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

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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)^{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 = 0.12, size = 79, normalized size = 0.73 \begin {gather*} -\frac {1390 (3 x+2)^4 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )-21 \left (83430 x^4+46227 x^3-7183 x^2-9606 x-4394\right )}{95256 \sqrt {1-2 x} (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.30, size = 79, normalized size = 0.73 \begin {gather*} \frac {\sqrt {1-2 x} \left (41715 (1-2 x)^3-213087 (1-2 x)^2+374605 (1-2 x)-238385\right )}{2268 (3 (1-2 x)-7)^4}+\frac {695 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-238385 + 374605*(1 - 2*x) - 213087*(1 - 2*x)^2 + 41715*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(2268*(-7 + 3*(1 - 2*x))
^4) + (695*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

________________________________________________________________________________________

fricas [A]  time = 1.46, size = 100, normalized size = 0.93 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 1.06, size = 100, normalized size = 0.93 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 66, normalized size = 0.61 \begin {gather*} \frac {695 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{47628}-\frac {1296 \left (-\frac {515 \left (-2 x +1\right )^{\frac {7}{2}}}{36288}+\frac {10147 \left (-2 x +1\right )^{\frac {5}{2}}}{139968}-\frac {53515 \left (-2 x +1\right )^{\frac {3}{2}}}{419904}+\frac {34055 \sqrt {-2 x +1}}{419904}\right )}{\left (-6 x -4\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.22, size = 110, normalized size = 1.02 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 1.18, size = 90, normalized size = 0.83 \begin {gather*} \frac {695\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{47628}-\frac {\frac {34055\,\sqrt {1-2\,x}}{26244}-\frac {53515\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {10147\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {515\,{\left (1-2\,x\right )}^{7/2}}{2268}}{\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(695*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/47628 - ((34055*(1 - 2*x)^(1/2))/26244 - (53515*(1 - 2*x)^(
3/2))/26244 + (10147*(1 - 2*x)^(5/2))/8748 - (515*(1 - 2*x)^(7/2))/2268)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (
28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________