3.19.24 \(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx\) [1824]

Optimal. Leaf size=107 \[ -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \]

[Out]

328715/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-53/378*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^3-1/12*(3+5
*x)^3*(1-2*x)^(1/2)/(2+3*x)^4-5/222264*(70429+110981*x)*(1-2*x)^(1/2)/(2+3*x)^2

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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 154, 150, 65, 212} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{378 (3 x+2)^3}-\frac {5 \sqrt {1-2 x} (110981 x+70429)}{222264 (3 x+2)^2}+\frac {328715 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)^3) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(12*(2 + 3*x)^4) - (5*Sqrt[1 -
2*x]*(70429 + 110981*x))/(222264*(2 + 3*x)^2) + (328715*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(111132*Sqrt[21])

Rule 65

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 99

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 150

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)/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(
n + 1), 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 154

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] && ILtQ[m, -1] && GtQ[n, 0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 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)^5} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}+\frac {1}{756} \int \frac {(445-3145 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}-\frac {328715 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{222264}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{222264}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 65, normalized size = 0.61 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (2469626+11657098 x+18358575 x^2+9646695 x^3\right )}{2 (2+3 x)^4}+328715 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2333772} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*Sqrt[1 - 2*x]*(2469626 + 11657098*x + 18358575*x^2 + 9646695*x^3))/(2*(2 + 3*x)^4) + 328715*Sqrt[21]*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2333772

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Maple [A]
time = 0.11, size = 66, normalized size = 0.62

method	result	size
risch	\(\frac {19293390 x^{4}[phone] x^{3}+4955621 x^{2}-6717846 x -2469626}{222264 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {328715 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\)	\(56\)
derivativedivides	\(-\frac {324 \left (-\frac {119095 \left (1-2 x \right )^{\frac {7}{2}}}{444528}+\frac {3126535 \left (1-2 x \right )^{\frac {5}{2}}}{1714608}-\frac {3040873 \left (1-2 x \right )^{\frac {3}{2}}}{734832}+\frac {328715 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{4}}+\frac {328715 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\)	\(66\)
default	\(-\frac {324 \left (-\frac {119095 \left (1-2 x \right )^{\frac {7}{2}}}{444528}+\frac {3126535 \left (1-2 x \right )^{\frac {5}{2}}}{1714608}-\frac {3040873 \left (1-2 x \right )^{\frac {3}{2}}}{734832}+\frac {328715 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{4}}+\frac {328715 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\)	\(66\)
trager	\(-\frac {\left (9646695 x^{3}[phone] x^{2}[phone] x +2469626\right ) \sqrt {1-2 x}}{222264 \left (2+3 x \right )^{4}}+\frac {328715 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{4667544}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^5,x,method=_RETURNVERBOSE)

[Out]

-324*(-119095/444528*(1-2*x)^(7/2)+3126535/1714608*(1-2*x)^(5/2)-3040873/734832*(1-2*x)^(3/2)+328715/104976*(1
-2*x)^(1/2))/(-4-6*x)^4+328715/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.53, size = 110, normalized size = 1.03 \begin {gather*} -\frac {328715}{4667544} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9646695 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 65657235 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 149002777 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 112749245 \, \sqrt {-2 \, x + 1}}{111132 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-328715/4667544*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/111132*(9646695
*(-2*x + 1)^(7/2) - 65657235*(-2*x + 1)^(5/2) + 149002777*(-2*x + 1)^(3/2) - 112749245*sqrt(-2*x + 1))/(81*(2*
x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]
time = 0.72, size = 100, normalized size = 0.93 \begin {gather*} \frac {328715 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (9646695 \, x^{3} + 18358575 \, x^{2} + 11657098 \, x + 2469626\right )} \sqrt {-2 \, x + 1}}{4667544 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4667544*(328715*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3
*x + 2)) - 21*(9646695*x^3 + 18358575*x^2 + 11657098*x + 2469626)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)

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Sympy [F(-1)] Timed out
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((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [A]
time = 1.46, size = 100, normalized size = 0.93 \begin {gather*} -\frac {328715}{4667544} \, \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 {9646695 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 65657235 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 149002777 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 112749245 \, \sqrt {-2 \, x + 1}}{1778112 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-328715/4667544*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/177811
2*(9646695*(2*x - 1)^3*sqrt(-2*x + 1) + 65657235*(2*x - 1)^2*sqrt(-2*x + 1) - 149002777*(-2*x + 1)^(3/2) + 112
749245*sqrt(-2*x + 1))/(3*x + 2)^4

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Mupad [B]
time = 1.18, size = 90, normalized size = 0.84 \begin {gather*} \frac {328715\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2333772}-\frac {\frac {328715\,\sqrt {1-2\,x}}{26244}-\frac {3040873\,{\left (1-2\,x\right )}^{3/2}}{183708}+\frac {3126535\,{\left (1-2\,x\right )}^{5/2}}{428652}-\frac {119095\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(328715*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2333772 - ((328715*(1 - 2*x)^(1/2))/26244 - (3040873*(1
- 2*x)^(3/2))/183708 + (3126535*(1 - 2*x)^(5/2))/428652 - (119095*(1 - 2*x)^(7/2))/111132)/((2744*x)/27 + (98*
(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

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