3.22.78 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx\) [2178]

Optimal. Leaf size=125 \[ -\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-1115/1617/(1-2*x)^(3/2)+3/14/(1-2*x)^(3/2)/(2+3*x)^2+33/14/(1-2*x)^(3/2)/(2+3*x)+3645/2401*arctanh(1/7*21^(1/
2)*(1-2*x)^(1/2))*21^(1/2)-1250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-12295/41503/(1-2*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {105, 156, 157, 162, 65, 212} \begin {gather*} -\frac {12295}{41503 \sqrt {1-2 x}}+\frac {33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac {1115}{1617 (1-2 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \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/((1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x)),x]

[Out]

-1115/(1617*(1 - 2*x)^(3/2)) - 12295/(41503*Sqrt[1 - 2*x]) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 33/(14*(1 -
2*x)^(3/2)*(2 + 3*x)) + (3645*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/121

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 157

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] && IntegersQ[2*m, 2*n, 2*p]

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}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {1}{14} \int \frac {7-105 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {1}{98} \int \frac {-1015-5775 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {\int \frac {-\frac {46515}{2}+\frac {351225 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{11319}\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {2 \int \frac {\frac {6679995}{4}-\frac {3872925 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{871563}\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {10935}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {3125}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {10935}{686} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {3125}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \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 = 0.25, size = 94, normalized size = 0.75 \begin {gather*} \frac {245383-594687 x-438840 x^2+1327860 x^3}{249018 (1-2 x)^{3/2} (2+3 x)^2}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \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/((1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x)),x]

[Out]

(245383 - 594687*x - 438840*x^2 + 1327860*x^3)/(249018*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (3645*Sqrt[3/7]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]
time = 0.16, size = 84, normalized size = 0.67

method	result	size
derivativedivides	\(-\frac {1250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {16}{11319 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2144}{290521 \sqrt {1-2 x}}-\frac {486 \left (\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{2}-\frac {581 \sqrt {1-2 x}}{18}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {3645 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(84\)
default	\(-\frac {1250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {16}{11319 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2144}{290521 \sqrt {1-2 x}}-\frac {486 \left (\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{2}-\frac {581 \sqrt {1-2 x}}{18}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {3645 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(84\)
trager	\(\frac {\left (1327860 x^{3}-438840 x^{2}-594687 x +245383\right ) \sqrt {1-2 x}}{249018 \left (6 x^{2}+x -2\right )^{2}}-\frac {3645 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4802}-\frac {625 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{1331}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+16/11319/(1-2*x)^(3/2)+2144/290521/(1-2*x)^(1/2)-486/
2401*(27/2*(1-2*x)^(3/2)-581/18*(1-2*x)^(1/2))/(-4-6*x)^2+3645/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/
2)

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Maxima [A]
time = 0.51, size = 128, normalized size = 1.02 \begin {gather*} \frac {625}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3645}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {331965 \, {\left (2 \, x - 1\right )}^{3} + 776475 \, {\left (2 \, x - 1\right )}^{2} - 75264 \, x + 46256}{124509 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3645/4802*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/124509*(331965*(2*x - 1)^3 + 776475*(2*x - 1)
^2 - 75264*x + 46256)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))

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Fricas [A]
time = 1.50, size = 162, normalized size = 1.30 \begin {gather*} \frac {9003750 \, \sqrt {11} \sqrt {5} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 14554485 \, \sqrt {7} \sqrt {3} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (1327860 \, x^{3} - 438840 \, x^{2} - 594687 \, x + 245383\right )} \sqrt {-2 \, x + 1}}{19174386 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19174386*(9003750*sqrt(11)*sqrt(5)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1)
+ 5*x - 8)/(5*x + 3)) + 14554485*sqrt(7)*sqrt(3)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(-(sqrt(7)*sqrt(3)*s
qrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(1327860*x^3 - 438840*x^2 - 594687*x + 245383)*sqrt(-2*x + 1))/(36*x^
4 + 12*x^3 - 23*x^2 - 4*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/(1-2*x)**(5/2)/(2+3*x)**3/(3+5*x),x)

[Out]

Exception raised: MellinTransformStripError >> Pole inside critical strip?

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Giac [A]
time = 1.15, size = 128, normalized size = 1.02 \begin {gather*} \frac {625}{1331} \, \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 {3645}{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 {16 \, {\left (804 \, x - 479\right )}}{871563 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {27 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 581 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3645/4802*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/871563*(804*x - 479)/((2*
x - 1)*sqrt(-2*x + 1)) - 27/9604*(243*(-2*x + 1)^(3/2) - 581*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.10, size = 89, normalized size = 0.71 \begin {gather*} \frac {3645\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {1250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {12325\,{\left (2\,x-1\right )}^2}{17787}-\frac {512\,x}{7623}+\frac {12295\,{\left (2\,x-1\right )}^3}{41503}+\frac {944}{22869}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3645*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (1250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11)
)/1331 + ((12325*(2*x - 1)^2)/17787 - (512*x)/7623 + (12295*(2*x - 1)^3)/41503 + 944/22869)/((49*(1 - 2*x)^(3/
2))/9 - (14*(1 - 2*x)^(5/2))/3 + (1 - 2*x)^(7/2))

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