3.20.41 \(\int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx\) [1941]

Optimal. Leaf size=148 \[ \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]

[Out]

1/126*(1-2*x)^(7/2)/(2+3*x)^6-41/378*(1-2*x)^(5/2)/(2+3*x)^5+205/4536*(1-2*x)^(3/2)/(2+3*x)^4+205/4667544*arct
anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-205/13608*(1-2*x)^(1/2)/(2+3*x)^3+205/190512*(1-2*x)^(1/2)/(2+3*x)^2+
205/444528*(1-2*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65, 212} \begin {gather*} \frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac {41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac {205 \sqrt {1-2 x}}{444528 (3 x+2)}+\frac {205 \sqrt {1-2 x}}{190512 (3 x+2)^2}-\frac {205 \sqrt {1-2 x}}{13608 (3 x+2)^3}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

(1 - 2*x)^(7/2)/(126*(2 + 3*x)^6) - (41*(1 - 2*x)^(5/2))/(378*(2 + 3*x)^5) + (205*(1 - 2*x)^(3/2))/(4536*(2 +
3*x)^4) - (205*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3) + (205*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (205*Sqrt[1 - 2
*x])/(444528*(2 + 3*x)) + (205*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 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+3 x)^7} \, dx &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}+\frac {205}{126} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}-\frac {205}{378} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {205 \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx}{1512}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{13608}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{63504}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{444528}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{444528}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.38, size = 75, normalized size = 0.51 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-51904+154312 x-176850 x^2-824526 x^3+204795 x^4+49815 x^5\right )}{2 (2+3 x)^6}+205 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4667544} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

((21*Sqrt[1 - 2*x]*(-51904 + 154312*x - 176850*x^2 - 824526*x^3 + 204795*x^4 + 49815*x^5))/(2*(2 + 3*x)^6) + 2
05*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/4667544

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 84, normalized size = 0.57

method	result	size
risch	\(-\frac {99630 x^{6}+359775 x^{5}-1853847 x^{4}+470826 x^{3}+485474 x^{2}-258120 x +51904}{444528 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {205 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\)	\(66\)
derivativedivides	\(-\frac {46656 \left (\frac {205 \left (1-2 x \right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (1-2 x \right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (1-2 x \right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (1-2 x \right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (1-2 x \right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {1-2 x}}{30233088}\right )}{\left (-4-6 x \right )^{6}}+\frac {205 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\)	\(84\)
default	\(-\frac {46656 \left (\frac {205 \left (1-2 x \right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (1-2 x \right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (1-2 x \right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (1-2 x \right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (1-2 x \right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {1-2 x}}{30233088}\right )}{\left (-4-6 x \right )^{6}}+\frac {205 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\)	\(84\)
trager	\(\frac {\left (49815 x^{5}+204795 x^{4}-824526 x^{3}-176850 x^{2}+154312 x -51904\right ) \sqrt {1-2 x}}{444528 \left (2+3 x \right )^{6}}+\frac {205 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{9335088}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

-46656*(205/42674688*(1-2*x)^(11/2)-3485/54867456*(1-2*x)^(9/2)-439/3919104*(1-2*x)^(7/2)+451/559872*(1-2*x)^(
5/2)-24395/30233088*(1-2*x)^(3/2)+10045/30233088*(1-2*x)^(1/2))/(-4-6*x)^6+205/4667544*arctanh(1/7*21^(1/2)*(1
-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 146, normalized size = 0.99 \begin {gather*} -\frac {205}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {49815 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 658665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 1161594 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 8353422 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 3445435 \, \sqrt {-2 \, x + 1}}{222264 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\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)^7,x, algorithm="maxima")

[Out]

-205/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/222264*(49815*(-2*
x + 1)^(11/2) - 658665*(-2*x + 1)^(9/2) - 1161594*(-2*x + 1)^(7/2) + 8353422*(-2*x + 1)^(5/2) - 8367485*(-2*x
+ 1)^(3/2) + 3445435*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x -
1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

________________________________________________________________________________________

Fricas [A]
time = 1.72, size = 130, normalized size = 0.88 \begin {gather*} \frac {205 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (49815 \, x^{5} + 204795 \, x^{4} - 824526 \, x^{3} - 176850 \, x^{2} + 154312 \, x - 51904\right )} \sqrt {-2 \, x + 1}}{9335088 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="fricas")

[Out]

1/9335088*(205*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((3*x - sqrt(21)
*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(49815*x^5 + 204795*x^4 - 824526*x^3 - 176850*x^2 + 154312*x - 51904)*sqr
t(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

________________________________________________________________________________________

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+3*x)**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.53, size = 132, normalized size = 0.89 \begin {gather*} -\frac {205}{9335088} \, \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 {49815 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 658665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 1161594 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 8353422 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3445435 \, \sqrt {-2 \, x + 1}}{14224896 \, {\left (3 \, x + 2\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

-205/9335088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/14224896*
(49815*(2*x - 1)^5*sqrt(-2*x + 1) + 658665*(2*x - 1)^4*sqrt(-2*x + 1) - 1161594*(2*x - 1)^3*sqrt(-2*x + 1) - 8
353422*(2*x - 1)^2*sqrt(-2*x + 1) + 8367485*(-2*x + 1)^(3/2) - 3445435*sqrt(-2*x + 1))/(3*x + 2)^6

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 126, normalized size = 0.85 \begin {gather*} \frac {205\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544}-\frac {\frac {10045\,\sqrt {1-2\,x}}{472392}-\frac {24395\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {451\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {439\,{\left (1-2\,x\right )}^{7/2}}{61236}-\frac {3485\,{\left (1-2\,x\right )}^{9/2}}{857304}+\frac {205\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \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)^7,x)

[Out]

(205*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4667544 - ((10045*(1 - 2*x)^(1/2))/472392 - (24395*(1 - 2*x
)^(3/2))/472392 + (451*(1 - 2*x)^(5/2))/8748 - (439*(1 - 2*x)^(7/2))/61236 - (3485*(1 - 2*x)^(9/2))/857304 + (
205*(1 - 2*x)^(11/2))/666792)/((67228*x)/81 + (12005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^
4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 184877/729)

________________________________________________________________________________________