3.21.51 \(\int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^2} \, dx\) [2051]

Optimal. Leaf size=61 \[ -\frac {9}{25} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{275 (3+5 x)}-\frac {134 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \]

[Out]

-134/15125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-9/25*(1-2*x)^(1/2)-1/275*(1-2*x)^(1/2)/(3+5*x)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 81, 65, 212} \begin {gather*} -\frac {9}{25} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{275 (5 x+3)}-\frac {134 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*Sqrt[1 - 2*x])/25 - Sqrt[1 - 2*x]/(275*(3 + 5*x)) - (134*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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 81

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

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}{\sqrt {1-2 x} (3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x}}{275 (3+5 x)}+\frac {1}{275} \int \frac {364+495 x}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {9}{25} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{275 (3+5 x)}+\frac {67}{275} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {9}{25} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{275 (3+5 x)}-\frac {67}{275} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {9}{25} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{275 (3+5 x)}-\frac {134 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 53, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {1-2 x} (298+495 x)}{275 (3+5 x)}-\frac {134 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/275*(Sqrt[1 - 2*x]*(298 + 495*x))/(3 + 5*x) - (134*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 45, normalized size = 0.74

method	result	size
derivativedivides	\(-\frac {9 \sqrt {1-2 x}}{25}+\frac {2 \sqrt {1-2 x}}{1375 \left (-\frac {6}{5}-2 x \right )}-\frac {134 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\)	\(45\)
default	\(-\frac {9 \sqrt {1-2 x}}{25}+\frac {2 \sqrt {1-2 x}}{1375 \left (-\frac {6}{5}-2 x \right )}-\frac {134 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\)	\(45\)
risch	\(\frac {990 x^{2}+101 x -298}{275 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {134 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\)	\(46\)
trager	\(-\frac {\left (298+495 x \right ) \sqrt {1-2 x}}{275 \left (3+5 x \right )}-\frac {67 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{15125}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/25*(1-2*x)^(1/2)+2/1375*(1-2*x)^(1/2)/(-6/5-2*x)-134/15125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 62, normalized size = 1.02 \begin {gather*} \frac {67}{15125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9}{25} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{275 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

67/15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/25*sqrt(-2*x + 1) - 1/
275*sqrt(-2*x + 1)/(5*x + 3)

________________________________________________________________________________________

Fricas [A]
time = 0.65, size = 59, normalized size = 0.97 \begin {gather*} \frac {67 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (495 \, x + 298\right )} \sqrt {-2 \, x + 1}}{15125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15125*(67*sqrt(55)*(5*x + 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(495*x + 298)*sqrt(-2*x
+ 1))/(5*x + 3)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.68, size = 65, normalized size = 1.07 \begin {gather*} \frac {67}{15125} \, \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 {9}{25} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{275 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

67/15125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/25*sqrt(-2*x
+ 1) - 1/275*sqrt(-2*x + 1)/(5*x + 3)

________________________________________________________________________________________

Mupad [B]
time = 1.23, size = 44, normalized size = 0.72 \begin {gather*} -\frac {134\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{15125}-\frac {2\,\sqrt {1-2\,x}}{1375\,\left (2\,x+\frac {6}{5}\right )}-\frac {9\,\sqrt {1-2\,x}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (134*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/15125 - (2*(1 - 2*x)^(1/2))/(1375*(2*x + 6/5)) - (9*(1 -
2*x)^(1/2))/25

________________________________________________________________________________________