3.2177 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=105 \[ -\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {190}{1617 (1-2 x)^{3/2}}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-190/1617/(1-2*x)^(3/2)+3/7/(1-2*x)^(3/2)/(2+3*x)+720/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/13
31*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1370/41503/(1-2*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 105, 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 = {103, 152, 156, 63, 206} \[ -\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {190}{1617 (1-2 x)^{3/2}}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-190/(1617*(1 - 2*x)^(3/2)) - 1370/(41503*Sqrt[1 - 2*x]) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)) + (720*Sqrt[3/7]*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

Rule 63

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 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

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 156

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 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)^2 (3+5 x)} \, dx &=\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {1}{7} \int \frac {-10-75 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {2 \int \frac {-555+\frac {4275 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1617}\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {4 \int \frac {\frac {55065}{2}-\frac {30825 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{124509}\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {1080}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {625}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {1080}{343} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {625}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 73, normalized size = 0.70 \[ -\frac {2640 (3 x+2) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-7 \left (350 (3 x+2) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+99\right )}{1617 (1-2 x)^{3/2} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1617*(2640*(2 + 3*x)*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7] - 7*(99 + 350*(2 + 3*x)*Hypergeometric
2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11]))/((1 - 2*x)^(3/2)*(2 + 3*x))

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fricas [A]  time = 0.99, size = 142, normalized size = 1.35 \[ \frac {900375 \, \sqrt {11} \sqrt {5} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1437480 \, \sqrt {7} \sqrt {3} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (24660 \, x^{2} - 39780 \, x + 15881\right )} \sqrt {-2 \, x + 1}}{9587193 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9587193*(900375*sqrt(11)*sqrt(5)*(12*x^3 - 4*x^2 - 5*x + 2)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/
(5*x + 3)) + 1437480*sqrt(7)*sqrt(3)*(12*x^3 - 4*x^2 - 5*x + 2)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5
)/(3*x + 2)) + 77*(24660*x^2 - 39780*x + 15881)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)

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giac [A]  time = 1.25, size = 116, normalized size = 1.10 \[ \frac {125}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {360}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (303 \, x - 190\right )}}{124509 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {27 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 360/2401*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/124509*(303*x - 190)/((2*x
- 1)*sqrt(-2*x + 1)) + 27/343*sqrt(-2*x + 1)/(3*x + 2)

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maple [A]  time = 0.02, size = 72, normalized size = 0.69 \[ \frac {720 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}-\frac {250 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {8}{1617 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {808}{41503 \sqrt {-2 x +1}}-\frac {18 \sqrt {-2 x +1}}{343 \left (-2 x -\frac {4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/1617/(-2*x+1)^(3/2)+808/41503/(-2*x+1)^(1/2)-250/1331*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-18/343*
(-2*x+1)^(1/2)/(-2*x-4/3)+720/2401*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.38, size = 110, normalized size = 1.05 \[ \frac {125}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {360}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (6165 \, {\left (2 \, x - 1\right )}^{2} - 15120 \, x + 9716\right )}}{124509 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 360/2401*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/124509*(6165*(2*x - 1)^2 - 15120*x + 9716)/(3*
(-2*x + 1)^(5/2) - 7*(-2*x + 1)^(3/2))

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mupad [B]  time = 0.10, size = 73, normalized size = 0.70 \[ \frac {\frac {1370\,{\left (2\,x-1\right )}^2}{41503}-\frac {480\,x}{5929}+\frac {2776}{53361}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}+\frac {720\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1370*(2*x - 1)^2)/41503 - (480*x)/5929 + 2776/53361)/((7*(1 - 2*x)^(3/2))/3 - (1 - 2*x)^(5/2)) + (720*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))/1331

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sympy [C]  time = 13.39, size = 1459, normalized size = 13.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

388962000*sqrt(55)*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(2070833688*I*(x - 1/2)**(11/2) + 724791
7908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 620991360*sqrt(21)*
(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/
2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 194481000*sqrt(55)*pi*(x - 1/2)**(11/2)/
(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I
*(x - 1/2)**(5/2)) + 310495680*sqrt(21)*pi*(x - 1/2)**(11/2)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x
- 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) + 1361367000*sqrt(55)*(x - 1/2
)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 84
55904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 2173469760*sqrt(21)*(x - 1/2)**(9/2)*atan(sqrt(
42)*sqrt(x - 1/2)/7)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)*
*(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 680683500*sqrt(55)*pi*(x - 1/2)**(9/2)/(2070833688*I*(x - 1/2)**(11/
2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) + 10867348
80*sqrt(21)*pi*(x - 1/2)**(9/2)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I
*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) + 1588261500*sqrt(55)*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(
x - 1/2)/11)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) +
3288407199*I*(x - 1/2)**(5/2)) - 2535714720*sqrt(21)*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(2070833
688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/
2)**(5/2)) - 794130750*sqrt(55)*pi*(x - 1/2)**(7/2)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**
(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) + 1267857360*sqrt(21)*pi*(x - 1/2)**(7/
2)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 328840719
9*I*(x - 1/2)**(5/2)) + 617657250*sqrt(55)*(x - 1/2)**(5/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(2070833688*I*(x
- 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)
) - 986111280*sqrt(21)*(x - 1/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(2070833688*I*(x - 1/2)**(11/2) + 72479
17908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 308828625*sqrt(55)
*pi*(x - 1/2)**(5/2)/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)*
*(7/2) + 3288407199*I*(x - 1/2)**(5/2)) + 493055640*sqrt(21)*pi*(x - 1/2)**(5/2)/(2070833688*I*(x - 1/2)**(11/
2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 34178760
*sqrt(2)*(x - 1/2)**5/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)
**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 58794120*sqrt(2)*(x - 1/2)**4/(2070833688*I*(x - 1/2)**(11/2) + 724
7917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 611226*sqrt(2)*(
x - 1/2)**3/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) +
3288407199*I*(x - 1/2)**(5/2)) + 21551376*sqrt(2)*(x - 1/2)**2/(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*
(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I*(x - 1/2)**(5/2)) - 4067294*sqrt(2)*(x - 1/2)/
(2070833688*I*(x - 1/2)**(11/2) + 7247917908*I*(x - 1/2)**(9/2) + 8455904226*I*(x - 1/2)**(7/2) + 3288407199*I
*(x - 1/2)**(5/2))

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