3.19.8 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{2+3 x} \, dx\) [1808]

Optimal. Leaf size=69 \[ \frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}-\frac {2}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]

[Out]

-155/54*(1-2*x)^(3/2)+5/6*(1-2*x)^(5/2)-2/81*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2/27*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {5}{6} (1-2 x)^{5/2}-\frac {155}{54} (1-2 x)^{3/2}+\frac {2}{27} \sqrt {1-2 x}-\frac {2}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - 2*x])/27 - (155*(1 - 2*x)^(3/2))/54 + (5*(1 - 2*x)^(5/2))/6 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/27

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 {\sqrt {1-2 x} (3+5 x)^2}{2+3 x} \, dx &=\int \left (\frac {155}{18} \sqrt {1-2 x}-\frac {25}{6} (1-2 x)^{3/2}+\frac {\sqrt {1-2 x}}{9 (2+3 x)}\right ) \, dx\\ &=-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}+\frac {1}{9} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}+\frac {7}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}-\frac {7}{27} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}-\frac {2}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.74 \begin {gather*} \frac {1}{81} \left (3 \sqrt {1-2 x} \left (-53+65 x+90 x^2\right )-2 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 - 2*x]*(-53 + 65*x + 90*x^2) - 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Maple [A]
time = 0.10, size = 47, normalized size = 0.68

method	result	size
risch	\(-\frac {\left (90 x^{2}+65 x -53\right ) \left (-1+2 x \right )}{27 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}\)	\(44\)
derivativedivides	\(-\frac {155 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}+\frac {2 \sqrt {1-2 x}}{27}\)	\(47\)
default	\(-\frac {155 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}+\frac {2 \sqrt {1-2 x}}{27}\)	\(47\)
trager	\(\left (\frac {10}{3} x^{2}+\frac {65}{27} x -\frac {53}{27}\right ) \sqrt {1-2 x}-\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{81}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-155/54*(1-2*x)^(3/2)+5/6*(1-2*x)^(5/2)-2/81*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2/27*(1-2*x)^(1/2)

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Maxima [A]
time = 0.48, size = 64, normalized size = 0.93 \begin {gather*} \frac {5}{6} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {155}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{81} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{27} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/6*(-2*x + 1)^(5/2) - 155/54*(-2*x + 1)^(3/2) + 1/81*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 2/27*sqrt(-2*x + 1)

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Fricas [A]
time = 1.37, size = 56, normalized size = 0.81 \begin {gather*} \frac {1}{81} \, \sqrt {7} \sqrt {3} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + \frac {1}{27} \, {\left (90 \, x^{2} + 65 \, x - 53\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/81*sqrt(7)*sqrt(3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) + 1/27*(90*x^2 + 65*x - 53)*sqr
t(-2*x + 1)

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Sympy [A]
time = 3.12, size = 95, normalized size = 1.38 \begin {gather*} \frac {5 \left (1 - 2 x\right )^{\frac {5}{2}}}{6} - \frac {155 \left (1 - 2 x\right )^{\frac {3}{2}}}{54} + \frac {2 \sqrt {1 - 2 x}}{27} + \frac {14 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*(1 - 2*x)**(5/2)/6 - 155*(1 - 2*x)**(3/2)/54 + 2*sqrt(1 - 2*x)/27 + 14*Piecewise((-sqrt(21)*acoth(sqrt(21)*s
qrt(1 - 2*x)/7)/21, x < -2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, x > -2/3))/27

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Giac [A]
time = 0.58, size = 74, normalized size = 1.07 \begin {gather*} \frac {5}{6} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {155}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{81} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2}{27} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/6*(2*x - 1)^2*sqrt(-2*x + 1) - 155/54*(-2*x + 1)^(3/2) + 1/81*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/27*sqrt(-2*x + 1)

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Mupad [B]
time = 0.07, size = 48, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{27}-\frac {155\,{\left (1-2\,x\right )}^{3/2}}{54}+\frac {5\,{\left (1-2\,x\right )}^{5/2}}{6}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2{}\mathrm {i}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*2i)/81 + (2*(1 - 2*x)^(1/2))/27 - (155*(1 - 2*x)^(3/2))/54 + (
5*(1 - 2*x)^(5/2))/6

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