3.22.25 \(\int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\) [2125]

Optimal. Leaf size=147 \[ \frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}} \]

[Out]

-33873/228765625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/11*(2+3*x)^5/(3+5*x)^2/(1-2*x)^(1/2)+377748/8
31875*(2+3*x)^2*(1-2*x)^(1/2)-71/1210*(2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)^2-2721/66550*(2+3*x)^3*(1-2*x)^(1/2)/(3+
5*x)+63/8318750*(2492512+831375*x)*(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 158, 152, 65, 212} \begin {gather*} \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {71 \sqrt {1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac {2721 \sqrt {1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac {377748 \sqrt {1-2 x} (3 x+2)^2}{831875}+\frac {63 \sqrt {1-2 x} (831375 x+2492512)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)^(3/2)*(3 + 5*x)^3),x]

[Out]

(377748*Sqrt[1 - 2*x]*(2 + 3*x)^2)/831875 - (71*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(1210*(3 + 5*x)^2) + (7*(2 + 3*x)^5
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (2721*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(66550*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(249
2512 + 831375*x))/8318750 - (33873*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(4159375*Sqrt[55])

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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 154

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*((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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 158

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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 {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{11} \int \frac {(2+3 x)^4 (173+312 x)}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\int \frac {(2+3 x)^3 (12450+21657 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (446523+755496 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{66550}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {\int \frac {(-31392102-52376625 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1663750}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}+\frac {33873 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{8318750}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8318750}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 73, normalized size = 0.50 \begin {gather*} \frac {-\frac {55 \left (-1702670584-4150263077 x+762244410 x^2+5682717810 x^3+1423105200 x^4+242574750 x^5\right )}{\sqrt {1-2 x} (3+5 x)^2}-67746 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{457531250} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)^(3/2)*(3 + 5*x)^3),x]

[Out]

((-55*(-1702670584 - 4150263077*x + 762244410*x^2 + 5682717810*x^3 + 1423105200*x^4 + 242574750*x^5))/(Sqrt[1
- 2*x]*(3 + 5*x)^2) - 67746*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/457531250

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Maple [A]
time = 0.13, size = 84, normalized size = 0.57

method	result	size
risch	\(-\frac {242574750 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-4150263077 x -1702670584}{8318750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {33873 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{228765625}\)	\(61\)
derivativedivides	\(\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {8991 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {333639 \sqrt {1-2 x}}{25000}+\frac {\frac {403 \left (1-2 x \right )^{\frac {3}{2}}}{831875}-\frac {81 \sqrt {1-2 x}}{75625}}{\left (-6-10 x \right )^{2}}-\frac {33873 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{228765625}+\frac {117649}{10648 \sqrt {1-2 x}}\)	\(84\)
default	\(\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {8991 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {333639 \sqrt {1-2 x}}{25000}+\frac {\frac {403 \left (1-2 x \right )^{\frac {3}{2}}}{831875}-\frac {81 \sqrt {1-2 x}}{75625}}{\left (-6-10 x \right )^{2}}-\frac {33873 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{228765625}+\frac {117649}{10648 \sqrt {1-2 x}}\)	\(84\)
trager	\(\frac {\left (242574750 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-4150263077 x -1702670584\right ) \sqrt {1-2 x}}{8318750 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {33873 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{457531250}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

729/5000*(1-2*x)^(5/2)-8991/5000*(1-2*x)^(3/2)+333639/25000*(1-2*x)^(1/2)+2/166375*(403/10*(1-2*x)^(3/2)-891/1
0*(1-2*x)^(1/2))/(-6-10*x)^2-33873/228765625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+117649/10648/(1-2*x
)^(1/2)

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Maxima [A]
time = 0.54, size = 110, normalized size = 0.75 \begin {gather*} \frac {729}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8991}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33873}{457531250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {333639}{25000} \, \sqrt {-2 \, x + 1} + \frac {1838268849 \, {\left (2 \, x - 1\right )}^{2} + 16176751756 \, x + 808829747}{6655000 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x)^3,x, algorithm="maxima")

[Out]

729/5000*(-2*x + 1)^(5/2) - 8991/5000*(-2*x + 1)^(3/2) + 33873/457531250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 333639/25000*sqrt(-2*x + 1) + 1/6655000*(1838268849*(2*x - 1)^2 + 1617
6751756*x + 808829747)/(25*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]
time = 0.92, size = 99, normalized size = 0.67 \begin {gather*} \frac {33873 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (242574750 \, x^{5} + 1423105200 \, x^{4} + 5682717810 \, x^{3} + 762244410 \, x^{2} - 4150263077 \, x - 1702670584\right )} \sqrt {-2 \, x + 1}}{457531250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/457531250*(33873*sqrt(55)*(50*x^3 + 35*x^2 - 12*x - 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) +
55*(242574750*x^5 + 1423105200*x^4 + 5682717810*x^3 + 762244410*x^2 - 4150263077*x - 1702670584)*sqrt(-2*x + 1
))/(50*x^3 + 35*x^2 - 12*x - 9)

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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((2+3*x)**6/(1-2*x)**(3/2)/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.71, size = 111, normalized size = 0.76 \begin {gather*} \frac {729}{5000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8991}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33873}{457531250} \, \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 {333639}{25000} \, \sqrt {-2 \, x + 1} + \frac {117649}{10648 \, \sqrt {-2 \, x + 1}} + \frac {403 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 891 \, \sqrt {-2 \, x + 1}}{3327500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x)^3,x, algorithm="giac")

[Out]

729/5000*(2*x - 1)^2*sqrt(-2*x + 1) - 8991/5000*(-2*x + 1)^(3/2) + 33873/457531250*sqrt(55)*log(1/2*abs(-2*sqr
t(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 333639/25000*sqrt(-2*x + 1) + 117649/10648/sqrt(-2
*x + 1) + 1/3327500*(403*(-2*x + 1)^(3/2) - 891*sqrt(-2*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 1.22, size = 91, normalized size = 0.62 \begin {gather*} \frac {\frac {367653449\,x}{3781250}+\frac {1838268849\,{\left (2\,x-1\right )}^2}{166375000}+\frac {73529977}{15125000}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}+\frac {333639\,\sqrt {1-2\,x}}{25000}-\frac {8991\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {729\,{\left (1-2\,x\right )}^{5/2}}{5000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,33873{}\mathrm {i}}{228765625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^6/((1 - 2*x)^(3/2)*(5*x + 3)^3),x)

[Out]

((367653449*x)/3781250 + (1838268849*(2*x - 1)^2)/166375000 + 73529977/15125000)/((121*(1 - 2*x)^(1/2))/25 - (
22*(1 - 2*x)^(3/2))/5 + (1 - 2*x)^(5/2)) + (55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*33873i)/228765625
+ (333639*(1 - 2*x)^(1/2))/25000 - (8991*(1 - 2*x)^(3/2))/5000 + (729*(1 - 2*x)^(5/2))/5000

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