \, x + 1}}{7546 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7546*(17150*sqrt(11)*sqrt(5)*(9*x^2 + 12*x + 4)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) +
27753*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 693*(73
*x + 51)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: MellinTransformStripError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: MellinTransformStripError >> Pole inside critical strip?

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Giac [A]
time = 1.82, size = 107, normalized size = 1.10 \begin {gather*} \frac {25}{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 {2523}{686} \, \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 {9 \, {\left (73 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 175 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2523/686*sqrt(21)
*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 9/196*(73*(-2*x + 1)^(3/2) - 175
*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 1.27, size = 71, normalized size = 0.73 \begin {gather*} \frac {2523\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {25\,\sqrt {1-2\,x}}{7}-\frac {73\,{\left (1-2\,x\right )}^{3/2}}{49}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2523*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - (50*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1
1 + ((25*(1 - 2*x)^(1/2))/7 - (73*(1 - 2*x)^(3/2))/49)/((28*x)/3 + (2*x - 1)^2 + 7/9)

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