3.1939 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ \frac{(1-2 x)^{7/2}}{84 (3 x+2)^4}-\frac{139 (1-2 x)^{5/2}}{756 (3 x+2)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (3 x+2)^2}-\frac{695 \sqrt{1-2 x}}{4536 (3 x+2)}+\frac{695 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

[Out]

(1 - 2*x)^(7/2)/(84*(2 + 3*x)^4) - (139*(1 - 2*x)^(5/2))/(756*(2 + 3*x)^3) + (695*(1 - 2*x)^(3/2))/(4536*(2 +
3*x)^2) - (695*Sqrt[1 - 2*x])/(4536*(2 + 3*x)) + (695*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

________________________________________________________________________________________

Rubi [A]  time = 0.0281605, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 47, 63, 206} \[ \frac{(1-2 x)^{7/2}}{84 (3 x+2)^4}-\frac{139 (1-2 x)^{5/2}}{756 (3 x+2)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (3 x+2)^2}-\frac{695 \sqrt{1-2 x}}{4536 (3 x+2)}+\frac{695 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(7/2)/(84*(2 + 3*x)^4) - (139*(1 - 2*x)^(5/2))/(756*(2 + 3*x)^3) + (695*(1 - 2*x)^(3/2))/(4536*(2 +
3*x)^2) - (695*Sqrt[1 - 2*x])/(4536*(2 + 3*x)) + (695*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2268*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 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+3 x)^5} \, dx &=\frac{(1-2 x)^{7/2}}{84 (2+3 x)^4}+\frac{139}{84} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{7/2}}{84 (2+3 x)^4}-\frac{139 (1-2 x)^{5/2}}{756 (2+3 x)^3}-\frac{695}{756} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{7/2}}{84 (2+3 x)^4}-\frac{139 (1-2 x)^{5/2}}{756 (2+3 x)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (2+3 x)^2}+\frac{695 \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx}{1512}\\ &=\frac{(1-2 x)^{7/2}}{84 (2+3 x)^4}-\frac{139 (1-2 x)^{5/2}}{756 (2+3 x)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (2+3 x)^2}-\frac{695 \sqrt{1-2 x}}{4536 (2+3 x)}-\frac{695 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{4536}\\ &=\frac{(1-2 x)^{7/2}}{84 (2+3 x)^4}-\frac{139 (1-2 x)^{5/2}}{756 (2+3 x)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (2+3 x)^2}-\frac{695 \sqrt{1-2 x}}{4536 (2+3 x)}+\frac{695 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{4536}\\ &=\frac{(1-2 x)^{7/2}}{84 (2+3 x)^4}-\frac{139 (1-2 x)^{5/2}}{756 (2+3 x)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (2+3 x)^2}-\frac{695 \sqrt{1-2 x}}{4536 (2+3 x)}+\frac{695 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0610729, size = 79, normalized size = 0.73 \[ \frac{21 \left (83430 x^4+46227 x^3-7183 x^2-9606 x-4394\right )+1390 \sqrt{21-42 x} (3 x+2)^4 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{95256 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*(-4394 - 9606*x - 7183*x^2 + 46227*x^3 + 83430*x^4) + 1390*Sqrt[21 - 42*x]*(2 + 3*x)^4*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(95256*Sqrt[1 - 2*x]*(2 + 3*x)^4)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 66, normalized size = 0.6 \begin{align*} -1296\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{515\, \left ( 1-2\,x \right ) ^{7/2}}{36288}}+{\frac{10147\, \left ( 1-2\,x \right ) ^{5/2}}{139968}}-{\frac{53515\, \left ( 1-2\,x \right ) ^{3/2}}{419904}}+{\frac{34055\,\sqrt{1-2\,x}}{419904}} \right ) }+{\frac{695\,\sqrt{21}}{47628}{\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)^(5/2)*(3+5*x)/(2+3*x)^5,x)

[Out]

-1296*(-515/36288*(1-2*x)^(7/2)+10147/139968*(1-2*x)^(5/2)-53515/419904*(1-2*x)^(3/2)+34055/419904*(1-2*x)^(1/
2))/(-6*x-4)^4+695/47628*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.04033, size = 149, normalized size = 1.38 \begin{align*} -\frac{695}{95256} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{41715 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 213087 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 374605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 238385 \, \sqrt{-2 \, x + 1}}{2268 \,{\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)^(5/2)*(3+5*x)/(2+3*x)^5,x, algorithm="maxima")

[Out]

-695/95256*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/2268*(41715*(-2*x +
1)^(7/2) - 213087*(-2*x + 1)^(5/2) + 374605*(-2*x + 1)^(3/2) - 238385*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2
*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

________________________________________________________________________________________

Fricas [A]  time = 1.53331, size = 294, normalized size = 2.72 \begin{align*} \frac{695 \, \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 (41715 \, x^{3} + 43971 \, x^{2} + 18394 \, x + 4394\right )} \sqrt{-2 \, x + 1}}{95256 \,{\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)^(5/2)*(3+5*x)/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/95256*(695*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*(41715*x^3 + 43971*x^2 + 18394*x + 4394)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

________________________________________________________________________________________

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)**(5/2)*(3+5*x)/(2+3*x)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.03605, size = 135, normalized size = 1.25 \begin{align*} -\frac{695}{95256} \, \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{41715 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 213087 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 374605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 238385 \, \sqrt{-2 \, x + 1}}{36288 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-695/95256*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/36288*(4171
5*(2*x - 1)^3*sqrt(-2*x + 1) + 213087*(2*x - 1)^2*sqrt(-2*x + 1) - 374605*(-2*x + 1)^(3/2) + 238385*sqrt(-2*x
+ 1))/(3*x + 2)^4