3.1826 \(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=147 \[ -\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{945 (3 x+2)^5}-\frac {\sqrt {1-2 x} (160029 x+98995)}{476280 (3 x+2)^4}+\frac {43957 \sqrt {1-2 x}}{3111696 (3 x+2)}+\frac {43957 \sqrt {1-2 x}}{1333584 (3 x+2)^2}+\frac {43957 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}} \]

[Out]

43957/32672808*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+43957/1333584*(1-2*x)^(1/2)/(2+3*x)^2+43957/311169
6*(1-2*x)^(1/2)/(2+3*x)-53/945*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^5-1/18*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^6-1/4762
80*(98995+160029*x)*(1-2*x)^(1/2)/(2+3*x)^4

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Rubi [A]  time = 0.05, 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 = {97, 149, 145, 51, 63, 206} \[ -\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{945 (3 x+2)^5}-\frac {\sqrt {1-2 x} (160029 x+98995)}{476280 (3 x+2)^4}+\frac {43957 \sqrt {1-2 x}}{3111696 (3 x+2)}+\frac {43957 \sqrt {1-2 x}}{1333584 (3 x+2)^2}+\frac {43957 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(43957*Sqrt[1 - 2*x])/(1333584*(2 + 3*x)^2) + (43957*Sqrt[1 - 2*x])/(3111696*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3
+ 5*x)^2)/(945*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(18*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(98995 + 160029*x
))/(476280*(2 + 3*x)^4) + (43957*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1555848*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 {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}+\frac {\int \frac {(247-3475 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{1890}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}-\frac {\sqrt {1-2 x} (98995+160029 x)}{476280 (2+3 x)^4}-\frac {43957 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{95256}\\ &=\frac {43957 \sqrt {1-2 x}}{1333584 (2+3 x)^2}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}-\frac {\sqrt {1-2 x} (98995+160029 x)}{476280 (2+3 x)^4}-\frac {43957 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{444528}\\ &=\frac {43957 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {43957 \sqrt {1-2 x}}{3111696 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}-\frac {\sqrt {1-2 x} (98995+160029 x)}{476280 (2+3 x)^4}-\frac {43957 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3111696}\\ &=\frac {43957 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {43957 \sqrt {1-2 x}}{3111696 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}-\frac {\sqrt {1-2 x} (98995+160029 x)}{476280 (2+3 x)^4}+\frac {43957 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3111696}\\ &=\frac {43957 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {43957 \sqrt {1-2 x}}{3111696 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}-\frac {\sqrt {1-2 x} (98995+160029 x)}{476280 (2+3 x)^4}+\frac {43957 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 52, normalized size = 0.35 \[ \frac {(1-2 x)^{3/2} \left (\frac {12005 \left (330750 x^2+439137 x+145793\right )}{(3 x+2)^6}-7033120 \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{476478450} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(3/2)*((12005*(145793 + 439137*x + 330750*x^2))/(2 + 3*x)^6 - 7033120*Hypergeometric2F1[3/2, 5, 5/2
, 3/7 - (6*x)/7]))/476478450

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fricas [A]  time = 1.02, size = 130, normalized size = 0.88 \[ \frac {219785 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (53407755 \, x^{5} + 219565215 \, x^{4} + 127601514 \, x^{3} - 139462938 \, x^{2} - 150340360 \, x - 36741296\right )} \sqrt {-2 \, x + 1}}{326728080 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/326728080*(219785*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((3*x - sqr
t(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(53407755*x^5 + 219565215*x^4 + 127601514*x^3 - 139462938*x^2 - 1503
40360*x - 36741296)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [A]  time = 1.30, size = 132, normalized size = 0.90 \[ -\frac {43957}{65345616} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {53407755 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 706169205 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 2801005326 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 3584374794 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 1082074105 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3693926495 \, \sqrt {-2 \, x + 1}}{497871360 \, {\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-43957/65345616*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/497871
360*(53407755*(2*x - 1)^5*sqrt(-2*x + 1) + 706169205*(2*x - 1)^4*sqrt(-2*x + 1) + 2801005326*(2*x - 1)^3*sqrt(
-2*x + 1) + 3584374794*(2*x - 1)^2*sqrt(-2*x + 1) + 1082074105*(-2*x + 1)^(3/2) - 3693926495*sqrt(-2*x + 1))/(
3*x + 2)^6

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maple [A]  time = 0.01, size = 84, normalized size = 0.57 \[ \frac {43957 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{32672808}-\frac {11664 \left (\frac {43957 \left (-2 x +1\right )^{\frac {11}{2}}}{74680704}-\frac {747269 \left (-2 x +1\right )^{\frac {9}{2}}}{96018048}+\frac {1058581 \left (-2 x +1\right )^{\frac {7}{2}}}{34292160}-\frac {1354639 \left (-2 x +1\right )^{\frac {5}{2}}}{34292160}-\frac {630947 \left (-2 x +1\right )^{\frac {3}{2}}}{52907904}+\frac {307699 \sqrt {-2 x +1}}{7558272}\right )}{\left (-6 x -4\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11664*(43957/74680704*(-2*x+1)^(11/2)-747269/96018048*(-2*x+1)^(9/2)+1058581/34292160*(-2*x+1)^(7/2)-1354639/
34292160*(-2*x+1)^(5/2)-630947/52907904*(-2*x+1)^(3/2)+307699/7558272*(-2*x+1)^(1/2))/(-6*x-4)^6+43957/3267280
8*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.39, size = 146, normalized size = 0.99 \[ -\frac {43957}{65345616} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {53407755 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 706169205 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2801005326 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 3584374794 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 1082074105 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 3693926495 \, \sqrt {-2 \, x + 1}}{7779240 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-43957/65345616*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/7779240*(534077
55*(-2*x + 1)^(11/2) - 706169205*(-2*x + 1)^(9/2) + 2801005326*(-2*x + 1)^(7/2) - 3584374794*(-2*x + 1)^(5/2)
- 1082074105*(-2*x + 1)^(3/2) + 3693926495*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)

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mupad [B]  time = 0.08, size = 126, normalized size = 0.86 \[ \frac {43957\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{32672808}-\frac {\frac {307699\,\sqrt {1-2\,x}}{472392}-\frac {630947\,{\left (1-2\,x\right )}^{3/2}}{3306744}-\frac {1354639\,{\left (1-2\,x\right )}^{5/2}}{2143260}+\frac {1058581\,{\left (1-2\,x\right )}^{7/2}}{2143260}-\frac {747269\,{\left (1-2\,x\right )}^{9/2}}{6001128}+\frac {43957\,{\left (1-2\,x\right )}^{11/2}}{4667544}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(43957*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/32672808 - ((307699*(1 - 2*x)^(1/2))/472392 - (630947*(1
- 2*x)^(3/2))/3306744 - (1354639*(1 - 2*x)^(5/2))/2143260 + (1058581*(1 - 2*x)^(7/2))/2143260 - (747269*(1 - 2
*x)^(9/2))/6001128 + (43957*(1 - 2*x)^(11/2))/4667544)/((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 - 184877/729)

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

[Out]

Timed out

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