3.1998 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=145 \[ \frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}+\frac {2311 \sqrt {1-2 x}}{5 x+3}+\frac {931 \sqrt {1-2 x}}{18 (3 x+2) (5 x+3)^2}-\frac {6899 \sqrt {1-2 x}}{18 (5 x+3)^2}+4555 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14073 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

7/6*(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2-14073/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+4555*arctanh(1/7*2
1^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6899/18*(1-2*x)^(1/2)/(3+5*x)^2+931/18*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+2311*(1
-2*x)^(1/2)/(3+5*x)

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Rubi [A]  time = 0.06, antiderivative size = 145, 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 = {98, 149, 151, 156, 63, 206} \[ \frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}+\frac {2311 \sqrt {1-2 x}}{5 x+3}+\frac {931 \sqrt {1-2 x}}{18 (3 x+2) (5 x+3)^2}-\frac {6899 \sqrt {1-2 x}}{18 (5 x+3)^2}+4555 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14073 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6899*Sqrt[1 - 2*x])/(18*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5*x)^2) + (931*Sqrt[1 - 2*x])
/(18*(2 + 3*x)*(3 + 5*x)^2) + (2311*Sqrt[1 - 2*x])/(3 + 5*x) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
- 14073*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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 98

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 149

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] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

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] && IntegerQ[m]

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 {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {1}{6} \int \frac {(199-167 x) \sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}-\frac {1}{18} \int \frac {-16591+22941 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {6899 \sqrt {1-2 x}}{18 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac {1}{396} \int \frac {-1193742+1366002 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {6899 \sqrt {1-2 x}}{18 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac {2311 \sqrt {1-2 x}}{3+5 x}-\frac {\int \frac {-49312098+30200148 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{4356}\\ &=-\frac {6899 \sqrt {1-2 x}}{18 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac {2311 \sqrt {1-2 x}}{3+5 x}-\frac {95655}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {154803}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {6899 \sqrt {1-2 x}}{18 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac {2311 \sqrt {1-2 x}}{3+5 x}+\frac {95655}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {154803}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {6899 \sqrt {1-2 x}}{18 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac {2311 \sqrt {1-2 x}}{3+5 x}+4555 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14073 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 119, normalized size = 0.82 \[ \frac {45550 \sqrt {21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-28146 \sqrt {55} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+5 \sqrt {1-2 x} \left (207990 x^3+395215 x^2+249939 x+52607\right )}{10 (3 x+2)^2 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(52607 + 249939*x + 395215*x^2 + 207990*x^3) + 45550*Sqrt[21]*(6 + 19*x + 15*x^2)^2*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]] - 28146*Sqrt[55]*(6 + 19*x + 15*x^2)^2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(10*(2 + 3*x
)^2*(3 + 5*x)^2)

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fricas [A]  time = 0.70, size = 156, normalized size = 1.08 \[ \frac {14073 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 22775 \, \sqrt {21} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 5 \, {\left (207990 \, x^{3} + 395215 \, x^{2} + 249939 \, x + 52607\right )} \sqrt {-2 \, x + 1}}{10 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(14073*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) +
5*x - 8)/(5*x + 3)) + 22775*sqrt(21)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((3*x - sqrt(21)*sqrt(-2*x
+ 1) - 5)/(3*x + 2)) + 5*(207990*x^3 + 395215*x^2 + 249939*x + 52607)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 54
1*x^2 + 228*x + 36)

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giac [A]  time = 1.01, size = 148, normalized size = 1.02 \[ \frac {14073}{10} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4555}{2} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (103995 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 707200 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1602293 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1209516 \, \sqrt {-2 \, x + 1}\right )}}{{\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14073/10*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4555/2*sqrt(21
)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2*(103995*(2*x - 1)^3*sqrt(-2*x
+ 1) + 707200*(2*x - 1)^2*sqrt(-2*x + 1) - 1602293*(-2*x + 1)^(3/2) + 1209516*sqrt(-2*x + 1))/(15*(2*x - 1)^2
+ 136*x + 9)^2

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maple [A]  time = 0.02, size = 94, normalized size = 0.65 \[ 4555 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )-\frac {14073 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{5}+\frac {-11385 \left (-2 x +1\right )^{\frac {3}{2}}+24805 \sqrt {-2 x +1}}{\left (-10 x -6\right )^{2}}-\frac {252 \left (\frac {67 \left (-2 x +1\right )^{\frac {3}{2}}}{4}-\frac {1421 \sqrt {-2 x +1}}{36}\right )}{\left (-6 x -4\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1100*(-207/20*(-2*x+1)^(3/2)+451/20*(-2*x+1)^(1/2))/(-10*x-6)^2-14073/5*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*
55^(1/2)-252*(67/4*(-2*x+1)^(3/2)-1421/36*(-2*x+1)^(1/2))/(-6*x-4)^2+4555*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))
*21^(1/2)

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maxima [A]  time = 1.13, size = 146, normalized size = 1.01 \[ \frac {14073}{10} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4555}{2} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (103995 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 707200 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 1602293 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1209516 \, \sqrt {-2 \, x + 1}\right )}}{225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14073/10*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4555/2*sqrt(21)*log(-(sq
rt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2*(103995*(-2*x + 1)^(7/2) - 707200*(-2*x + 1)^(5/
2) + 1602293*(-2*x + 1)^(3/2) - 1209516*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*(2*x - 1)^2
+ 20944*x - 4543)

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mupad [B]  time = 0.09, size = 107, normalized size = 0.74 \[ 4555\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )-\frac {14073\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}+\frac {\frac {806344\,\sqrt {1-2\,x}}{75}-\frac {3204586\,{\left (1-2\,x\right )}^{3/2}}{225}+\frac {56576\,{\left (1-2\,x\right )}^{5/2}}{9}-\frac {4622\,{\left (1-2\,x\right )}^{7/2}}{5}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4555*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7) - (14073*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/5 +
((806344*(1 - 2*x)^(1/2))/75 - (3204586*(1 - 2*x)^(3/2))/225 + (56576*(1 - 2*x)^(5/2))/9 - (4622*(1 - 2*x)^(7/
2))/5)/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2*x - 1)^3)/15 + (2*x - 1)^4 - 4543/225)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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