3.22.96 \(\int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\) [2196]

Optimal. Leaf size=96 \[ \frac {35}{3993 (1-2 x)^{3/2}}+\frac {175}{14641 \sqrt {1-2 x}}-\frac {1}{22 (1-2 x)^{3/2} (3+5 x)^2}-\frac {7}{242 (1-2 x)^{3/2} (3+5 x)}-\frac {175 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

[Out]

35/3993/(1-2*x)^(3/2)-1/22/(1-2*x)^(3/2)/(3+5*x)^2-7/242/(1-2*x)^(3/2)/(3+5*x)-175/161051*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)+175/14641/(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {175}{14641 \sqrt {1-2 x}}-\frac {7}{242 (1-2 x)^{3/2} (5 x+3)}+\frac {35}{3993 (1-2 x)^{3/2}}-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}-\frac {175 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

35/(3993*(1 - 2*x)^(3/2)) + 175/(14641*Sqrt[1 - 2*x]) - 1/(22*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 7/(242*(1 - 2*x)^
(3/2)*(3 + 5*x)) - (175*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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)^{5/2} (3+5 x)^3} \, dx &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {35}{33} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}+\frac {875}{363} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}+\frac {875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{2662}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{29282 (3+5 x)}+\frac {875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{29282 (3+5 x)}-\frac {875 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{29282 (3+5 x)}-\frac {175 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 65, normalized size = 0.68 \begin {gather*} \frac {-\frac {11 \left (-4764-22995 x+17500 x^2+52500 x^3\right )}{2 (1-2 x)^{3/2} (3+5 x)^2}-525 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{483153} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-4764 - 22995*x + 17500*x^2 + 52500*x^3))/(2*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 525*Sqrt[55]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/483153

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Maple [A]
time = 0.11, size = 66, normalized size = 0.69

method	result	size
risch	\(\frac {52500 x^{3}+17500 x^{2}-22995 x -4764}{87846 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {175 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}\)	\(58\)
derivativedivides	\(\frac {\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {325 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {175 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {8}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {120}{14641 \sqrt {1-2 x}}\)	\(66\)
default	\(\frac {\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {325 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {175 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {8}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {120}{14641 \sqrt {1-2 x}}\)	\(66\)
trager	\(-\frac {\left (52500 x^{3}+17500 x^{2}-22995 x -4764\right ) \sqrt {1-2 x}}{87846 \left (10 x^{2}+x -3\right )^{2}}+\frac {175 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{322102}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5000/14641*(11/40*(1-2*x)^(3/2)-143/200*(1-2*x)^(1/2))/(-6-10*x)^2-175/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1
/2))*55^(1/2)+8/3993/(1-2*x)^(3/2)+120/14641/(1-2*x)^(1/2)

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Maxima [A]
time = 0.50, size = 92, normalized size = 0.96 \begin {gather*} \frac {175}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {13125 \, {\left (2 \, x - 1\right )}^{3} + 48125 \, {\left (2 \, x - 1\right )}^{2} + 67760 \, x - 44528}{43923 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

175/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/43923*(13125*(2*x -
1)^3 + 48125*(2*x - 1)^2 + 67760*x - 44528)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2)
)

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Fricas [A]
time = 0.83, size = 105, normalized size = 1.09 \begin {gather*} \frac {525 \, \sqrt {11} \sqrt {5} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (52500 \, x^{3} + 17500 \, x^{2} - 22995 \, x - 4764\right )} \sqrt {-2 \, x + 1}}{966306 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/966306*(525*sqrt(11)*sqrt(5)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*
x - 8)/(5*x + 3)) - 11*(52500*x^3 + 17500*x^2 - 22995*x - 4764)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6
*x + 9)

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Sympy [C] Result contains complex when optimal does not.
time = 6.57, size = 983, normalized size = 10.24 \begin {gather*} \begin {cases} - \frac {105000 \sqrt {55} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} + \frac {52500 \sqrt {55} i \pi \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} + \frac {115500 \sqrt {55} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} - \frac {57750 \sqrt {55} i \pi \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} + \frac {577500 \sqrt {2} \left (x + \frac {3}{5}\right )^{77}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} - \frac {847000 \sqrt {2} \left (x + \frac {3}{5}\right )^{76}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} + \frac {139755 \sqrt {2} \left (x + \frac {3}{5}\right )^{75}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} + \frac {43923 \sqrt {2} \left (x + \frac {3}{5}\right )^{74}}{96630600 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\frac {105000 \sqrt {55} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{96630600 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} - \frac {115500 \sqrt {55} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}} \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{96630600 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} - \frac {577500 \sqrt {2} i \left (x + \frac {3}{5}\right )^{77}}{96630600 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} + \frac {847000 \sqrt {2} i \left (x + \frac {3}{5}\right )^{76}}{96630600 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} - \frac {139755 \sqrt {2} i \left (x + \frac {3}{5}\right )^{75}}{96630600 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} - \frac {43923 \sqrt {2} i \left (x + \frac {3}{5}\right )^{74}}{96630600 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {155}{2}} - 106293660 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {153}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-105000*sqrt(55)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5))
)/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)
**(153/2)) + 52500*sqrt(55)*I*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)/(96630600*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 115500*sqrt(55)*sq
rt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*sqrt(-1 + 11/(10*(
x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 57750*sqrt(55)*I*
pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) -
106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 577500*sqrt(2)*(x + 3/5)**77/(96630600*sqrt(-1 +
11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 847000*s
qrt(2)*(x + 3/5)**76/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(153/2)) + 139755*sqrt(2)*(x + 3/5)**75/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)*
*(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 43923*sqrt(2)*(x + 3/5)**74/(96630600*
sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)),
1/Abs(x + 3/5) > 10/11), (105000*sqrt(55)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)*asin(sqrt(110)/(10*
sqrt(x + 3/5)))/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)
))*(x + 3/5)**(153/2)) - 115500*sqrt(55)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*asin(sqrt(110)/(10*s
qrt(x + 3/5)))/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5))
)*(x + 3/5)**(153/2)) - 577500*sqrt(2)*I*(x + 3/5)**77/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2
) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 847000*sqrt(2)*I*(x + 3/5)**76/(96630600*sqrt(
1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 139755
*sqrt(2)*I*(x + 3/5)**75/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*
(x + 3/5)))*(x + 3/5)**(153/2)) - 43923*sqrt(2)*I*(x + 3/5)**74/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5
)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)), True))

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Giac [A]
time = 0.83, size = 89, normalized size = 0.93 \begin {gather*} \frac {175}{322102} \, \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 {16 \, {\left (45 \, x - 28\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {25 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13 \, \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)^(5/2)/(3+5*x)^3,x, algorithm="giac")

[Out]

175/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 16/43923*(45
*x - 28)/((2*x - 1)*sqrt(-2*x + 1)) + 25/5324*(5*(-2*x + 1)^(3/2) - 13*sqrt(-2*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 1.21, size = 72, normalized size = 0.75 \begin {gather*} -\frac {175\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}-\frac {\frac {112\,x}{1815}+\frac {175\,{\left (2\,x-1\right )}^2}{3993}+\frac {175\,{\left (2\,x-1\right )}^3}{14641}-\frac {368}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (175*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/161051 - ((112*x)/1815 + (175*(2*x - 1)^2)/3993 + (175*(
2*x - 1)^3)/14641 - 368/9075)/((121*(1 - 2*x)^(3/2))/25 - (22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2))

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