3.18.100 \(\int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^4} \, dx\) [1800]

Optimal. Leaf size=88 \[ \frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]

[Out]

1/63*(1-2*x)^(3/2)/(2+3*x)^3+34/9261*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-17/63*(1-2*x)^(1/2)/(2+3*x)^
2+17/441*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, 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}}{63 (3 x+2)^3}+\frac {17 \sqrt {1-2 x}}{441 (3 x+2)}-\frac {17 \sqrt {1-2 x}}{63 (3 x+2)^2}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(1 - 2*x)^(3/2)/(63*(2 + 3*x)^3) - (17*Sqrt[1 - 2*x])/(63*(2 + 3*x)^2) + (17*Sqrt[1 - 2*x])/(441*(2 + 3*x)) +
(34*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*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 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)^4} \, dx &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}+\frac {34}{21} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}-\frac {17}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}-\frac {17}{441} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}+\frac {17}{441} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 58, normalized size = 0.66 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-163-167 x+153 x^2\right )}{(2+3 x)^3}+34 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

((21*Sqrt[1 - 2*x]*(-163 - 167*x + 153*x^2))/(2 + 3*x)^3 + 34*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9261

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Maple [A]
time = 0.12, size = 57, normalized size = 0.65

method	result	size
risch	\(-\frac {306 x^{3}-487 x^{2}-159 x +163}{441 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {34 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\)	\(51\)
derivativedivides	\(\frac {-\frac {34 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{63}+\frac {34 \sqrt {1-2 x}}{9}}{\left (-4-6 x \right )^{3}}+\frac {34 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\)	\(57\)
default	\(\frac {-\frac {34 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{63}+\frac {34 \sqrt {1-2 x}}{9}}{\left (-4-6 x \right )^{3}}+\frac {34 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\)	\(57\)
trager	\(\frac {\left (153 x^{2}-167 x -163\right ) \sqrt {1-2 x}}{441 \left (2+3 x \right )^{3}}-\frac {17 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{9261}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)*(1-2*x)^(1/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

216*(-17/5292*(1-2*x)^(5/2)-1/1701*(1-2*x)^(3/2)+17/972*(1-2*x)^(1/2))/(-4-6*x)^3+34/9261*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.91, size = 92, normalized size = 1.05 \begin {gather*} -\frac {17}{9261} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (153 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 28 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 833 \, \sqrt {-2 \, x + 1}\right )}}{441 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\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)^4,x, algorithm="maxima")

[Out]

-17/9261*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/441*(153*(-2*x + 1)^(5
/2) + 28*(-2*x + 1)^(3/2) - 833*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]
time = 1.03, size = 85, normalized size = 0.97 \begin {gather*} \frac {17 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (153 \, x^{2} - 167 \, x - 163\right )} \sqrt {-2 \, x + 1}}{9261 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

1/9261*(17*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(153*
x^2 - 167*x - 163)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]
time = 228.26, size = 500, normalized size = 5.68 \begin {gather*} \frac {40 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} + \frac {296 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} + \frac {112 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \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)**4,x)

[Out]

40*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqr
t(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt
(1 - 2*x) < sqrt(21)/3)))/9 + 296*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)
*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) +
3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt
(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/9 + 112*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32
+ 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*
x)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt
(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3)
& (sqrt(1 - 2*x) < sqrt(21)/3)))/9

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Giac [A]
time = 1.33, size = 84, normalized size = 0.95 \begin {gather*} -\frac {17}{9261} \, \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 {153 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 28 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 833 \, \sqrt {-2 \, x + 1}}{1764 \, {\left (3 \, x + 2\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

-17/9261*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1764*(153*(2*
x - 1)^2*sqrt(-2*x + 1) + 28*(-2*x + 1)^(3/2) - 833*sqrt(-2*x + 1))/(3*x + 2)^3

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Mupad [B]
time = 1.19, size = 71, normalized size = 0.81 \begin {gather*} \frac {34\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9261}+\frac {\frac {8\,{\left (1-2\,x\right )}^{3/2}}{1701}-\frac {34\,\sqrt {1-2\,x}}{243}+\frac {34\,{\left (1-2\,x\right )}^{5/2}}{1323}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \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)^4,x)

[Out]

(34*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/9261 + ((8*(1 - 2*x)^(3/2))/1701 - (34*(1 - 2*x)^(1/2))/243
+ (34*(1 - 2*x)^(5/2))/1323)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)

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