3.22.95 \(\int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\) [2195]

Optimal. Leaf size=96 \[ \frac {73}{3993 (1-2 x)^{3/2}}+\frac {365}{14641 \sqrt {1-2 x}}-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac {73}{1210 (1-2 x)^{3/2} (3+5 x)}-\frac {365 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

[Out]

73/3993/(1-2*x)^(3/2)-1/110/(1-2*x)^(3/2)/(3+5*x)^2-73/1210/(1-2*x)^(3/2)/(3+5*x)-365/161051*arctanh(1/11*55^(
1/2)*(1-2*x)^(1/2))*55^(1/2)+365/14641/(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 = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65, 212} \begin {gather*} \frac {365}{14641 \sqrt {1-2 x}}-\frac {73}{1210 (1-2 x)^{3/2} (5 x+3)}+\frac {73}{3993 (1-2 x)^{3/2}}-\frac {1}{110 (1-2 x)^{3/2} (5 x+3)^2}-\frac {365 \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[(2 + 3*x)/((1 - 2*x)^(5/2)*(3 + 5*x)^3),x]

[Out]

73/(3993*(1 - 2*x)^(3/2)) + 365/(14641*Sqrt[1 - 2*x]) - 1/(110*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 73/(1210*(1 - 2*
x)^(3/2)*(3 + 5*x)) - (365*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 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 {2+3 x}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}+\frac {73}{110} \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\\ &=-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac {73}{1210 (1-2 x)^{3/2} (3+5 x)}+\frac {73}{242} \int \frac {1}{(1-2 x)^{5/2} (3+5 x)} \, dx\\ &=\frac {73}{3993 (1-2 x)^{3/2}}-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac {73}{1210 (1-2 x)^{3/2} (3+5 x)}+\frac {365 \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx}{2662}\\ &=\frac {73}{3993 (1-2 x)^{3/2}}+\frac {365}{14641 \sqrt {1-2 x}}-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac {73}{1210 (1-2 x)^{3/2} (3+5 x)}+\frac {1825 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=\frac {73}{3993 (1-2 x)^{3/2}}+\frac {365}{14641 \sqrt {1-2 x}}-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac {73}{1210 (1-2 x)^{3/2} (3+5 x)}-\frac {1825 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=\frac {73}{3993 (1-2 x)^{3/2}}+\frac {365}{14641 \sqrt {1-2 x}}-\frac {1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac {73}{1210 (1-2 x)^{3/2} (3+5 x)}-\frac {365 \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 (-17466-47961 x+36500 x^2+109500 x^3\right )}{2 (1-2 x)^{3/2} (3+5 x)^2}-1095 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{483153} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-17466 - 47961*x + 36500*x^2 + 109500*x^3))/(2*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 1095*Sqrt[55]*ArcTanh[Sqr
t[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	\(\frac {109500 x^{3}+36500 x^{2}-47961 x -17466}{87846 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {365 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}\)	\(58\)
derivativedivides	\(\frac {\frac {175 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {395 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {365 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {28}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {288}{14641 \sqrt {1-2 x}}\)	\(66\)
default	\(\frac {\frac {175 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {395 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {365 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {28}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {288}{14641 \sqrt {1-2 x}}\)	\(66\)
trager	\(-\frac {\left (109500 x^{3}+36500 x^{2}-47961 x -17466\right ) \sqrt {1-2 x}}{87846 \left (10 x^{2}+x -3\right )^{2}}+\frac {365 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{322102}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/14641*(77/20*(1-2*x)^(3/2)-869/100*(1-2*x)^(1/2))/(-6-10*x)^2-365/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/
2))*55^(1/2)+28/3993/(1-2*x)^(3/2)+288/14641/(1-2*x)^(1/2)

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Maxima [A]
time = 0.51, size = 92, normalized size = 0.96 \begin {gather*} \frac {365}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27375 \, {\left (2 \, x - 1\right )}^{3} + 100375 \, {\left (2 \, x - 1\right )}^{2} + 141328 \, x - 107932}{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((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="maxima")

[Out]

365/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/43923*(27375*(2*x -
1)^3 + 100375*(2*x - 1)^2 + 141328*x - 107932)/(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 = 1.03, size = 105, normalized size = 1.09 \begin {gather*} \frac {1095 \, \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 (109500 \, x^{3} + 36500 \, x^{2} - 47961 \, x - 17466\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((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/966306*(1095*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*(109500*x^3 + 36500*x^2 - 47961*x - 17466)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2
- 6*x + 9)

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

[Out]

Timed out

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Giac [A]
time = 0.81, size = 89, normalized size = 0.93 \begin {gather*} \frac {365}{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 {4 \, {\left (432 \, x - 293\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {5 \, {\left (35 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 79 \, \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((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="giac")

[Out]

365/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4/43923*(432
*x - 293)/((2*x - 1)*sqrt(-2*x + 1)) + 5/5324*(35*(-2*x + 1)^(3/2) - 79*sqrt(-2*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 0.08, size = 72, normalized size = 0.75 \begin {gather*} -\frac {365\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}-\frac {\frac {1168\,x}{9075}+\frac {365\,{\left (2\,x-1\right )}^2}{3993}+\frac {365\,{\left (2\,x-1\right )}^3}{14641}-\frac {892}{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((3*x + 2)/((1 - 2*x)^(5/2)*(5*x + 3)^3),x)

[Out]

- (365*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/161051 - ((1168*x)/9075 + (365*(2*x - 1)^2)/3993 + (365*
(2*x - 1)^3)/14641 - 892/9075)/((121*(1 - 2*x)^(3/2))/25 - (22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2))

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