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

Optimal. Leaf size=148 \[ \frac {83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac {263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac {1315 \sqrt {1-2 x}}{148176 (3 x+2)}-\frac {1315 \sqrt {1-2 x}}{21168 (3 x+2)^2}+\frac {1315 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}} \]

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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} \begin {gather*} \frac {83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac {263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac {1315 \sqrt {1-2 x}}{148176 (3 x+2)}-\frac {1315 \sqrt {1-2 x}}{21168 (3 x+2)^2}+\frac {1315 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - 2*x)^(7/2)/(378*(2 + 3*x)^6) + (83*(1 - 2*x)^(7/2))/(2646*(2 + 3*x)^5) - (263*(1 - 2*x)^(5/2))/(1176*(2
+ 3*x)^4) + (1315*(1 - 2*x)^(3/2))/(10584*(2 + 3*x)^3) - (1315*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (1315*Sqrt
[1 - 2*x])/(148176*(2 + 3*x)) + (1315*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(74088*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 {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {1}{378} \int \frac {(1-2 x)^{5/2} (1685+3150 x)}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}+\frac {263}{98} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}-\frac {1315 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx}{1176}\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}+\frac {1315 \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx}{3528}\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}-\frac {1315 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{21168}\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}-\frac {1315 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{148176}\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}+\frac {1315 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{148176}\\ &=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}+\frac {1315 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 47, normalized size = 0.32 \begin {gather*} \frac {(1-2 x)^{7/2} \left (\frac {352947 (83 x+53)}{(3 x+2)^6}-227232 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{311299254} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(7/2)*((352947*(53 + 83*x))/(2 + 3*x)^6 - 227232*Hypergeometric2F1[7/2, 5, 9/2, 3/7 - (6*x)/7]))/31
1299254

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IntegrateAlgebraic [A]  time = 0.48, size = 97, normalized size = 0.66 \begin {gather*} \frac {1315 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}}-\frac {\left (319545 (1-2 x)^5+2360505 (1-2 x)^4-22080870 (1-2 x)^3+53584146 (1-2 x)^2-53674355 (1-2 x)[phone]\right ) \sqrt {1-2 x}}{74088 (3 (1-2 x)-7)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/74088*((22101205 - 53674355*(1 - 2*x) + 53584146*(1 - 2*x)^2 - 22080870*(1 - 2*x)^3 + 2360505*(1 - 2*x)^4 +
319545*(1 - 2*x)^5)*Sqrt[1 - 2*x])/(-7 + 3*(1 - 2*x))^6 + (1315*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(74088*Sqrt
[21])

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fricas [A]  time = 1.54, size = 130, normalized size = 0.88 \begin {gather*} \frac {1315 \, \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 (319545 \, x^{5} - 1979115 \, x^{4} - 2360850 \, x^{3} - 587502 \, x^{2} - 106808 \, x - 81568\right )} \sqrt {-2 \, x + 1}}{3111696 \, {\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/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/3111696*(1315*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*(319545*x^5 - 1979115*x^4 - 2360850*x^3 - 587502*x^2 - 106808*x - 81568)
*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 = 1.19, size = 132, normalized size = 0.89 \begin {gather*} -\frac {1315}{3111696} \, \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 {319545 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 2360505 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 22080870 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 53584146 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 53674355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 22101205 \, \sqrt {-2 \, x + 1}}{4741632 \, {\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/(2+3*x)^7,x, algorithm="giac")

[Out]

-1315/3111696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/4741632*
(319545*(2*x - 1)^5*sqrt(-2*x + 1) - 2360505*(2*x - 1)^4*sqrt(-2*x + 1) - 22080870*(2*x - 1)^3*sqrt(-2*x + 1)
- 53584146*(2*x - 1)^2*sqrt(-2*x + 1) + 53674355*(-2*x + 1)^(3/2) - 22101205*sqrt(-2*x + 1))/(3*x + 2)^6

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maple [A]  time = 0.01, size = 84, normalized size = 0.57 \begin {gather*} \frac {1315 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1555848}+\frac {-\frac {11835 \left (-2 x +1\right )^{\frac {11}{2}}}{2744}-\frac {112405 \left (-2 x +1\right )^{\frac {9}{2}}}{3528}+\frac {8345 \left (-2 x +1\right )^{\frac {7}{2}}}{28}-\frac {2893 \left (-2 x +1\right )^{\frac {5}{2}}}{4}+\frac {156485 \left (-2 x +1\right )^{\frac {3}{2}}}{216}-\frac {64435 \sqrt {-2 x +1}}{216}}{\left (-6 x -4\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

23328*(-1315/7112448*(-2*x+1)^(11/2)-112405/82301184*(-2*x+1)^(9/2)+8345/653184*(-2*x+1)^(7/2)-2893/93312*(-2*
x+1)^(5/2)+156485/5038848*(-2*x+1)^(3/2)-64435/5038848*(-2*x+1)^(1/2))/(-6*x-4)^6+1315/1555848*arctanh(1/7*21^
(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.17, size = 146, normalized size = 0.99 \begin {gather*} -\frac {1315}{3111696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {319545 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 2360505 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 22080870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 53584146 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 53674355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 22101205 \, \sqrt {-2 \, x + 1}}{74088 \, {\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/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1315/3111696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/74088*(319545*(-2
*x + 1)^(11/2) + 2360505*(-2*x + 1)^(9/2) - 22080870*(-2*x + 1)^(7/2) + 53584146*(-2*x + 1)^(5/2) - 53674355*(
-2*x + 1)^(3/2) + 22101205*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 = 1.18, size = 126, normalized size = 0.85 \begin {gather*} \frac {1315\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1555848}-\frac {\frac {64435\,\sqrt {1-2\,x}}{157464}-\frac {156485\,{\left (1-2\,x\right )}^{3/2}}{157464}+\frac {2893\,{\left (1-2\,x\right )}^{5/2}}{2916}-\frac {8345\,{\left (1-2\,x\right )}^{7/2}}{20412}+\frac {112405\,{\left (1-2\,x\right )}^{9/2}}{2571912}+\frac {1315\,{\left (1-2\,x\right )}^{11/2}}{222264}}{\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)^2)/(3*x + 2)^7,x)

[Out]

(1315*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1555848 - ((64435*(1 - 2*x)^(1/2))/157464 - (156485*(1 - 2
*x)^(3/2))/157464 + (2893*(1 - 2*x)^(5/2))/2916 - (8345*(1 - 2*x)^(7/2))/20412 + (112405*(1 - 2*x)^(9/2))/2571
912 + (1315*(1 - 2*x)^(11/2))/222264)/((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)

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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/(2+3*x)**7,x)

[Out]

Timed out

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