3.20.13 \(\int \frac {(1-2 x)^{3/2}}{(3+5 x)^2} \, dx\) [1913]

Optimal. Leaf size=63 \[ -\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}+\frac {6}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-1/5*(1-2*x)^(3/2)/(3+5*x)+6/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-6/25*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{3/2}}{5 (5 x+3)}-\frac {6}{25} \sqrt {1-2 x}+\frac {6}{25} \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)^(3/2)/(3 + 5*x)^2,x]

[Out]

(-6*Sqrt[1 - 2*x])/25 - (1 - 2*x)^(3/2)/(5*(3 + 5*x)) + (6*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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 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 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)^{3/2}}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}-\frac {3}{5} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=-\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}-\frac {33}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}+\frac {33}{25} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}+\frac {6}{25} \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 = 0.10, size = 53, normalized size = 0.84 \begin {gather*} \frac {1}{125} \left (-\frac {5 \sqrt {1-2 x} (23+20 x)}{3+5 x}+6 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*Sqrt[1 - 2*x]*(23 + 20*x))/(3 + 5*x) + 6*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

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Maple [A]
time = 0.10, size = 45, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {4 \sqrt {1-2 x}}{25}+\frac {22 \sqrt {1-2 x}}{125 \left (-\frac {6}{5}-2 x \right )}+\frac {6 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\)	\(45\)
default	\(-\frac {4 \sqrt {1-2 x}}{25}+\frac {22 \sqrt {1-2 x}}{125 \left (-\frac {6}{5}-2 x \right )}+\frac {6 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\)	\(45\)
risch	\(\frac {40 x^{2}+26 x -23}{25 \left (3+5 x \right ) \sqrt {1-2 x}}+\frac {6 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\)	\(46\)
trager	\(-\frac {\left (23+20 x \right ) \sqrt {1-2 x}}{25 \left (3+5 x \right )}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{125}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/25*(1-2*x)^(1/2)+22/125*(1-2*x)^(1/2)/(-6/5-2*x)+6/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]
time = 0.51, size = 62, normalized size = 0.98 \begin {gather*} -\frac {3}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4}{25} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4/25*sqrt(-2*x + 1) - 11/2
5*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]
time = 1.10, size = 66, normalized size = 1.05 \begin {gather*} \frac {3 \, \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 (20 \, x + 23\right )} \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/125*(3*sqrt(11)*sqrt(5)*(5*x + 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) - 5*(20*x + 23
)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [C] Result contains complex when optimal does not.
time = 1.46, size = 238, normalized size = 3.78 \begin {gather*} \begin {cases} \frac {6 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} + \frac {4 \sqrt {2} \sqrt {x + \frac {3}{5}}}{25 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} - \frac {11 \sqrt {2}}{125 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {121 \sqrt {2}}{1250 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {6 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} - \frac {4 \sqrt {2} i \sqrt {x + \frac {3}{5}}}{25 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} + \frac {11 \sqrt {2} i}{125 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {121 \sqrt {2} i}{1250 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((6*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/125 + 4*sqrt(2)*sqrt(x + 3/5)/(25*sqrt(-1 + 11/(10*(
x + 3/5)))) - 11*sqrt(2)/(125*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - 121*sqrt(2)/(1250*sqrt(-1 + 11/(10
*(x + 3/5)))*(x + 3/5)**(3/2)), 1/Abs(x + 3/5) > 10/11), (-6*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/125
- 4*sqrt(2)*I*sqrt(x + 3/5)/(25*sqrt(1 - 11/(10*(x + 3/5)))) + 11*sqrt(2)*I/(125*sqrt(1 - 11/(10*(x + 3/5)))*
sqrt(x + 3/5)) + 121*sqrt(2)*I/(1250*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), True))

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Giac [A]
time = 0.57, size = 65, normalized size = 1.03 \begin {gather*} -\frac {3}{125} \, \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 {4}{25} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4/25*sqrt(-2*x +
1) - 11/25*sqrt(-2*x + 1)/(5*x + 3)

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Mupad [B]
time = 0.06, size = 44, normalized size = 0.70 \begin {gather*} \frac {6\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{125}-\frac {22\,\sqrt {1-2\,x}}{125\,\left (2\,x+\frac {6}{5}\right )}-\frac {4\,\sqrt {1-2\,x}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)/(5*x + 3)^2,x)

[Out]

(6*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/125 - (22*(1 - 2*x)^(1/2))/(125*(2*x + 6/5)) - (4*(1 - 2*x)^
(1/2))/25

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