3.20.83 \(\int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^2} \, dx\) [1983]

Optimal. Leaf size=89 \[ \frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}-\frac {616}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

56/375*(1-2*x)^(3/2)+56/1375*(1-2*x)^(5/2)-1/55*(1-2*x)^(7/2)/(3+5*x)-616/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(
1/2))*55^(1/2)+616/625*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 89, 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, 52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{7/2}}{55 (5 x+3)}+\frac {56 (1-2 x)^{5/2}}{1375}+\frac {56}{375} (1-2 x)^{3/2}+\frac {616}{625} \sqrt {1-2 x}-\frac {616}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(616*Sqrt[1 - 2*x])/625 + (56*(1 - 2*x)^(3/2))/375 + (56*(1 - 2*x)^(5/2))/1375 - (1 - 2*x)^(7/2)/(55*(3 + 5*x)
) - (616*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {28}{55} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {28}{25} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {308}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {3388}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}-\frac {3388}{625} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}-\frac {616}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 63, normalized size = 0.71 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (6579+8630 x-3820 x^2+1800 x^3\right )}{3+5 x}-1848 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{9375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*Sqrt[1 - 2*x]*(6579 + 8630*x - 3820*x^2 + 1800*x^3))/(3 + 5*x) - 1848*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/9375

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Maple [A]
time = 0.12, size = 63, normalized size = 0.71

method	result	size
risch	\(-\frac {3600 x^{4}-9440 x^{3}+21080 x^{2}+4528 x -6579}{1875 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {616 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(56\)
derivativedivides	\(\frac {6 \left (1-2 x \right )^{\frac {5}{2}}}{125}+\frac {62 \left (1-2 x \right )^{\frac {3}{2}}}{375}+\frac {638 \sqrt {1-2 x}}{625}+\frac {242 \sqrt {1-2 x}}{3125 \left (-\frac {6}{5}-2 x \right )}-\frac {616 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(63\)
default	\(\frac {6 \left (1-2 x \right )^{\frac {5}{2}}}{125}+\frac {62 \left (1-2 x \right )^{\frac {3}{2}}}{375}+\frac {638 \sqrt {1-2 x}}{625}+\frac {242 \sqrt {1-2 x}}{3125 \left (-\frac {6}{5}-2 x \right )}-\frac {616 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(63\)
trager	\(\frac {\left (1800 x^{3}-3820 x^{2}+8630 x +6579\right ) \sqrt {1-2 x}}{5625+9375 x}+\frac {308 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{3125}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(1-2*x)^(5/2)+62/375*(1-2*x)^(3/2)+638/625*(1-2*x)^(1/2)+242/3125*(1-2*x)^(1/2)/(-6/5-2*x)-616/3125*arct
anh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]
time = 0.56, size = 80, normalized size = 0.90 \begin {gather*} \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {62}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {308}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {638}{625} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(-2*x + 1)^(5/2) + 62/375*(-2*x + 1)^(3/2) + 308/3125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(
55) + 5*sqrt(-2*x + 1))) + 638/625*sqrt(-2*x + 1) - 121/625*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]
time = 1.83, size = 75, normalized size = 0.84 \begin {gather*} \frac {924 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (1800 \, x^{3} - 3820 \, x^{2} + 8630 \, x + 6579\right )} \sqrt {-2 \, x + 1}}{9375 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9375*(924*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 5*(1800*x^
3 - 3820*x^2 + 8630*x + 6579)*sqrt(-2*x + 1))/(5*x + 3)

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

[Out]

Timed out

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Giac [A]
time = 0.76, size = 90, normalized size = 1.01 \begin {gather*} \frac {6}{125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {62}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {308}{3125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {638}{625} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(2*x - 1)^2*sqrt(-2*x + 1) + 62/375*(-2*x + 1)^(3/2) + 308/3125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sq
rt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 638/625*sqrt(-2*x + 1) - 121/625*sqrt(-2*x + 1)/(5*x + 3)

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Mupad [B]
time = 1.18, size = 64, normalized size = 0.72 \begin {gather*} \frac {638\,\sqrt {1-2\,x}}{625}-\frac {242\,\sqrt {1-2\,x}}{3125\,\left (2\,x+\frac {6}{5}\right )}+\frac {62\,{\left (1-2\,x\right )}^{3/2}}{375}+\frac {6\,{\left (1-2\,x\right )}^{5/2}}{125}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,616{}\mathrm {i}}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*616i)/3125 - (242*(1 - 2*x)^(1/2))/(3125*(2*x + 6/5)) + (638*
(1 - 2*x)^(1/2))/625 + (62*(1 - 2*x)^(3/2))/375 + (6*(1 - 2*x)^(5/2))/125

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