3.19.12 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\) [1812]

Optimal. Leaf size=108 \[ -\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \]

[Out]

-1/252*(1-2*x)^(3/2)/(2+3*x)^4+275/5292*(1-2*x)^(3/2)/(2+3*x)^3+4625/777924*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2)
)*21^(1/2)-4625/10584*(1-2*x)^(1/2)/(2+3*x)^2+4625/74088*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 43, 44, 65, 212} \begin {gather*} \frac {275 (1-2 x)^{3/2}}{5292 (3 x+2)^3}-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}+\frac {4625 \sqrt {1-2 x}}{74088 (3 x+2)}-\frac {4625 \sqrt {1-2 x}}{10584 (3 x+2)^2}+\frac {4625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/252*(1 - 2*x)^(3/2)/(2 + 3*x)^4 + (275*(1 - 2*x)^(3/2))/(5292*(2 + 3*x)^3) - (4625*Sqrt[1 - 2*x])/(10584*(2
+ 3*x)^2) + (4625*Sqrt[1 - 2*x])/(74088*(2 + 3*x)) + (4625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {\sqrt {1-2 x} (1125+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}+\frac {4625 \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx}{1764}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}-\frac {4625 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}-\frac {4625 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{74088}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{74088}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 65, normalized size = 0.60 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-85094-225262 x-64725 x^2+124875 x^3\right )}{2 (2+3 x)^4}+4625 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{777924} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*Sqrt[1 - 2*x]*(-85094 - 225262*x - 64725*x^2 + 124875*x^3))/(2*(2 + 3*x)^4) + 4625*Sqrt[21]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/777924

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Maple [A]
time = 0.10, size = 66, normalized size = 0.61

method	result	size
risch	\(-\frac {249750 x^{4}-254325 x^{3}-385799 x^{2}+55074 x +85094}{74088 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {4625 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\)	\(56\)
derivativedivides	\(\frac {-\frac {4625 \left (1-2 x \right )^{\frac {7}{2}}}{1372}+\frac {11675 \left (1-2 x \right )^{\frac {5}{2}}}{1764}+\frac {16027 \left (1-2 x \right )^{\frac {3}{2}}}{756}-\frac {4625 \sqrt {1-2 x}}{108}}{\left (-4-6 x \right )^{4}}+\frac {4625 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\)	\(66\)
default	\(\frac {-\frac {4625 \left (1-2 x \right )^{\frac {7}{2}}}{1372}+\frac {11675 \left (1-2 x \right )^{\frac {5}{2}}}{1764}+\frac {16027 \left (1-2 x \right )^{\frac {3}{2}}}{756}-\frac {4625 \sqrt {1-2 x}}{108}}{\left (-4-6 x \right )^{4}}+\frac {4625 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\)	\(66\)
trager	\(\frac {\left (124875 x^{3}-64725 x^{2}-225262 x -85094\right ) \sqrt {1-2 x}}{74088 \left (2+3 x \right )^{4}}+\frac {4625 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1555848}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

648*(-4625/889056*(1-2*x)^(7/2)+11675/1143072*(1-2*x)^(5/2)+16027/489888*(1-2*x)^(3/2)-4625/69984*(1-2*x)^(1/2
))/(-4-6*x)^4+4625/777924*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.48, size = 110, normalized size = 1.02 \begin {gather*} -\frac {4625}{1555848} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {124875 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 245175 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 785323 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1586375 \, \sqrt {-2 \, x + 1}}{37044 \, {\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)^2*(1-2*x)^(1/2)/(2+3*x)^5,x, algorithm="maxima")

[Out]

-4625/1555848*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/37044*(124875*(-2
*x + 1)^(7/2) - 245175*(-2*x + 1)^(5/2) - 785323*(-2*x + 1)^(3/2) + 1586375*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 = 0.88, size = 100, normalized size = 0.93 \begin {gather*} \frac {4625 \, \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 (124875 \, x^{3} - 64725 \, x^{2} - 225262 \, x - 85094\right )} \sqrt {-2 \, x + 1}}{1555848 \, {\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)^2*(1-2*x)^(1/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/1555848*(4625*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*(124875*x^3 - 64725*x^2 - 225262*x - 85094)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 1
6)

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

[Out]

Timed out

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Giac [A]
time = 0.54, size = 100, normalized size = 0.93 \begin {gather*} -\frac {4625}{1555848} \, \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 {124875 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 245175 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 785323 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1586375 \, \sqrt {-2 \, x + 1}}{592704 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4625/1555848*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/592704*(
124875*(2*x - 1)^3*sqrt(-2*x + 1) + 245175*(2*x - 1)^2*sqrt(-2*x + 1) + 785323*(-2*x + 1)^(3/2) - 1586375*sqrt
(-2*x + 1))/(3*x + 2)^4

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Mupad [B]
time = 1.20, size = 90, normalized size = 0.83 \begin {gather*} \frac {4625\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{777924}-\frac {\frac {4625\,\sqrt {1-2\,x}}{8748}-\frac {16027\,{\left (1-2\,x\right )}^{3/2}}{61236}-\frac {11675\,{\left (1-2\,x\right )}^{5/2}}{142884}+\frac {4625\,{\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)^2)/(3*x + 2)^5,x)

[Out]

(4625*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/777924 - ((4625*(1 - 2*x)^(1/2))/8748 - (16027*(1 - 2*x)^(
3/2))/61236 - (11675*(1 - 2*x)^(5/2))/142884 + (4625*(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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