3.21.11 \(\int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx\) [2011]

Optimal. Leaf size=108 \[ \frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]

[Out]

-95/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/84*(1-2*x)^(1/2)/(2+3*x)^4-19/252*(1-2*x)^(1/2)/(2+3*
x)^3-95/3528*(1-2*x)^(1/2)/(2+3*x)^2-95/8232*(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 = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 44, 65, 212} \begin {gather*} -\frac {95 \sqrt {1-2 x}}{8232 (3 x+2)}-\frac {95 \sqrt {1-2 x}}{3528 (3 x+2)^2}-\frac {19 \sqrt {1-2 x}}{252 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(84*(2 + 3*x)^4) - (19*Sqrt[1 - 2*x])/(252*(2 + 3*x)^3) - (95*Sqrt[1 - 2*x])/(3528*(2 + 3*x)^2)
- (95*Sqrt[1 - 2*x])/(8232*(2 + 3*x)) - (95*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(4116*Sqrt[21])

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 {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}+\frac {19}{12} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}+\frac {95}{252} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}+\frac {95 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}+\frac {95 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8232}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 65, normalized size = 0.60 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (2790+7942 x+7125 x^2+2565 x^3\right )}{2 (2+3 x)^4}-95 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{86436} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*Sqrt[1 - 2*x]*(2790 + 7942*x + 7125*x^2 + 2565*x^3))/(2*(2 + 3*x)^4) - 95*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/86436

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Maple [A]
time = 0.12, size = 66, normalized size = 0.61

method	result	size
risch	\(\frac {5130 x^{4}+11685 x^{3}+8759 x^{2}-2362 x -2790}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {95 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\)	\(56\)
derivativedivides	\(-\frac {1296 \left (-\frac {95 \left (1-2 x \right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (1-2 x \right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{4}}-\frac {95 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\)	\(66\)
default	\(-\frac {1296 \left (-\frac {95 \left (1-2 x \right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (1-2 x \right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{4}}-\frac {95 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\)	\(66\)
trager	\(-\frac {\left (2565 x^{3}+7125 x^{2}+7942 x +2790\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}-\frac {95 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{172872}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1296*(-95/197568*(1-2*x)^(7/2)+1045/254016*(1-2*x)^(5/2)-1387/108864*(1-2*x)^(3/2)+1447/108864*(1-2*x)^(1/2))
/(-4-6*x)^4-95/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.49, size = 110, normalized size = 1.02 \begin {gather*} \frac {95}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2565 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 21945 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 70903 \, \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)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

95/172872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/4116*(2565*(-2*x + 1)
^(7/2) - 21945*(-2*x + 1)^(5/2) + 67963*(-2*x + 1)^(3/2) - 70903*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 = 1.10, size = 99, normalized size = 0.92 \begin {gather*} \frac {95 \, \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 (2565 \, x^{3} + 7125 \, x^{2} + 7942 \, x + 2790\right )} \sqrt {-2 \, x + 1}}{172872 \, {\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+3*x)^5/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/172872*(95*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*(2565*x^3 + 7125*x^2 + 7942*x + 2790)*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)/(2+3*x)**5/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 0.96, size = 100, normalized size = 0.93 \begin {gather*} \frac {95}{172872} \, \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 {2565 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 21945 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 70903 \, \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)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

95/172872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/65856*(2565*
(2*x - 1)^3*sqrt(-2*x + 1) + 21945*(2*x - 1)^2*sqrt(-2*x + 1) - 67963*(-2*x + 1)^(3/2) + 70903*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 {95\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {1447\,\sqrt {1-2\,x}}{6804}-\frac {1387\,{\left (1-2\,x\right )}^{3/2}}{6804}+\frac {1045\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {95\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\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((5*x + 3)/((1 - 2*x)^(1/2)*(3*x + 2)^5),x)

[Out]

- (95*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/86436 - ((1447*(1 - 2*x)^(1/2))/6804 - (1387*(1 - 2*x)^(3/
2))/6804 + (1045*(1 - 2*x)^(5/2))/15876 - (95*(1 - 2*x)^(7/2))/12348)/((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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