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]

47434/200120949*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+47434/2917215*(1-2*x)^(1/2)/(2+3*x)^3+23717/40841
01*(1-2*x)^(1/2)/(2+3*x)^2+23717/9529569*(1-2*x)^(1/2)/(2+3*x)-53/1323*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^6-1/21*
(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^7-2/972405*(54227+88099*x)*(1-2*x)^(1/2)/(2+3*x)^5

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Rubi [A]  time = 0.06, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac {2 \sqrt {1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac {23717 \sqrt {1-2 x}}{9529569 (3 x+2)}+\frac {23717 \sqrt {1-2 x}}{4084101 (3 x+2)^2}+\frac {47434 \sqrt {1-2 x}}{2917215 (3 x+2)^3}+\frac {47434 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9529569 \sqrt {21}} \]

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 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)^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*}

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Mathematica [C]  time = 0.04, 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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fricas [A]  time = 0.94, size = 145, normalized size = 0.87 \[ \frac {118585 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (86448465 \, x^{6} + 413031555 \, x^{5} + 863203932 \, x^{4} + 473987484 \, x^{3} - 306463011 \, x^{2} - 361589428 \, x - 88036937\right )} \sqrt {-2 \, x + 1}}{1000604745 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

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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giac [A]  time = 1.31, size = 148, normalized size = 0.89 \[ -\frac {23717}{200120949} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {86448465 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 1344753900 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 8879858253 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 27592763184 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 36746543883 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 8458290820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13951406665 \, \sqrt {-2 \, x + 1}}{3049462080 \, {\left (3 \, x + 2\right )}^{7}} \]

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

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maple [A]  time = 0.01, size = 93, normalized size = 0.56 \[ \frac {47434 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{200120949}+\frac {-\frac {426906 \left (-2 x +1\right )^{\frac {13}{2}}}{117649}+\frac {948680 \left (-2 x +1\right )^{\frac {11}{2}}}{16807}-\frac {13423822 \left (-2 x +1\right )^{\frac {9}{2}}}{36015}+\frac {41712416 \left (-2 x +1\right )^{\frac {7}{2}}}{36015}-\frac {10203122 \left (-2 x +1\right )^{\frac {5}{2}}}{6615}+\frac {201304 \left (-2 x +1\right )^{\frac {3}{2}}}{567}+\frac {47434 \sqrt {-2 x +1}}{81}}{\left (-6 x -4\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3*(-2*x+1)^(1/2)/(3*x+2)^8,x)

[Out]

69984*(-23717/457419312*(-2*x+1)^(13/2)+118585/147027636*(-2*x+1)^(11/2)-6711911/1260236880*(-2*x+1)^(9/2)+130
3513/78764805*(-2*x+1)^(7/2)-5101561/231472080*(-2*x+1)^(5/2)+25163/4960116*(-2*x+1)^(3/2)+23717/2834352*(-2*x
+1)^(1/2))/(-6*x-4)^7+47434/200120949*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.30, size = 164, normalized size = 0.98 \[ -\frac {23717}{200120949} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (86448465 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 1344753900 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 8879858253 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 27592763184 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 36746543883 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8458290820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13951406665 \, \sqrt {-2 \, x + 1}\right )}}{47647845 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]

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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mupad [B]  time = 0.07, size = 144, normalized size = 0.86 \[ \frac {47434\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{200120949}-\frac {\frac {47434\,\sqrt {1-2\,x}}{177147}+\frac {201304\,{\left (1-2\,x\right )}^{3/2}}{1240029}-\frac {10203122\,{\left (1-2\,x\right )}^{5/2}}{14467005}+\frac {41712416\,{\left (1-2\,x\right )}^{7/2}}{78764805}-\frac {13423822\,{\left (1-2\,x\right )}^{9/2}}{78764805}+\frac {948680\,{\left (1-2\,x\right )}^{11/2}}{36756909}-\frac {47434\,{\left (1-2\,x\right )}^{13/2}}{28588707}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^3)/(3*x + 2)^8,x)

[Out]

(47434*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/200120949 - ((47434*(1 - 2*x)^(1/2))/177147 + (201304*(1
- 2*x)^(3/2))/1240029 - (10203122*(1 - 2*x)^(5/2))/14467005 + (41712416*(1 - 2*x)^(7/2))/78764805 - (13423822*
(1 - 2*x)^(9/2))/78764805 + (948680*(1 - 2*x)^(11/2))/36756909 - (47434*(1 - 2*x)^(13/2))/28588707)/((1647086*
x)/729 + (117649*(2*x - 1)^2)/81 + (84035*(2*x - 1)^3)/81 + (12005*(2*x - 1)^4)/27 + (343*(2*x - 1)^5)/3 + (49
*(2*x - 1)^6)/3 + (2*x - 1)^7 - 1647086/2187)

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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)**8,x)

[Out]

Timed out

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