3.1868 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx\)

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

[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])

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Rubi [A]  time = 0.0278284, antiderivative size = 108, normalized size of antiderivative = 1., 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} \[ \frac{(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac{137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac{137 \sqrt{1-2 x}}{10584 (3 x+2)}+\frac{137 \sqrt{1-2 x}}{1512 (3 x+2)^2}-\frac{137 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{5292 \sqrt{21}} \]

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 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 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 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*}

Mathematica [C]  time = 0.0142215, size = 42, normalized size = 0.39 \[ \frac{(1-2 x)^{5/2} \left (\frac{12005}{(3 x+2)^4}-2192 \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{1008420} \]

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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Maple [A]  time = 0.009, size = 66, normalized size = 0.6 \begin{align*} -1296\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{137\, \left ( 1-2\,x \right ) ^{7/2}}{254016}}-{\frac{733\, \left ( 1-2\,x \right ) ^{5/2}}{326592}}+{\frac{1507\, \left ( 1-2\,x \right ) ^{3/2}}{139968}}-{\frac{959\,\sqrt{1-2\,x}}{139968}} \right ) }-{\frac{137\,\sqrt{21}}{111132}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.63578, size = 149, normalized size = 1.38 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.37725, size = 292, normalized size = 2.7 \begin{align*} \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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 2.05511, size = 135, normalized size = 1.25 \begin{align*} \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{align*}

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