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

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

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Rubi [A]  time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 ) \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.09, size = 63, normalized size = 0.66 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.07, size = 79, normalized size = 0.83 \begin {gather*} \frac {\left (7875 (1-2 x)^4-20925 (1-2 x)^3+252 (1-2 x)^2+980 (1-2 x)+6860\right ) \sqrt {1-2 x}}{17010}-\frac {98}{243} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6860 + 980*(1 - 2*x) + 252*(1 - 2*x)^2 - 20925*(1 - 2*x)^3 + 7875*(1 - 2*x)^4)*Sqrt[1 - 2*x])/17010 - (98*Sq
rt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243

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fricas [A]  time = 1.64, size = 66, normalized size = 0.69 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.88, size = 106, normalized size = 1.12 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 65, normalized size = 0.68 \begin {gather*} -\frac {98 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{729}+\frac {14 \left (-2 x +1\right )^{\frac {3}{2}}}{243}+\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{135}-\frac {155 \left (-2 x +1\right )^{\frac {7}{2}}}{126}+\frac {25 \left (-2 x +1\right )^{\frac {9}{2}}}{54}+\frac {98 \sqrt {-2 x +1}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 82, normalized size = 0.86 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.19, size = 66, normalized size = 0.69 \begin {gather*} \frac {98\,\sqrt {1-2\,x}}{243}+\frac {14\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{135}-\frac {155\,{\left (1-2\,x\right )}^{7/2}}{126}+\frac {25\,{\left (1-2\,x\right )}^{9/2}}{54}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*98i)/729 + (98*(1 - 2*x)^(1/2))/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

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sympy [A]  time = 54.90, size = 126, normalized size = 1.33 \begin {gather*} \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 {gather*}

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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