3.19.91 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx\) [1891]

Optimal. Leaf size=131 \[ \frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}} \]

[Out]

-1/12*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^4+13243/12348*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+13/56*(3+5*x)
^2*(1-2*x)^(1/2)/(2+3*x)^2+(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^3-1/1176*(18187+26775*x)*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.03, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 154, 151, 65, 212} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{56 (3 x+2)^2}-\frac {\sqrt {1-2 x} (26775 x+18187)}{1176 (3 x+2)}+\frac {13243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(13*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(56*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (Sqrt[1 - 2
*x]*(3 + 5*x)^3)/(2 + 3*x)^3 - (Sqrt[1 - 2*x]*(18187 + 26775*x))/(1176*(2 + 3*x)) + (13243*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(588*Sqrt[21])

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 99

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {1}{108} \int \frac {(-189-810 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\int \frac {(-14013-61965 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{4536}\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}-\frac {13243 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1176}\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1176}\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 68, normalized size = 0.52 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (74810+401850 x+788415 x^2+661639 x^3+196000 x^4\right )}{(2+3 x)^4}+26486 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*Sqrt[1 - 2*x]*(74810 + 401850*x + 788415*x^2 + 661639*x^3 + 196000*x^4))/(2 + 3*x)^4 + 26486*Sqrt[21]*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/24696

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Maple [A]
time = 0.13, size = 75, normalized size = 0.57

method	result	size
risch	\(\frac {392000 x^{5}+1127278 x^{4}+915191 x^{3}+15285 x^{2}-252230 x -74810}{1176 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {13243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\)	\(61\)
derivativedivides	\(-\frac {500 \sqrt {1-2 x}}{243}-\frac {4 \left (-\frac {416917 \left (1-2 x \right )^{\frac {7}{2}}}{2352}+\frac {406463 \left (1-2 x \right )^{\frac {5}{2}}}{336}-\frac {1189171 \left (1-2 x \right )^{\frac {3}{2}}}{432}+\frac {2706781 \sqrt {1-2 x}}{1296}\right )}{3 \left (-4-6 x \right )^{4}}+\frac {13243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\)	\(75\)
default	\(-\frac {500 \sqrt {1-2 x}}{243}-\frac {4 \left (-\frac {416917 \left (1-2 x \right )^{\frac {7}{2}}}{2352}+\frac {406463 \left (1-2 x \right )^{\frac {5}{2}}}{336}-\frac {1189171 \left (1-2 x \right )^{\frac {3}{2}}}{432}+\frac {2706781 \sqrt {1-2 x}}{1296}\right )}{3 \left (-4-6 x \right )^{4}}+\frac {13243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\)	\(75\)
trager	\(-\frac {\left (196000 x^{4}+661639 x^{3}+788415 x^{2}+401850 x +74810\right ) \sqrt {1-2 x}}{1176 \left (2+3 x \right )^{4}}+\frac {13243 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{24696}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-500/243*(1-2*x)^(1/2)-4/3*(-416917/2352*(1-2*x)^(7/2)+406463/336*(1-2*x)^(5/2)-1189171/432*(1-2*x)^(3/2)+2706
781/1296*(1-2*x)^(1/2))/(-4-6*x)^4+13243/12348*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.55, size = 119, normalized size = 0.91 \begin {gather*} -\frac {13243}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {500}{243} \, \sqrt {-2 \, x + 1} + \frac {11256759 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 76821507 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 174808137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 132632269 \, \sqrt {-2 \, x + 1}}{47628 \, {\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)^3/(2+3*x)^5,x, algorithm="maxima")

[Out]

-13243/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 500/243*sqrt(-2*x +
1) + 1/47628*(11256759*(-2*x + 1)^(7/2) - 76821507*(-2*x + 1)^(5/2) + 174808137*(-2*x + 1)^(3/2) - 132632269*s
qrt(-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 = 1.35, size = 105, normalized size = 0.80 \begin {gather*} \frac {13243 \, \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 (196000 \, x^{4} + 661639 \, x^{3} + 788415 \, x^{2} + 401850 \, x + 74810\right )} \sqrt {-2 \, x + 1}}{24696 \, {\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)^3/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/24696*(13243*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*(196000*x^4 + 661639*x^3 + 788415*x^2 + 401850*x + 74810)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x
^2 + 96*x + 16)

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

[Out]

Timed out

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Giac [A]
time = 1.19, size = 109, normalized size = 0.83 \begin {gather*} -\frac {13243}{24696} \, \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 {500}{243} \, \sqrt {-2 \, x + 1} - \frac {11256759 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 76821507 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 174808137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 132632269 \, \sqrt {-2 \, x + 1}}{762048 \, {\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)^3/(2+3*x)^5,x, algorithm="giac")

[Out]

-13243/24696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 500/243*sqr
t(-2*x + 1) - 1/762048*(11256759*(2*x - 1)^3*sqrt(-2*x + 1) + 76821507*(2*x - 1)^2*sqrt(-2*x + 1) - 174808137*
(-2*x + 1)^(3/2) + 132632269*sqrt(-2*x + 1))/(3*x + 2)^4

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Mupad [B]
time = 1.18, size = 99, normalized size = 0.76 \begin {gather*} \frac {13243\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-\frac {500\,\sqrt {1-2\,x}}{243}-\frac {\frac {2706781\,\sqrt {1-2\,x}}{78732}-\frac {1189171\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {406463\,{\left (1-2\,x\right )}^{5/2}}{20412}-\frac {416917\,{\left (1-2\,x\right )}^{7/2}}{142884}}{\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)^(3/2)*(5*x + 3)^3)/(3*x + 2)^5,x)

[Out]

(13243*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/12348 - (500*(1 - 2*x)^(1/2))/243 - ((2706781*(1 - 2*x)^(
1/2))/78732 - (1189171*(1 - 2*x)^(3/2))/26244 + (406463*(1 - 2*x)^(5/2))/20412 - (416917*(1 - 2*x)^(7/2))/1428
84)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

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