3.1814 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac {137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac {1613 \sqrt {1-2 x}}{1037232 (3 x+2)}+\frac {1613 \sqrt {1-2 x}}{444528 (3 x+2)^2}+\frac {1613 \sqrt {1-2 x}}{158760 (3 x+2)^3}-\frac {1613 \sqrt {1-2 x}}{7560 (3 x+2)^4}+\frac {1613 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}} \]

[Out]

-1/378*(1-2*x)^(3/2)/(2+3*x)^6+137/4410*(1-2*x)^(3/2)/(2+3*x)^5+1613/10890936*arctanh(1/7*21^(1/2)*(1-2*x)^(1/
2))*21^(1/2)-1613/7560*(1-2*x)^(1/2)/(2+3*x)^4+1613/158760*(1-2*x)^(1/2)/(2+3*x)^3+1613/444528*(1-2*x)^(1/2)/(
2+3*x)^2+1613/1037232*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.05, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac {137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac {1613 \sqrt {1-2 x}}{1037232 (3 x+2)}+\frac {1613 \sqrt {1-2 x}}{444528 (3 x+2)^2}+\frac {1613 \sqrt {1-2 x}}{158760 (3 x+2)^3}-\frac {1613 \sqrt {1-2 x}}{7560 (3 x+2)^4}+\frac {1613 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - 2*x)^(3/2)/(378*(2 + 3*x)^6) + (137*(1 - 2*x)^(3/2))/(4410*(2 + 3*x)^5) - (1613*Sqrt[1 - 2*x])/(7560*(2
+ 3*x)^4) + (1613*Sqrt[1 - 2*x])/(158760*(2 + 3*x)^3) + (1613*Sqrt[1 - 2*x])/(444528*(2 + 3*x)^2) + (1613*Sqrt
[1 - 2*x])/(1037232*(2 + 3*x)) + (1613*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(518616*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 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 89

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 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 {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {1}{378} \int \frac {\sqrt {1-2 x} (1689+3150 x)}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}+\frac {1613}{630} \int \frac {\sqrt {1-2 x}}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{7560}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{31752}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{148176}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1037232}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}+\frac {1613 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1037232}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}+\frac {1613 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 47, normalized size = 0.32 \[ \frac {(1-2 x)^{3/2} \left (\frac {2401 (1233 x+787)}{(3 x+2)^6}-51616 \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{31765230} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(3/2)*((2401*(787 + 1233*x))/(2 + 3*x)^6 - 51616*Hypergeometric2F1[3/2, 5, 5/2, 3/7 - (6*x)/7]))/31
765230

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fricas [A]  time = 0.88, size = 130, normalized size = 0.88 \[ \frac {8065 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1959795 \, x^{5} + 8056935 \, x^{4} + 14197626 \, x^{3} + 1791558 \, x^{2} - 7772840 \, x - 3136864\right )} \sqrt {-2 \, x + 1}}{108909360 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/108909360*(8065*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*(1959795*x^5 + 8056935*x^4 + 14197626*x^3 + 1791558*x^2 - 7772840*x -
3136864)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [A]  time = 1.26, size = 132, normalized size = 0.89 \[ -\frac {1613}{21781872} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1959795 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 25912845 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 140843934 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 300985146 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 148925455 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 135548455 \, \sqrt {-2 \, x + 1}}{165957120 \, {\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1613/21781872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1659571
20*(1959795*(2*x - 1)^5*sqrt(-2*x + 1) + 25912845*(2*x - 1)^4*sqrt(-2*x + 1) + 140843934*(2*x - 1)^3*sqrt(-2*x
+ 1) + 300985146*(2*x - 1)^2*sqrt(-2*x + 1) - 148925455*(-2*x + 1)^(3/2) - 135548455*sqrt(-2*x + 1))/(3*x + 2
)^6

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maple [A]  time = 0.01, size = 84, normalized size = 0.57 \[ \frac {1613 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{10890936}+\frac {-\frac {14517 \left (-2 x +1\right )^{\frac {11}{2}}}{19208}+\frac {27421 \left (-2 x +1\right )^{\frac {9}{2}}}{2744}-\frac {53229 \left (-2 x +1\right )^{\frac {7}{2}}}{980}+\frac {113751 \left (-2 x +1\right )^{\frac {5}{2}}}{980}-\frac {86837 \left (-2 x +1\right )^{\frac {3}{2}}}{1512}-\frac {11291 \sqrt {-2 x +1}}{216}}{\left (-6 x -4\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

23328*(-1613/49787136*(-2*x+1)^(11/2)+27421/64012032*(-2*x+1)^(9/2)-17743/7620480*(-2*x+1)^(7/2)+4213/846720*(
-2*x+1)^(5/2)-86837/35271936*(-2*x+1)^(3/2)-11291/5038848*(-2*x+1)^(1/2))/(-6*x-4)^6+1613/10890936*arctanh(1/7
*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.32, size = 146, normalized size = 0.99 \[ -\frac {1613}{21781872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1959795 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 25912845 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 140843934 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 300985146 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 148925455 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 135548455 \, \sqrt {-2 \, x + 1}}{2593080 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1613/21781872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/2593080*(1959795
*(-2*x + 1)^(11/2) - 25912845*(-2*x + 1)^(9/2) + 140843934*(-2*x + 1)^(7/2) - 300985146*(-2*x + 1)^(5/2) + 148
925455*(-2*x + 1)^(3/2) + 135548455*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)

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mupad [B]  time = 0.08, size = 126, normalized size = 0.85 \[ \frac {1613\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{10890936}-\frac {\frac {11291\,\sqrt {1-2\,x}}{157464}+\frac {86837\,{\left (1-2\,x\right )}^{3/2}}{1102248}-\frac {4213\,{\left (1-2\,x\right )}^{5/2}}{26460}+\frac {17743\,{\left (1-2\,x\right )}^{7/2}}{238140}-\frac {27421\,{\left (1-2\,x\right )}^{9/2}}{2000376}+\frac {1613\,{\left (1-2\,x\right )}^{11/2}}{1555848}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1613*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/10890936 - ((11291*(1 - 2*x)^(1/2))/157464 + (86837*(1 - 2
*x)^(3/2))/1102248 - (4213*(1 - 2*x)^(5/2))/26460 + (17743*(1 - 2*x)^(7/2))/238140 - (27421*(1 - 2*x)^(9/2))/2
000376 + (1613*(1 - 2*x)^(11/2))/1555848)/((67228*x)/81 + (12005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (24
5*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 184877/729)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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