3.18.25 \(\int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=69 \[ -\frac {1}{3} (1-2 x)^{5/2}-\frac {2}{27} (1-2 x)^{3/2}-\frac {14}{27} \sqrt {1-2 x}+\frac {14}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {80, 50, 63, 206} \begin {gather*} -\frac {1}{3} (1-2 x)^{5/2}-\frac {2}{27} (1-2 x)^{3/2}-\frac {14}{27} \sqrt {1-2 x}+\frac {14}{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[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x),x]

[Out]

(-14*Sqrt[1 - 2*x])/27 - (2*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(5/2)/3 + (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/27

Rule 50

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 63

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 80

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 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} (3+5 x)}{2+3 x} \, dx &=-\frac {1}{3} (1-2 x)^{5/2}-\frac {1}{3} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {2}{27} (1-2 x)^{3/2}-\frac {1}{3} (1-2 x)^{5/2}-\frac {7}{9} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {14}{27} \sqrt {1-2 x}-\frac {2}{27} (1-2 x)^{3/2}-\frac {1}{3} (1-2 x)^{5/2}-\frac {49}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {14}{27} \sqrt {1-2 x}-\frac {2}{27} (1-2 x)^{3/2}-\frac {1}{3} (1-2 x)^{5/2}+\frac {49}{27} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {14}{27} \sqrt {1-2 x}-\frac {2}{27} (1-2 x)^{3/2}-\frac {1}{3} (1-2 x)^{5/2}+\frac {14}{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 = 0.03, size = 51, normalized size = 0.74 \begin {gather*} \frac {1}{81} \left (14 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-3 \sqrt {1-2 x} \left (36 x^2-40 x+25\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[1 - 2*x]*(25 - 40*x + 36*x^2) + 14*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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IntegrateAlgebraic [A]  time = 0.06, size = 61, normalized size = 0.88 \begin {gather*} \frac {14}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1}{27} \left (9 (1-2 x)^2+2 (1-2 x)+14\right ) \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27*((14 + 2*(1 - 2*x) + 9*(1 - 2*x)^2)*Sqrt[1 - 2*x]) + (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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fricas [A]  time = 0.81, size = 57, normalized size = 0.83 \begin {gather*} \frac {7}{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 (36 \, x^{2} - 40 \, x + 25\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/81*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 1/27*(36*x^2 - 40*x + 25)*sq
rt(-2*x + 1)

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giac [A]  time = 1.02, size = 74, normalized size = 1.07 \begin {gather*} -\frac {1}{3} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {2}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{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 {14}{27} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*x - 1)^2*sqrt(-2*x + 1) - 2/27*(-2*x + 1)^(3/2) - 7/81*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/27*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 47, normalized size = 0.68 \begin {gather*} \frac {14 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{81}-\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{27}-\frac {\left (-2 x +1\right )^{\frac {5}{2}}}{3}-\frac {14 \sqrt {-2 x +1}}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*(-2*x+1)^(3/2)-1/3*(-2*x+1)^(5/2)+14/81*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)-14/27*(-2*x+1)^(1/
2)

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maxima [A]  time = 1.23, size = 64, normalized size = 0.93 \begin {gather*} -\frac {1}{3} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{81} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14}{27} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-2*x + 1)^(5/2) - 2/27*(-2*x + 1)^(3/2) - 7/81*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) - 14/27*sqrt(-2*x + 1)

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mupad [B]  time = 0.07, size = 48, normalized size = 0.70 \begin {gather*} -\frac {14\,\sqrt {1-2\,x}}{27}-\frac {2\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {{\left (1-2\,x\right )}^{5/2}}{3}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,14{}\mathrm {i}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*14i)/81 - (14*(1 - 2*x)^(1/2))/27 - (2*(1 - 2*x)^(3/2))/27 -
(1 - 2*x)^(5/2)/3

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sympy [A]  time = 21.90, size = 102, normalized size = 1.48 \begin {gather*} - \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{3} - \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{27} - \frac {14 \sqrt {1 - 2 x}}{27} - \frac {98 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1 - 2*x)**(5/2)/3 - 2*(1 - 2*x)**(3/2)/27 - 14*sqrt(1 - 2*x)/27 - 98*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqr
t(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3))/27

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