3.22.31 \(\int \frac {1}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\) [2131]

Optimal. Leaf size=83 \[ \frac {15}{1331 \sqrt {1-2 x}}-\frac {1}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {5}{242 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]

[Out]

-15/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+15/1331/(1-2*x)^(1/2)-1/22/(3+5*x)^2/(1-2*x)^(1/2)-5/2
42/(3+5*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {15}{1331 \sqrt {1-2 x}}-\frac {5}{242 \sqrt {1-2 x} (5 x+3)}-\frac {1}{22 \sqrt {1-2 x} (5 x+3)^2}-\frac {15 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

15/(1331*Sqrt[1 - 2*x]) - 1/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) - 5/(242*Sqrt[1 - 2*x]*(3 + 5*x)) - (15*Sqrt[5/11]*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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)^3} \, dx &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^2}+\frac {25}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {25 \sqrt {1-2 x}}{242 (3+5 x)^2}+\frac {75}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {25 \sqrt {1-2 x}}{242 (3+5 x)^2}-\frac {75 \sqrt {1-2 x}}{2662 (3+5 x)}+\frac {75 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{2662}\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {25 \sqrt {1-2 x}}{242 (3+5 x)^2}-\frac {75 \sqrt {1-2 x}}{2662 (3+5 x)}-\frac {75 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2662}\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {25 \sqrt {1-2 x}}{242 (3+5 x)^2}-\frac {75 \sqrt {1-2 x}}{2662 (3+5 x)}-\frac {15 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 58, normalized size = 0.70 \begin {gather*} \frac {\frac {11 \left (-16+625 x+750 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-30 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{29282} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(-16 + 625*x + 750*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 30*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/292
82

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Maple [A]
time = 0.11, size = 57, normalized size = 0.69

method	result	size
risch	\(\frac {750 x^{2}+625 x -16}{2662 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {15 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}\)	\(46\)
derivativedivides	\(\frac {\frac {175 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {45 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {15 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {8}{1331 \sqrt {1-2 x}}\)	\(57\)
default	\(\frac {\frac {175 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {45 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {15 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {8}{1331 \sqrt {1-2 x}}\)	\(57\)
trager	\(-\frac {\left (750 x^{2}+625 x -16\right ) \sqrt {1-2 x}}{2662 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {15 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{29282}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000/1331*(7/40*(1-2*x)^(3/2)-99/200*(1-2*x)^(1/2))/(-6-10*x)^2-15/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*
55^(1/2)+8/1331/(1-2*x)^(1/2)

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Maxima [A]
time = 0.53, size = 83, normalized size = 1.00 \begin {gather*} \frac {15}{29282} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {375 \, {\left (2 \, x - 1\right )}^{2} + 2750 \, x - 407}{1331 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \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)^3,x, algorithm="maxima")

[Out]

15/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1/1331*(375*(2*x - 1)^2
+ 2750*x - 407)/(25*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]
time = 0.56, size = 90, normalized size = 1.08 \begin {gather*} \frac {15 \, \sqrt {11} \sqrt {5} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (750 \, x^{2} + 625 \, x - 16\right )} \sqrt {-2 \, x + 1}}{29282 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/29282*(15*sqrt(11)*sqrt(5)*(50*x^3 + 35*x^2 - 12*x - 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x
+ 3)) - 11*(750*x^2 + 625*x - 16)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [C] Result contains complex when optimal does not.
time = 4.75, size = 231, normalized size = 2.78 \begin {gather*} \begin {cases} - \frac {15 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{14641} + \frac {15 \sqrt {2}}{2662 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{484 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} - \frac {\sqrt {2}}{1100 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\frac {15 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{14641} - \frac {15 \sqrt {2} i}{2662 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{484 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} + \frac {\sqrt {2} i}{1100 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{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)**3,x)

[Out]

Piecewise((-15*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/14641 + 15*sqrt(2)/(2662*sqrt(-1 + 11/(10*(x + 3/5
)))*sqrt(x + 3/5)) - sqrt(2)/(484*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) - sqrt(2)/(1100*sqrt(-1 + 11/
(10*(x + 3/5)))*(x + 3/5)**(5/2)), 1/Abs(x + 3/5) > 10/11), (15*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/
14641 - 15*sqrt(2)*I/(2662*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) + sqrt(2)*I/(484*sqrt(1 - 11/(10*(x + 3/
5)))*(x + 3/5)**(3/2)) + sqrt(2)*I/(1100*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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Giac [A]
time = 1.27, size = 77, normalized size = 0.93 \begin {gather*} \frac {15}{29282} \, \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 {8}{1331 \, \sqrt {-2 \, x + 1}} + \frac {5 \, {\left (35 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 99 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 8/1331/sqrt(-2
*x + 1) + 5/5324*(35*(-2*x + 1)^(3/2) - 99*sqrt(-2*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 0.08, size = 62, normalized size = 0.75 \begin {gather*} \frac {\frac {10\,x}{121}+\frac {15\,{\left (2\,x-1\right )}^2}{1331}-\frac {37}{3025}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}-\frac {15\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((10*x)/121 + (15*(2*x - 1)^2)/1331 - 37/3025)/((121*(1 - 2*x)^(1/2))/25 - (22*(1 - 2*x)^(3/2))/5 + (1 - 2*x)^
(5/2)) - (15*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641

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