3.20.11 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^2} \, dx\) [1911]

Optimal. Leaf size=89 \[ \frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}-\frac {126}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

42/1375*(1-2*x)^(3/2)-9/125*(1-2*x)^(5/2)-1/275*(1-2*x)^(5/2)/(3+5*x)-126/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(
1/2))*55^(1/2)+126/625*(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 81, 52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{5/2}}{275 (5 x+3)}-\frac {9}{125} (1-2 x)^{5/2}+\frac {42 (1-2 x)^{3/2}}{1375}+\frac {126}{625} \sqrt {1-2 x}-\frac {126}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(126*Sqrt[1 - 2*x])/625 + (42*(1 - 2*x)^(3/2))/1375 - (9*(1 - 2*x)^(5/2))/125 - (1 - 2*x)^(5/2)/(275*(3 + 5*x)
) - (126*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 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 {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {1}{275} \int \frac {(1-2 x)^{3/2} (360+495 x)}{3+5 x} \, dx\\ &=-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {63}{275} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {63}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {693}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}-\frac {693}{625} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}-\frac {126}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 63, normalized size = 0.71 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (298+935 x+160 x^2-900 x^3\right )}{3+5 x}-126 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*Sqrt[1 - 2*x]*(298 + 935*x + 160*x^2 - 900*x^3))/(3 + 5*x) - 126*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
])/3125

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 63, normalized size = 0.71

method	result	size
risch	\(\frac {1800 x^{4}-1220 x^{3}-1710 x^{2}+339 x +298}{625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {126 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(56\)
derivativedivides	\(-\frac {9 \left (1-2 x \right )^{\frac {5}{2}}}{125}+\frac {4 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {128 \sqrt {1-2 x}}{625}+\frac {22 \sqrt {1-2 x}}{3125 \left (-\frac {6}{5}-2 x \right )}-\frac {126 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(63\)
default	\(-\frac {9 \left (1-2 x \right )^{\frac {5}{2}}}{125}+\frac {4 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {128 \sqrt {1-2 x}}{625}+\frac {22 \sqrt {1-2 x}}{3125 \left (-\frac {6}{5}-2 x \right )}-\frac {126 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(63\)
trager	\(-\frac {\left (900 x^{3}-160 x^{2}-935 x -298\right ) \sqrt {1-2 x}}{625 \left (3+5 x \right )}-\frac {63 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{3125}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/125*(1-2*x)^(5/2)+4/125*(1-2*x)^(3/2)+128/625*(1-2*x)^(1/2)+22/3125*(1-2*x)^(1/2)/(-6/5-2*x)-126/3125*arcta
nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 80, normalized size = 0.90 \begin {gather*} -\frac {9}{125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {4}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {63}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {128}{625} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/125*(-2*x + 1)^(5/2) + 4/125*(-2*x + 1)^(3/2) + 63/3125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(5
5) + 5*sqrt(-2*x + 1))) + 128/625*sqrt(-2*x + 1) - 11/625*sqrt(-2*x + 1)/(5*x + 3)

________________________________________________________________________________________

Fricas [A]
time = 0.93, size = 75, normalized size = 0.84 \begin {gather*} \frac {63 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (900 \, x^{3} - 160 \, x^{2} - 935 \, x - 298\right )} \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3125*(63*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 5*(900*x^3
- 160*x^2 - 935*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((1-2*x)**(3/2)*(2+3*x)**2/(3+5*x)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.66, size = 90, normalized size = 1.01 \begin {gather*} -\frac {9}{125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {4}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {63}{3125} \, \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 {128}{625} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/125*(2*x - 1)^2*sqrt(-2*x + 1) + 4/125*(-2*x + 1)^(3/2) + 63/3125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqr
t(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 128/625*sqrt(-2*x + 1) - 11/625*sqrt(-2*x + 1)/(5*x + 3)

________________________________________________________________________________________

Mupad [B]
time = 1.17, size = 64, normalized size = 0.72 \begin {gather*} \frac {128\,\sqrt {1-2\,x}}{625}-\frac {22\,\sqrt {1-2\,x}}{3125\,\left (2\,x+\frac {6}{5}\right )}+\frac {4\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {9\,{\left (1-2\,x\right )}^{5/2}}{125}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,126{}\mathrm {i}}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*126i)/3125 - (22*(1 - 2*x)^(1/2))/(3125*(2*x + 6/5)) + (128*(
1 - 2*x)^(1/2))/625 + (4*(1 - 2*x)^(3/2))/125 - (9*(1 - 2*x)^(5/2))/125

________________________________________________________________________________________