3.19.99 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx\) [1899]

Optimal. Leaf size=82 \[ \frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}-\frac {22}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

2/375*(1-2*x)^(3/2)-111/250*(1-2*x)^(5/2)+9/70*(1-2*x)^(7/2)-22/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(
1/2)+22/625*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212} \begin {gather*} \frac {9}{70} (1-2 x)^{7/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {2}{375} (1-2 x)^{3/2}+\frac {22}{625} \sqrt {1-2 x}-\frac {22}{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),x]

[Out]

(22*Sqrt[1 - 2*x])/625 + (2*(1 - 2*x)^(3/2))/375 - (111*(1 - 2*x)^(5/2))/250 + (9*(1 - 2*x)^(7/2))/70 - (22*Sq
rt[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 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -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} \, dx &=\int \left (\frac {111}{50} (1-2 x)^{3/2}-\frac {9}{10} (1-2 x)^{5/2}+\frac {(1-2 x)^{3/2}}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}+\frac {1}{25} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}+\frac {11}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}+\frac {121}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}-\frac {121}{625} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}-\frac {22}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.68 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (3608-13045 x+3060 x^2+13500 x^3\right )-462 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{65625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x]*(3608 - 13045*x + 3060*x^2 + 13500*x^3) - 462*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/65
625

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Maple [A]
time = 0.10, size = 56, normalized size = 0.68

method	result	size
risch	\(\frac {\left (13500 x^{3}+3060 x^{2}-13045 x +3608\right ) \left (-1+2 x \right )}{13125 \sqrt {1-2 x}}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\)	\(49\)
derivativedivides	\(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{375}-\frac {111 \left (1-2 x \right )^{\frac {5}{2}}}{250}+\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{70}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}+\frac {22 \sqrt {1-2 x}}{625}\)	\(56\)
default	\(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{375}-\frac {111 \left (1-2 x \right )^{\frac {5}{2}}}{250}+\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{70}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}+\frac {22 \sqrt {1-2 x}}{625}\)	\(56\)
trager	\(\left (-\frac {36}{35} x^{3}-\frac {204}{875} x^{2}+\frac {2609}{2625} x -\frac {3608}{13125}\right ) \sqrt {1-2 x}+\frac {11 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{3125}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/375*(1-2*x)^(3/2)-111/250*(1-2*x)^(5/2)+9/70*(1-2*x)^(7/2)-22/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(
1/2)+22/625*(1-2*x)^(1/2)

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Maxima [A]
time = 0.50, size = 73, normalized size = 0.89 \begin {gather*} \frac {9}{70} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {111}{250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{625} \, \sqrt {-2 \, x + 1} \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),x, algorithm="maxima")

[Out]

9/70*(-2*x + 1)^(7/2) - 111/250*(-2*x + 1)^(5/2) + 2/375*(-2*x + 1)^(3/2) + 11/3125*sqrt(55)*log(-(sqrt(55) -
5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/625*sqrt(-2*x + 1)

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Fricas [A]
time = 1.41, size = 61, normalized size = 0.74 \begin {gather*} \frac {11}{3125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{13125} \, {\left (13500 \, x^{3} + 3060 \, x^{2} - 13045 \, x + 3608\right )} \sqrt {-2 \, x + 1} \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),x, algorithm="fricas")

[Out]

11/3125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 1/13125*(13500*x^3 + 306
0*x^2 - 13045*x + 3608)*sqrt(-2*x + 1)

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Sympy [A]
time = 22.47, size = 107, normalized size = 1.30 \begin {gather*} \frac {9 \left (1 - 2 x\right )^{\frac {7}{2}}}{70} - \frac {111 \left (1 - 2 x\right )^{\frac {5}{2}}}{250} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{375} + \frac {22 \sqrt {1 - 2 x}}{625} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{625} \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),x)

[Out]

9*(1 - 2*x)**(7/2)/70 - 111*(1 - 2*x)**(5/2)/250 + 2*(1 - 2*x)**(3/2)/375 + 22*sqrt(1 - 2*x)/625 + 242*Piecewi
se((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, x < -3/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55,
x > -3/5))/625

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Giac [A]
time = 0.72, size = 90, normalized size = 1.10 \begin {gather*} -\frac {9}{70} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {111}{250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{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 {22}{625} \, \sqrt {-2 \, x + 1} \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),x, algorithm="giac")

[Out]

-9/70*(2*x - 1)^3*sqrt(-2*x + 1) - 111/250*(2*x - 1)^2*sqrt(-2*x + 1) + 2/375*(-2*x + 1)^(3/2) + 11/3125*sqrt(
55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/625*sqrt(-2*x + 1)

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Mupad [B]
time = 1.18, size = 57, normalized size = 0.70 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{375}-\frac {111\,{\left (1-2\,x\right )}^{5/2}}{250}+\frac {9\,{\left (1-2\,x\right )}^{7/2}}{70}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\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),x)

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/3125 + (22*(1 - 2*x)^(1/2))/625 + (2*(1 - 2*x)^(3/2))/37
5 - (111*(1 - 2*x)^(5/2))/250 + (9*(1 - 2*x)^(7/2))/70

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