3.22.13 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx\) [2113]

Optimal. Leaf size=112 \[ -\frac {2525}{3773 \sqrt {1-2 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}+\frac {8025}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

8025/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-2
525/3773/(1-2*x)^(1/2)+3/14/(2+3*x)^2/(1-2*x)^(1/2)+225/98/(2+3*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {2525}{3773 \sqrt {1-2 x}}+\frac {225}{98 \sqrt {1-2 x} (3 x+2)}+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {8025}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{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/((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)),x]

[Out]

-2525/(3773*Sqrt[1 - 2*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 225/(98*Sqrt[1 - 2*x]*(2 + 3*x)) + (8025*Sqrt[
3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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)^{3/2} (2+3 x)^3 (3+5 x)} \, dx &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {1}{14} \int \frac {25-75 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}+\frac {1}{98} \int \frac {425-3375 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {2525}{3773 \sqrt {1-2 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {63025}{2}+\frac {37875 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{3773}\\ &=-\frac {2525}{3773 \sqrt {1-2 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}-\frac {24075}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {625}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {2525}{3773 \sqrt {1-2 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}+\frac {24075}{686} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {625}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {2525}{3773 \sqrt {1-2 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}+\frac {8025}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{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 = 0.24, size = 89, normalized size = 0.79 \begin {gather*} \frac {16067-8625 x-45450 x^2}{7546 \sqrt {1-2 x} (2+3 x)^2}+\frac {8025}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{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/((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)),x]

[Out]

(16067 - 8625*x - 45450*x^2)/(7546*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (8025*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]
time = 0.18, size = 75, normalized size = 0.67

method	result	size
risch	\(-\frac {45450 x^{2}+8625 x -16067}{7546 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {8025 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(64\)
derivativedivides	\(-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {16}{3773 \sqrt {1-2 x}}-\frac {486 \left (\frac {77 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {553 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {8025 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(75\)
default	\(-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {16}{3773 \sqrt {1-2 x}}-\frac {486 \left (\frac {77 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {553 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {8025 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(75\)
trager	\(\frac {\left (45450 x^{2}+8625 x -16067\right ) \sqrt {1-2 x}}{7546 \left (2+3 x \right )^{2} \left (-1+2 x \right )}+\frac {75 \RootOf \left (\textit {\_Z}^{2}-240429\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-240429\right ) x +2247 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-240429\right )}{2+3 x}\right )}{4802}-\frac {125 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{121}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+16/3773/(1-2*x)^(1/2)-486/343*(77/18*(1-2*x)^(3/2)-553/
54*(1-2*x)^(1/2))/(-4-6*x)^2+8025/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.50, size = 119, normalized size = 1.06 \begin {gather*} \frac {125}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8025}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {22725 \, {\left (2 \, x - 1\right )}^{2} + 108150 \, x - 54859}{3773 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 49 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8025/4802*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3773*(22725*(2*x - 1)^2 + 108150*x - 54859)/(9
*(-2*x + 1)^(5/2) - 42*(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))

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Fricas [A]
time = 0.76, size = 142, normalized size = 1.27 \begin {gather*} \frac {600250 \, \sqrt {11} \sqrt {5} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 971025 \, \sqrt {7} \sqrt {3} {\left (18 \, x^{3} + 15 \, 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 (45450 \, x^{2} + 8625 \, x - 16067\right )} \sqrt {-2 \, x + 1}}{581042 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/581042*(600250*sqrt(11)*sqrt(5)*(18*x^3 + 15*x^2 - 4*x - 4)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/
(5*x + 3)) + 971025*sqrt(7)*sqrt(3)*(18*x^3 + 15*x^2 - 4*x - 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5
)/(3*x + 2)) + 77*(45450*x^2 + 8625*x - 16067)*sqrt(-2*x + 1))/(18*x^3 + 15*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)**(3/2)/(2+3*x)**3/(3+5*x),x)

[Out]

Exception raised: MellinTransformStripError >> Pole inside critical strip?

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Giac [A]
time = 1.63, size = 116, normalized size = 1.04 \begin {gather*} \frac {125}{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 {8025}{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}{3773 \, \sqrt {-2 \, x + 1}} - \frac {9 \, {\left (33 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 79 \, \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/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x),x, algorithm="giac")

[Out]

125/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8025/4802*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/3773/sqrt(-2*x + 1) - 9/19
6*(33*(-2*x + 1)^(3/2) - 79*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.10, size = 81, normalized size = 0.72 \begin {gather*} \frac {8025\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {\frac {5150\,x}{1617}+\frac {2525\,{\left (2\,x-1\right )}^2}{3773}-\frac {7837}{4851}}{\frac {49\,\sqrt {1-2\,x}}{9}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{3}+{\left (1-2\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8025*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))
/121 - ((5150*x)/1617 + (2525*(2*x - 1)^2)/3773 - 7837/4851)/((49*(1 - 2*x)^(1/2))/9 - (14*(1 - 2*x)^(3/2))/3
+ (1 - 2*x)^(5/2))

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