3.22.84 \(\int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\) [2184]

Optimal. Leaf size=81 \[ \frac {142}{6655 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac {1231}{3630 \sqrt {1-2 x} (3+5 x)}-\frac {142 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \]

[Out]

49/66/(1-2*x)^(3/2)/(3+5*x)-142/73205*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+142/6655/(1-2*x)^(1/2)-123
1/3630/(3+5*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 53, 65, 212} \begin {gather*} \frac {142}{6655 \sqrt {1-2 x}}-\frac {1231}{3630 \sqrt {1-2 x} (5 x+3)}+\frac {49}{66 (1-2 x)^{3/2} (5 x+3)}-\frac {142 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

142/(6655*Sqrt[1 - 2*x]) + 49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)) - 1231/(3630*Sqrt[1 - 2*x]*(3 + 5*x)) - (142*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqrt[55])

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 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{66} \int \frac {-68+297 x}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac {1231}{3630 \sqrt {1-2 x} (3+5 x)}+\frac {71}{605} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {142}{6655 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac {1231}{3630 \sqrt {1-2 x} (3+5 x)}+\frac {71 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac {142}{6655 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac {1231}{3630 \sqrt {1-2 x} (3+5 x)}-\frac {71 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1331}\\ &=\frac {142}{6655 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac {1231}{3630 \sqrt {1-2 x} (3+5 x)}-\frac {142 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 58, normalized size = 0.72 \begin {gather*} \frac {-\frac {55 \left (-1866-2623 x+852 x^2\right )}{(1-2 x)^{3/2} (3+5 x)}-426 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{219615} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*(-1866 - 2623*x + 852*x^2))/((1 - 2*x)^(3/2)*(3 + 5*x)) - 426*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/219615

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Maple [A]
time = 0.11, size = 54, normalized size = 0.67

method	result	size
risch	\(\frac {852 x^{2}-2623 x -1866}{3993 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {142 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}\)	\(53\)
derivativedivides	\(\frac {2 \sqrt {1-2 x}}{6655 \left (-\frac {6}{5}-2 x \right )}-\frac {142 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {49}{363 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {28}{1331 \sqrt {1-2 x}}\)	\(54\)
default	\(\frac {2 \sqrt {1-2 x}}{6655 \left (-\frac {6}{5}-2 x \right )}-\frac {142 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {49}{363 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {28}{1331 \sqrt {1-2 x}}\)	\(54\)
trager	\(-\frac {\left (852 x^{2}-2623 x -1866\right ) \sqrt {1-2 x}}{3993 \left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {71 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{73205}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6655*(1-2*x)^(1/2)/(-6/5-2*x)-142/73205*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+49/363/(1-2*x)^(3/2)+2
8/1331/(1-2*x)^(1/2)

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Maxima [A]
time = 0.51, size = 74, normalized size = 0.91 \begin {gather*} \frac {71}{73205} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (213 \, {\left (2 \, x - 1\right )}^{2} - 1771 \, x - 2079\right )}}{3993 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\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)^2/(1-2*x)^(5/2)/(3+5*x)^2,x, algorithm="maxima")

[Out]

71/73205*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/3993*(213*(2*x - 1)^2
- 1771*x - 2079)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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Fricas [A]
time = 1.20, size = 84, normalized size = 1.04 \begin {gather*} \frac {213 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (852 \, x^{2} - 2623 \, x - 1866\right )} \sqrt {-2 \, x + 1}}{219615 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/219615*(213*sqrt(55)*(20*x^3 - 8*x^2 - 7*x + 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(852
*x^2 - 2623*x - 1866)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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

[Out]

Timed out

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Giac [A]
time = 1.92, size = 77, normalized size = 0.95 \begin {gather*} \frac {71}{73205} \, \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 {7 \, {\left (24 \, x - 89\right )}}{3993 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

71/73205*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 7/3993*(24*x -
89)/((2*x - 1)*sqrt(-2*x + 1)) - 1/1331*sqrt(-2*x + 1)/(5*x + 3)

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Mupad [B]
time = 0.06, size = 55, normalized size = 0.68 \begin {gather*} \frac {\frac {322\,x}{1815}-\frac {142\,{\left (2\,x-1\right )}^2}{6655}+\frac {126}{605}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {142\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{73205} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((322*x)/1815 - (142*(2*x - 1)^2)/6655 + 126/605)/((11*(1 - 2*x)^(3/2))/5 - (1 - 2*x)^(5/2)) - (142*55^(1/2)*a
tanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/73205

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