3.20.51 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^5} \, dx\) [1951]

Optimal. Leaf size=121 \[ \frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac {24965 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}} \]

[Out]

-1/252*(1-2*x)^(7/2)/(2+3*x)^4+31/588*(1-2*x)^(7/2)/(2+3*x)^3-4993/10584*(1-2*x)^(5/2)/(2+3*x)^2+24965/31752*(
1-2*x)^(3/2)/(2+3*x)-24965/47628*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+24965/15876*(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 43, 52, 65, 212} \begin {gather*} \frac {31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac {(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac {4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac {24965 \sqrt {1-2 x}}{15876}-\frac {24965 \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)/(2 + 3*x)^5,x]

[Out]

(24965*Sqrt[1 - 2*x])/15876 - (1 - 2*x)^(7/2)/(252*(2 + 3*x)^4) + (31*(1 - 2*x)^(7/2))/(588*(2 + 3*x)^3) - (49
93*(1 - 2*x)^(5/2))/(10584*(2 + 3*x)^2) + (24965*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)) - (24965*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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 65

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 79

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] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 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)^5} \, dx &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {(1-2 x)^{5/2} (1121+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}+\frac {4993 \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3} \, dx}{1764}\\ &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}-\frac {24965 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}+\frac {24965 \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx}{10584}\\ &=\frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}+\frac {24965 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{4536}\\ &=\frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac {24965 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{4536}\\ &=\frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac {24965 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 70, normalized size = 0.58 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (134558+762598 x+1526937 x^2+1231065 x^3+302400 x^4\right )}{2 (2+3 x)^4}-24965 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{47628} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*Sqrt[1 - 2*x]*(134558 + 762598*x + 1526937*x^2 + 1231065*x^3 + 302400*x^4))/(2*(2 + 3*x)^4) - 24965*Sqrt[
21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/47628

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 75, normalized size = 0.62

method	result	size
risch	\(-\frac {604800 x^{5}+2159730 x^{4}+1822809 x^{3}-1741 x^{2}-493482 x -134558}{4536 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {24965 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{47628}\)	\(61\)
derivativedivides	\(\frac {200 \sqrt {1-2 x}}{243}+\frac {-\frac {47185 \left (1-2 x \right )^{\frac {7}{2}}}{252}+\frac {129289 \left (1-2 x \right )^{\frac {5}{2}}}{108}-\frac {824705 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {1749055 \sqrt {1-2 x}}{972}}{\left (-4-6 x \right )^{4}}-\frac {24965 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{47628}\)	\(75\)
default	\(\frac {200 \sqrt {1-2 x}}{243}+\frac {-\frac {47185 \left (1-2 x \right )^{\frac {7}{2}}}{252}+\frac {129289 \left (1-2 x \right )^{\frac {5}{2}}}{108}-\frac {824705 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {1749055 \sqrt {1-2 x}}{972}}{\left (-4-6 x \right )^{4}}-\frac {24965 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{47628}\)	\(75\)
trager	\(\frac {\left (302400 x^{4}+1231065 x^{3}+1526937 x^{2}+762598 x +134558\right ) \sqrt {1-2 x}}{4536 \left (2+3 x \right )^{4}}-\frac {24965 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{95256}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

200/243*(1-2*x)^(1/2)+8/3*(-47185/672*(1-2*x)^(7/2)+129289/288*(1-2*x)^(5/2)-824705/864*(1-2*x)^(3/2)+1749055/
2592*(1-2*x)^(1/2))/(-4-6*x)^4-24965/47628*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 119, normalized size = 0.98 \begin {gather*} \frac {24965}{95256} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {200}{243} \, \sqrt {-2 \, x + 1} - \frac {1273995 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 8145207 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 17318805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12243385 \, \sqrt {-2 \, x + 1}}{6804 \, {\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/(2+3*x)^5,x, algorithm="maxima")

[Out]

24965/95256*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 200/243*sqrt(-2*x + 1
) - 1/6804*(1273995*(-2*x + 1)^(7/2) - 8145207*(-2*x + 1)^(5/2) + 17318805*(-2*x + 1)^(3/2) - 12243385*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 = 1.09, size = 104, normalized size = 0.86 \begin {gather*} \frac {24965 \, \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 (302400 \, x^{4} + 1231065 \, x^{3} + 1526937 \, x^{2} + 762598 \, x + 134558\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/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/95256*(24965*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*(302400*x^4 + 1231065*x^3 + 1526937*x^2 + 762598*x + 134558)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 21
6*x^2 + 96*x + 16)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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/(2+3*x)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.84, size = 109, normalized size = 0.90 \begin {gather*} \frac {24965}{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 {200}{243} \, \sqrt {-2 \, x + 1} + \frac {1273995 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 8145207 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 17318805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 12243385 \, \sqrt {-2 \, x + 1}}{108864 \, {\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/(2+3*x)^5,x, algorithm="giac")

[Out]

24965/95256*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 200/243*sqrt
(-2*x + 1) + 1/108864*(1273995*(2*x - 1)^3*sqrt(-2*x + 1) + 8145207*(2*x - 1)^2*sqrt(-2*x + 1) - 17318805*(-2*
x + 1)^(3/2) + 12243385*sqrt(-2*x + 1))/(3*x + 2)^4

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 98, normalized size = 0.81 \begin {gather*} \frac {200\,\sqrt {1-2\,x}}{243}-\frac {24965\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{47628}+\frac {\frac {1749055\,\sqrt {1-2\,x}}{78732}-\frac {824705\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {129289\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {47185\,{\left (1-2\,x\right )}^{7/2}}{20412}}{\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)^2)/(3*x + 2)^5,x)

[Out]

(200*(1 - 2*x)^(1/2))/243 - (24965*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/47628 + ((1749055*(1 - 2*x)^(
1/2))/78732 - (824705*(1 - 2*x)^(3/2))/26244 + (129289*(1 - 2*x)^(5/2))/8748 - (47185*(1 - 2*x)^(7/2))/20412)/
((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

________________________________________________________________________________________