3.1947 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx\)

Optimal. Leaf size=95 \[ \frac{25}{54} (1-2 x)^{9/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{2}{135} (1-2 x)^{5/2}+\frac{14}{243} (1-2 x)^{3/2}+\frac{98}{243} \sqrt{1-2 x}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(98*Sqrt[1 - 2*x])/243 + (14*(1 - 2*x)^(3/2))/243 + (2*(1 - 2*x)^(5/2))/135 - (155*(1 - 2*x)^(7/2))/126 + (25*
(1 - 2*x)^(9/2))/54 - (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243

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Rubi [A]  time = 0.0367094, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ \frac{25}{54} (1-2 x)^{9/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{2}{135} (1-2 x)^{5/2}+\frac{14}{243} (1-2 x)^{3/2}+\frac{98}{243} \sqrt{1-2 x}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(98*Sqrt[1 - 2*x])/243 + (14*(1 - 2*x)^(3/2))/243 + (2*(1 - 2*x)^(5/2))/135 - (155*(1 - 2*x)^(7/2))/126 + (25*
(1 - 2*x)^(9/2))/54 - (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243

Rule 88

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 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 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)^{5/2} (3+5 x)^2}{2+3 x} \, dx &=\int \left (\frac{155}{18} (1-2 x)^{5/2}-\frac{25}{6} (1-2 x)^{7/2}+\frac{(1-2 x)^{5/2}}{9 (2+3 x)}\right ) \, dx\\ &=-\frac{155}{126} (1-2 x)^{7/2}+\frac{25}{54} (1-2 x)^{9/2}+\frac{1}{9} \int \frac{(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac{2}{135} (1-2 x)^{5/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{25}{54} (1-2 x)^{9/2}+\frac{7}{27} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{14}{243} (1-2 x)^{3/2}+\frac{2}{135} (1-2 x)^{5/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{25}{54} (1-2 x)^{9/2}+\frac{49}{81} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{98}{243} \sqrt{1-2 x}+\frac{14}{243} (1-2 x)^{3/2}+\frac{2}{135} (1-2 x)^{5/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{25}{54} (1-2 x)^{9/2}+\frac{343}{243} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{98}{243} \sqrt{1-2 x}+\frac{14}{243} (1-2 x)^{3/2}+\frac{2}{135} (1-2 x)^{5/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{25}{54} (1-2 x)^{9/2}-\frac{343}{243} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{98}{243} \sqrt{1-2 x}+\frac{14}{243} (1-2 x)^{3/2}+\frac{2}{135} (1-2 x)^{5/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{25}{54} (1-2 x)^{9/2}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0636819, size = 63, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (63000 x^4-42300 x^3-30546 x^2+29791 x-2479\right )}{8505}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-2479 + 29791*x - 30546*x^2 - 42300*x^3 + 63000*x^4))/8505 - (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/243

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Maple [A]  time = 0.005, size = 65, normalized size = 0.7 \begin{align*}{\frac{14}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{135} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{155}{126} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{25}{54} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{98\,\sqrt{21}}{729}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{98}{243}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14/243*(1-2*x)^(3/2)+2/135*(1-2*x)^(5/2)-155/126*(1-2*x)^(7/2)+25/54*(1-2*x)^(9/2)-98/729*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)+98/243*(1-2*x)^(1/2)

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Maxima [A]  time = 4.31601, size = 111, normalized size = 1.17 \begin{align*} \frac{25}{54} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{155}{126} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{135} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{14}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{49}{729} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{98}{243} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/54*(-2*x + 1)^(9/2) - 155/126*(-2*x + 1)^(7/2) + 2/135*(-2*x + 1)^(5/2) + 14/243*(-2*x + 1)^(3/2) + 49/729*
sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 98/243*sqrt(-2*x + 1)

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Fricas [A]  time = 1.49389, size = 215, normalized size = 2.26 \begin{align*} \frac{49}{729} \, \sqrt{7} \sqrt{3} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + \frac{1}{8505} \,{\left (63000 \, x^{4} - 42300 \, x^{3} - 30546 \, x^{2} + 29791 \, x - 2479\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/729*sqrt(7)*sqrt(3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) + 1/8505*(63000*x^4 - 42300*x
^3 - 30546*x^2 + 29791*x - 2479)*sqrt(-2*x + 1)

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Sympy [A]  time = 38.4241, size = 126, normalized size = 1.33 \begin{align*} \frac{25 \left (1 - 2 x\right )^{\frac{9}{2}}}{54} - \frac{155 \left (1 - 2 x\right )^{\frac{7}{2}}}{126} + \frac{2 \left (1 - 2 x\right )^{\frac{5}{2}}}{135} + \frac{14 \left (1 - 2 x\right )^{\frac{3}{2}}}{243} + \frac{98 \sqrt{1 - 2 x}}{243} + \frac{686 \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 )}{243} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*(1 - 2*x)**(9/2)/54 - 155*(1 - 2*x)**(7/2)/126 + 2*(1 - 2*x)**(5/2)/135 + 14*(1 - 2*x)**(3/2)/243 + 98*sqrt
(1 - 2*x)/243 + 686*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(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))/243

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Giac [A]  time = 1.57364, size = 143, normalized size = 1.51 \begin{align*} \frac{25}{54} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{155}{126} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{135} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{14}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{49}{729} \, \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{98}{243} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/54*(2*x - 1)^4*sqrt(-2*x + 1) + 155/126*(2*x - 1)^3*sqrt(-2*x + 1) + 2/135*(2*x - 1)^2*sqrt(-2*x + 1) + 14/
243*(-2*x + 1)^(3/2) + 49/729*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 98/243*sqrt(-2*x + 1)