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

Optimal. Leaf size=108 \[ \frac {(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac {137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac {137 \sqrt {1-2 x}}{10584 (3 x+2)}+\frac {137 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \]

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Rubi [A]  time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \begin {gather*} \frac {(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac {137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac {137 \sqrt {1-2 x}}{10584 (3 x+2)}+\frac {137 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(5/2)/(84*(2 + 3*x)^4) - (137*(1 - 2*x)^(3/2))/(756*(2 + 3*x)^3) + (137*Sqrt[1 - 2*x])/(1512*(2 + 3*
x)^2) - (137*Sqrt[1 - 2*x])/(10584*(2 + 3*x)) - (137*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5292*Sqrt[21])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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)^5} \, dx &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}+\frac {137}{84} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}-\frac {137}{252} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1512}\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{10584}\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{10584}\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 42, normalized size = 0.39 \begin {gather*} \frac {(1-2 x)^{5/2} \left (\frac {12005}{(3 x+2)^4}-2192 \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{1008420} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(5/2)*(12005/(2 + 3*x)^4 - 2192*Hypergeometric2F1[5/2, 4, 7/2, 3/7 - (6*x)/7]))/1008420

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IntegrateAlgebraic [A]  time = 0.30, size = 79, normalized size = 0.73 \begin {gather*} \frac {\left (3699 (1-2 x)^3+15393 (1-2 x)^2-73843 (1-2 x)+46991\right ) \sqrt {1-2 x}}{5292 (3 (1-2 x)-7)^4}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((46991 - 73843*(1 - 2*x) + 15393*(1 - 2*x)^2 + 3699*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(5292*(-7 + 3*(1 - 2*x))^4) -
(137*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5292*Sqrt[21])

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fricas [A]  time = 1.45, size = 99, normalized size = 0.92 \begin {gather*} \frac {137 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (3699 \, x^{3} - 13245 \, x^{2} - 7990 \, x + 970\right )} \sqrt {-2 \, x + 1}}{222264 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \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)^5,x, algorithm="fricas")

[Out]

1/222264*(137*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x +
2)) - 21*(3699*x^3 - 13245*x^2 - 7990*x + 970)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [A]  time = 0.78, size = 100, normalized size = 0.93 \begin {gather*} \frac {137}{222264} \, \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 {3699 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 15393 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 46991 \, \sqrt {-2 \, x + 1}}{84672 \, {\left (3 \, x + 2\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

137/222264*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/84672*(3699
*(2*x - 1)^3*sqrt(-2*x + 1) - 15393*(2*x - 1)^2*sqrt(-2*x + 1) + 73843*(-2*x + 1)^(3/2) - 46991*sqrt(-2*x + 1)
)/(3*x + 2)^4

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maple [A]  time = 0.02, size = 66, normalized size = 0.61 \begin {gather*} -\frac {137 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{111132}-\frac {1296 \left (-\frac {137 \left (-2 x +1\right )^{\frac {7}{2}}}{254016}-\frac {733 \left (-2 x +1\right )^{\frac {5}{2}}}{326592}+\frac {1507 \left (-2 x +1\right )^{\frac {3}{2}}}{139968}-\frac {959 \sqrt {-2 x +1}}{139968}\right )}{\left (-6 x -4\right )^{4}} \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)^5,x)

[Out]

-1296*(-137/254016*(-2*x+1)^(7/2)-733/326592*(-2*x+1)^(5/2)+1507/139968*(-2*x+1)^(3/2)-959/139968*(-2*x+1)^(1/
2))/(-6*x-4)^4-137/111132*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.44, size = 110, normalized size = 1.02 \begin {gather*} \frac {137}{222264} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3699 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 15393 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 46991 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \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)^5,x, algorithm="maxima")

[Out]

137/222264*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/5292*(3699*(-2*x + 1
)^(7/2) + 15393*(-2*x + 1)^(5/2) - 73843*(-2*x + 1)^(3/2) + 46991*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x -
1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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mupad [B]  time = 1.19, size = 89, normalized size = 0.82 \begin {gather*} \frac {\frac {959\,\sqrt {1-2\,x}}{8748}-\frac {1507\,{\left (1-2\,x\right )}^{3/2}}{8748}+\frac {733\,{\left (1-2\,x\right )}^{5/2}}{20412}+\frac {137\,{\left (1-2\,x\right )}^{7/2}}{15876}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {137\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{111132} \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)^5,x)

[Out]

((959*(1 - 2*x)^(1/2))/8748 - (1507*(1 - 2*x)^(3/2))/8748 + (733*(1 - 2*x)^(5/2))/20412 + (137*(1 - 2*x)^(7/2)
)/15876)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81) - (137*21^(1/2)*atanh
((21^(1/2)*(1 - 2*x)^(1/2))/7))/111132

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**5,x)

[Out]

Timed out

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