3.19.1 \(\int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx\) [1801]

Optimal. Leaf size=110 \[ \frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {5 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372} \]

[Out]

1/84*(1-2*x)^(3/2)/(2+3*x)^4+5/9604*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-5/28*(1-2*x)^(1/2)/(2+3*x)^3+
5/392*(1-2*x)^(1/2)/(2+3*x)^2+15/2744*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.02, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65, 212} \begin {gather*} \frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}+\frac {15 \sqrt {1-2 x}}{2744 (3 x+2)}+\frac {5 \sqrt {1-2 x}}{392 (3 x+2)^2}-\frac {5 \sqrt {1-2 x}}{28 (3 x+2)^3}+\frac {5 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(3/2)/(84*(2 + 3*x)^4) - (5*Sqrt[1 - 2*x])/(28*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x])/(392*(2 + 3*x)^2) +
(15*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (5*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1372

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 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+3 x)^5} \, dx &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}+\frac {45}{28} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}-\frac {5}{28} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}-\frac {15}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}-\frac {15 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2744}\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {15 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2744}\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {5 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 65, normalized size = 0.59 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \left (-2062-1726 x+3375 x^2+1215 x^3\right )}{2 (2+3 x)^4}+15 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*(-2062 - 1726*x + 3375*x^2 + 1215*x^3))/(2*(2 + 3*x)^4) + 15*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/28812

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Maple [A]
time = 0.12, size = 66, normalized size = 0.60

method	result	size
risch	\(-\frac {2430 x^{4}+5535 x^{3}-6827 x^{2}-2398 x +2062}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {5 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\)	\(56\)
derivativedivides	\(-\frac {1296 \left (\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21952}-\frac {55 \left (1-2 x \right )^{\frac {5}{2}}}{28224}+\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {5 \sqrt {1-2 x}}{1728}\right )}{\left (-4-6 x \right )^{4}}+\frac {5 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\)	\(66\)
default	\(-\frac {1296 \left (\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21952}-\frac {55 \left (1-2 x \right )^{\frac {5}{2}}}{28224}+\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {5 \sqrt {1-2 x}}{1728}\right )}{\left (-4-6 x \right )^{4}}+\frac {5 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\)	\(66\)
trager	\(\frac {\left (1215 x^{3}+3375 x^{2}-1726 x -2062\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}+\frac {5 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{19208}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1296*(5/21952*(1-2*x)^(7/2)-55/28224*(1-2*x)^(5/2)+209/108864*(1-2*x)^(3/2)+5/1728*(1-2*x)^(1/2))/(-4-6*x)^4+
5/9604*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.85, size = 110, normalized size = 1.00 \begin {gather*} -\frac {5}{19208} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1215 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10395 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 10241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15435 \, \sqrt {-2 \, x + 1}}{4116 \, {\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)*(1-2*x)^(1/2)/(2+3*x)^5,x, algorithm="maxima")

[Out]

-5/19208*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/4116*(1215*(-2*x + 1)^
(7/2) - 10395*(-2*x + 1)^(5/2) + 10241*(-2*x + 1)^(3/2) + 15435*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 = 1.14, size = 106, normalized size = 0.96 \begin {gather*} \frac {15 \, \sqrt {7} \sqrt {3} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 7 \, {\left (1215 \, x^{3} + 3375 \, x^{2} - 1726 \, x - 2062\right )} \sqrt {-2 \, x + 1}}{57624 \, {\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)*(1-2*x)^(1/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/57624*(15*sqrt(7)*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*
x + 5)/(3*x + 2)) + 7*(1215*x^3 + 3375*x^2 - 1726*x - 2062)*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)*(1-2*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [A]
time = 1.05, size = 100, normalized size = 0.91 \begin {gather*} -\frac {5}{19208} \, \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 {1215 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10395 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 10241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 15435 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/19208*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/65856*(1215*(
2*x - 1)^3*sqrt(-2*x + 1) + 10395*(2*x - 1)^2*sqrt(-2*x + 1) - 10241*(-2*x + 1)^(3/2) - 15435*sqrt(-2*x + 1))/
(3*x + 2)^4

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Mupad [B]
time = 0.08, size = 90, normalized size = 0.82 \begin {gather*} \frac {5\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9604}-\frac {\frac {5\,\sqrt {1-2\,x}}{108}+\frac {209\,{\left (1-2\,x\right )}^{3/2}}{6804}-\frac {55\,{\left (1-2\,x\right )}^{5/2}}{1764}+\frac {5\,{\left (1-2\,x\right )}^{7/2}}{1372}}{\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*x + 2)^5,x)

[Out]

(5*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/9604 - ((5*(1 - 2*x)^(1/2))/108 + (209*(1 - 2*x)^(3/2))/6804
- (55*(1 - 2*x)^(5/2))/1764 + (5*(1 - 2*x)^(7/2))/1372)/((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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