3.18.54 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=154 \[ \frac {2 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{2520 (3 x+2)^4}-\frac {\sqrt {1-2 x} (664915 x+413424)}{317520 (3 x+2)^3}-\frac {15313 \sqrt {1-2 x}}{444528 (3 x+2)}-\frac {15313 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 154, 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 = {97, 149, 145, 51, 63, 206} \begin {gather*} \frac {2 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{2520 (3 x+2)^4}-\frac {\sqrt {1-2 x} (664915 x+413424)}{317520 (3 x+2)^3}-\frac {15313 \sqrt {1-2 x}}{444528 (3 x+2)}-\frac {15313 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

(-15313*Sqrt[1 - 2*x])/(444528*(2 + 3*x)) - (653*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2520*(2 + 3*x)^4) - ((1 - 2*x)^(3
/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^6) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(5*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(413424 + 6
64915*x))/(317520*(2 + 3*x)^3) - (15313*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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 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)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {1}{270} \int \frac {(3+5 x)^2 (-1179+1170 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=-\frac {653 \sqrt {1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {\int \frac {(3+5 x) (-83907+75645 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{22680}\\ &=-\frac {653 \sqrt {1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {\sqrt {1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}+\frac {15313 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{63504}\\ &=-\frac {15313 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {653 \sqrt {1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {\sqrt {1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}+\frac {15313 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{444528}\\ &=-\frac {15313 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {653 \sqrt {1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {\sqrt {1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}-\frac {15313 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{444528}\\ &=-\frac {15313 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {653 \sqrt {1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {\sqrt {1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}-\frac {15313 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.04, size = 52, normalized size = 0.34 \begin {gather*} \frac {(1-2 x)^{5/2} \left (\frac {16807 \left (26250 x^2+34911 x+11609\right )}{(3 x+2)^6}-490016 \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{31765230} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

((1 - 2*x)^(5/2)*((16807*(11609 + 34911*x + 26250*x^2))/(2 + 3*x)^6 - 490016*Hypergeometric2F1[5/2, 5, 7/2, 3/
7 - (6*x)/7]))/31765230

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.49, size = 97, normalized size = 0.63 \begin {gather*} \frac {\left (18605295 (1-2 x)^5+956655 (1-2 x)^4-678093066 (1-2 x)^3+2443710654 (1-2 x)^2-3125153605 (1-2 x)[phone]\right ) \sqrt {1-2 x}}{1111320 (3 (1-2 x)-7)^6}-\frac {15313 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

((1286827955 - 3125153605*(1 - 2*x) + 2443710654*(1 - 2*x)^2 - 678093066*(1 - 2*x)^3 + 956655*(1 - 2*x)^4 + 18
605295*(1 - 2*x)^5)*Sqrt[1 - 2*x])/(1111320*(-7 + 3*(1 - 2*x))^6) - (15313*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
222264*Sqrt[21])

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 129, normalized size = 0.84 \begin {gather*} \frac {76565 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (18605295 \, x^{5} - 46991565 \, x^{4} - 122053374 \, x^{3} - 75153042 \, x^{2} - 10947400 \, x + 1660816\right )} \sqrt {-2 \, x + 1}}{46675440 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/46675440*(76565*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((3*x + sqrt(
21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(18605295*x^5 - 46991565*x^4 - 122053374*x^3 - 75153042*x^2 - 10947400
*x + 1660816)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

________________________________________________________________________________________

giac [A]  time = 1.03, size = 132, normalized size = 0.86 \begin {gather*} \frac {15313}{9335088} \, \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 {18605295 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 956655 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 678093066 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 2443710654 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 3125153605 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1286827955 \, \sqrt {-2 \, x + 1}}{71124480 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^7,x, algorithm="giac")

[Out]

15313/9335088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/71124480
*(18605295*(2*x - 1)^5*sqrt(-2*x + 1) - 956655*(2*x - 1)^4*sqrt(-2*x + 1) - 678093066*(2*x - 1)^3*sqrt(-2*x +
1) - 2443710654*(2*x - 1)^2*sqrt(-2*x + 1) + 3125153605*(-2*x + 1)^(3/2) - 1286827955*sqrt(-2*x + 1))/(3*x + 2
)^6

________________________________________________________________________________________

maple [A]  time = 0.01, size = 84, normalized size = 0.55 \begin {gather*} -\frac {15313 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{4667544}-\frac {11664 \left (-\frac {15313 \left (-2 x +1\right )^{\frac {11}{2}}}{10668672}-\frac {3037 \left (-2 x +1\right )^{\frac {9}{2}}}{41150592}+\frac {256271 \left (-2 x +1\right )^{\frac {7}{2}}}{4898880}-\frac {923549 \left (-2 x +1\right )^{\frac {5}{2}}}{4898880}+\frac {1822247 \left (-2 x +1\right )^{\frac {3}{2}}}{7558272}-\frac {750337 \sqrt {-2 x +1}}{7558272}\right )}{\left (-6 x -4\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(5*x+3)^3/(3*x+2)^7,x)

[Out]

-11664*(-15313/10668672*(-2*x+1)^(11/2)-3037/41150592*(-2*x+1)^(9/2)+256271/4898880*(-2*x+1)^(7/2)-923549/4898
880*(-2*x+1)^(5/2)+1822247/7558272*(-2*x+1)^(3/2)-750337/7558272*(-2*x+1)^(1/2))/(-6*x-4)^6-15313/4667544*arct
anh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.23, size = 146, normalized size = 0.95 \begin {gather*} \frac {15313}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {18605295 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 956655 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 678093066 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2443710654 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 3125153605 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1286827955 \, \sqrt {-2 \, x + 1}}{1111320 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^7,x, algorithm="maxima")

[Out]

15313/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1111320*(18605295
*(-2*x + 1)^(11/2) + 956655*(-2*x + 1)^(9/2) - 678093066*(-2*x + 1)^(7/2) + 2443710654*(-2*x + 1)^(5/2) - 3125
153605*(-2*x + 1)^(3/2) + 1286827955*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4
+ 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

________________________________________________________________________________________

mupad [B]  time = 1.19, size = 125, normalized size = 0.81 \begin {gather*} \frac {\frac {750337\,\sqrt {1-2\,x}}{472392}-\frac {1822247\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {923549\,{\left (1-2\,x\right )}^{5/2}}{306180}-\frac {256271\,{\left (1-2\,x\right )}^{7/2}}{306180}+\frac {3037\,{\left (1-2\,x\right )}^{9/2}}{2571912}+\frac {15313\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}}-\frac {15313\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^3)/(3*x + 2)^7,x)

[Out]

((750337*(1 - 2*x)^(1/2))/472392 - (1822247*(1 - 2*x)^(3/2))/472392 + (923549*(1 - 2*x)^(5/2))/306180 - (25627
1*(1 - 2*x)^(7/2))/306180 + (3037*(1 - 2*x)^(9/2))/2571912 + (15313*(1 - 2*x)^(11/2))/666792)/((67228*x)/81 +
(12005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 184877/7
29) - (15313*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4667544

________________________________________________________________________________________

sympy [F(-1)]  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((1-2*x)**(3/2)*(3+5*x)**3/(2+3*x)**7,x)

[Out]

Timed out

________________________________________________________________________________________