3.1827 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=167 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac{2 \sqrt{1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac{23717 \sqrt{1-2 x}}{9529569 (3 x+2)}+\frac{23717 \sqrt{1-2 x}}{4084101 (3 x+2)^2}+\frac{47434 \sqrt{1-2 x}}{2917215 (3 x+2)^3}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}} \]

[Out]

(47434*Sqrt[1 - 2*x])/(2917215*(2 + 3*x)^3) + (23717*Sqrt[1 - 2*x])/(4084101*(2 + 3*x)^2) + (23717*Sqrt[1 - 2*
x])/(9529569*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(1323*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(21*
(2 + 3*x)^7) - (2*Sqrt[1 - 2*x]*(54227 + 88099*x))/(972405*(2 + 3*x)^5) + (47434*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(9529569*Sqrt[21])

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Rubi [A]  time = 0.0552626, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac{2 \sqrt{1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac{23717 \sqrt{1-2 x}}{9529569 (3 x+2)}+\frac{23717 \sqrt{1-2 x}}{4084101 (3 x+2)^2}+\frac{47434 \sqrt{1-2 x}}{2917215 (3 x+2)^3}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

(47434*Sqrt[1 - 2*x])/(2917215*(2 + 3*x)^3) + (23717*Sqrt[1 - 2*x])/(4084101*(2 + 3*x)^2) + (23717*Sqrt[1 - 2*
x])/(9529569*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(1323*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(21*
(2 + 3*x)^7) - (2*Sqrt[1 - 2*x]*(54227 + 88099*x))/(972405*(2 + 3*x)^5) + (47434*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(9529569*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)^8} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}+\frac{1}{21} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^7} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}+\frac{\int \frac{(148-3640 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^6} \, dx}{2646}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{47434 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx}{138915}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{47434 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{583443}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{23717 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{1361367}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}+\frac{23717 \sqrt{1-2 x}}{9529569 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{23717 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{9529569}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}+\frac{23717 \sqrt{1-2 x}}{9529569 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}+\frac{23717 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{9529569}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}+\frac{23717 \sqrt{1-2 x}}{9529569 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0296301, size = 52, normalized size = 0.31 \[ \frac{(1-2 x)^{3/2} \left (\frac{235298 \left (165375 x^2+219414 x+72797\right )}{(3 x+2)^7}-24286208 \, _2F_1\left (\frac{3}{2},6;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{6537284334} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

((1 - 2*x)^(3/2)*((235298*(72797 + 219414*x + 165375*x^2))/(2 + 3*x)^7 - 24286208*Hypergeometric2F1[3/2, 6, 5/
2, 3/7 - (6*x)/7]))/6537284334

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Maple [A]  time = 0.011, size = 93, normalized size = 0.6 \begin{align*} 69984\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{7}} \left ( -{\frac{23717\, \left ( 1-2\,x \right ) ^{13/2}}{457419312}}+{\frac{118585\, \left ( 1-2\,x \right ) ^{11/2}}{147027636}}-{\frac{6711911\, \left ( 1-2\,x \right ) ^{9/2}}{1260236880}}+{\frac{1303513\, \left ( 1-2\,x \right ) ^{7/2}}{78764805}}-{\frac{5101561\, \left ( 1-2\,x \right ) ^{5/2}}{231472080}}+{\frac{25163\, \left ( 1-2\,x \right ) ^{3/2}}{4960116}}+{\frac{23717\,\sqrt{1-2\,x}}{2834352}} \right ) }+{\frac{47434\,\sqrt{21}}{200120949}{\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)^8,x)

[Out]

69984*(-23717/457419312*(1-2*x)^(13/2)+118585/147027636*(1-2*x)^(11/2)-6711911/1260236880*(1-2*x)^(9/2)+130351
3/78764805*(1-2*x)^(7/2)-5101561/231472080*(1-2*x)^(5/2)+25163/4960116*(1-2*x)^(3/2)+23717/2834352*(1-2*x)^(1/
2))/(-6*x-4)^7+47434/200120949*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.9789, size = 221, normalized size = 1.32 \begin{align*} -\frac{23717}{200120949} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (86448465 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 1344753900 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 8879858253 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 27592763184 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 36746543883 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 8458290820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13951406665 \, \sqrt{-2 \, x + 1}\right )}}{47647845 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\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)^8,x, algorithm="maxima")

[Out]

-23717/200120949*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/47647845*(8644
8465*(-2*x + 1)^(13/2) - 1344753900*(-2*x + 1)^(11/2) + 8879858253*(-2*x + 1)^(9/2) - 27592763184*(-2*x + 1)^(
7/2) + 36746543883*(-2*x + 1)^(5/2) - 8458290820*(-2*x + 1)^(3/2) - 13951406665*sqrt(-2*x + 1))/(2187*(2*x - 1
)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2
+ 4941258*x - 1647086)

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Fricas [A]  time = 1.594, size = 509, normalized size = 3.05 \begin{align*} \frac{118585 \, \sqrt{21}{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (86448465 \, x^{6} + 413031555 \, x^{5} + 863203932 \, x^{4} + 473987484 \, x^{3} - 306463011 \, x^{2} - 361589428 \, x - 88036937\right )} \sqrt{-2 \, x + 1}}{1000604745 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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)^8,x, algorithm="fricas")

[Out]

1/1000604745*(118585*sqrt(21)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x +
128)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(86448465*x^6 + 413031555*x^5 + 863203932*x^4 + 4
73987484*x^3 - 306463011*x^2 - 361589428*x - 88036937)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 226
80*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.66353, size = 200, normalized size = 1.2 \begin{align*} -\frac{23717}{200120949} \, \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{86448465 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + 1344753900 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 8879858253 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 27592763184 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 36746543883 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 8458290820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13951406665 \, \sqrt{-2 \, x + 1}}{3049462080 \,{\left (3 \, x + 2\right )}^{7}} \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)^8,x, algorithm="giac")

[Out]

-23717/200120949*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/30494
62080*(86448465*(2*x - 1)^6*sqrt(-2*x + 1) + 1344753900*(2*x - 1)^5*sqrt(-2*x + 1) + 8879858253*(2*x - 1)^4*sq
rt(-2*x + 1) + 27592763184*(2*x - 1)^3*sqrt(-2*x + 1) + 36746543883*(2*x - 1)^2*sqrt(-2*x + 1) - 8458290820*(-
2*x + 1)^(3/2) - 13951406665*sqrt(-2*x + 1))/(3*x + 2)^7