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*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])

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Rubi [A]  time = 0.0475691, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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 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 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 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 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*}

Mathematica [C]  time = 0.0264708, 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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Maple [A]  time = 0.011, size = 84, normalized size = 0.6 \begin{align*} -11664\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ({\frac{43957\, \left ( 1-2\,x \right ) ^{11/2}}{74680704}}-{\frac{747269\, \left ( 1-2\,x \right ) ^{9/2}}{96018048}}+{\frac{1058581\, \left ( 1-2\,x \right ) ^{7/2}}{34292160}}-{\frac{1354639\, \left ( 1-2\,x \right ) ^{5/2}}{34292160}}-{\frac{630947\, \left ( 1-2\,x \right ) ^{3/2}}{52907904}}+{\frac{307699\,\sqrt{1-2\,x}}{7558272}} \right ) }+{\frac{43957\,\sqrt{21}}{32672808}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11664*(43957/74680704*(1-2*x)^(11/2)-747269/96018048*(1-2*x)^(9/2)+1058581/34292160*(1-2*x)^(7/2)-1354639/342
92160*(1-2*x)^(5/2)-630947/52907904*(1-2*x)^(3/2)+307699/7558272*(1-2*x)^(1/2))/(-6*x-4)^6+43957/32672808*arct
anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.31494, size = 197, normalized size = 1.34 \begin{align*} -\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 )}} \end{align*}

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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Fricas [A]  time = 1.56931, size = 437, normalized size = 2.97 \begin{align*} \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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.53146, size = 178, normalized size = 1.21 \begin{align*} -\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}} \end{align*}

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