3.22.21 \(\int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\) [2121]

Optimal. Leaf size=63 \[ \frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (3+5 x)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-6/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+6/121/(1-2*x)^(1/2)-1/11/(3+5*x)/(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 = {44, 53, 65, 212} \begin {gather*} \frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

6/(121*Sqrt[1 - 2*x]) - 1/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (6*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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 53

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}+\frac {15}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {1-2 x}}{121 (3+5 x)}+\frac {15}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {1-2 x}}{121 (3+5 x)}-\frac {15}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {1-2 x}}{121 (3+5 x)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 63, normalized size = 1.00 \begin {gather*} \frac {2 (-22+15 (1-2 x))}{121 (-11+5 (1-2 x)) \sqrt {1-2 x}}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-22 + 15*(1 - 2*x)))/(121*(-11 + 5*(1 - 2*x))*Sqrt[1 - 2*x]) - (6*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/121

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Maple [A]
time = 0.16, size = 45, normalized size = 0.71

method	result	size
risch	\(\frac {7+30 x}{121 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {6 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\)	\(41\)
derivativedivides	\(\frac {2 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}-\frac {6 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {4}{121 \sqrt {1-2 x}}\)	\(45\)
default	\(\frac {2 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}-\frac {6 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {4}{121 \sqrt {1-2 x}}\)	\(45\)
trager	\(-\frac {\left (7+30 x \right ) \sqrt {1-2 x}}{121 \left (10 x^{2}+x -3\right )}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/121*(1-2*x)^(1/2)/(-6/5-2*x)-6/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+4/121/(1-2*x)^(1/2)

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Maxima [A]
time = 0.60, size = 65, normalized size = 1.03 \begin {gather*} \frac {3}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/121*(30*x + 7)/(5*(-2*x
+ 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]
time = 0.77, size = 71, normalized size = 1.13 \begin {gather*} \frac {3 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (30 \, x + 7\right )} \sqrt {-2 \, x + 1}}{1331 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1331*(3*sqrt(11)*sqrt(5)*(10*x^2 + x - 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 11*(3
0*x + 7)*sqrt(-2*x + 1))/(10*x^2 + x - 3)

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Sympy [C] Result contains complex when optimal does not.
time = 1.43, size = 175, normalized size = 2.78 \begin {gather*} \begin {cases} - \frac {6 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} + \frac {3 \sqrt {2}}{121 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{110 \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 )}}{1331} - \frac {3 \sqrt {2} i}{121 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{110 \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/(1-2*x)**(3/2)/(3+5*x)**2,x)

[Out]

Piecewise((-6*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/1331 + 3*sqrt(2)/(121*sqrt(-1 + 11/(10*(x + 3/5)))*
sqrt(x + 3/5)) - sqrt(2)/(110*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)))/1331 - 3*sqrt(2)*I/(121*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) +
sqrt(2)*I/(110*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), True))

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Giac [A]
time = 2.00, size = 68, normalized size = 1.08 \begin {gather*} \frac {3}{1331} \, \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 {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/121*(30*x + 7)
/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Mupad [B]
time = 1.22, size = 46, normalized size = 0.73 \begin {gather*} \frac {\frac {12\,x}{121}+\frac {14}{605}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {6\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12*x)/121 + 14/605)/((11*(1 - 2*x)^(1/2))/5 - (1 - 2*x)^(3/2)) - (6*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2)
)/11))/1331

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