3.19.11 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=114 \[ \frac {211 (1-2 x)^{7/2}}{2646 (3 x+2)^2}-\frac {(1-2 x)^{7/2}}{189 (3 x+2)^3}-\frac {887 (1-2 x)^{5/2}}{882 (3 x+2)}-\frac {4435 (1-2 x)^{3/2}}{3969}-\frac {4435}{567} \sqrt {1-2 x}+\frac {4435 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]

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Rubi [A]  time = 0.03, antiderivative size = 114, 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 = {89, 78, 47, 50, 63, 206} \begin {gather*} \frac {211 (1-2 x)^{7/2}}{2646 (3 x+2)^2}-\frac {(1-2 x)^{7/2}}{189 (3 x+2)^3}-\frac {887 (1-2 x)^{5/2}}{882 (3 x+2)}-\frac {4435 (1-2 x)^{3/2}}{3969}-\frac {4435}{567} \sqrt {1-2 x}+\frac {4435 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(-4435*Sqrt[1 - 2*x])/567 - (4435*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(7/2)/(189*(2 + 3*x)^3) + (211*(1 - 2*x)^(
7/2))/(2646*(2 + 3*x)^2) - (887*(1 - 2*x)^(5/2))/(882*(2 + 3*x)) + (4435*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81
*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 50

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 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)^4} \, dx &=-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {1}{189} \int \frac {(1-2 x)^{5/2} (839+1575 x)}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}+\frac {887}{294} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac {887 (1-2 x)^{5/2}}{882 (2+3 x)}-\frac {4435}{882} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {4435 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac {887 (1-2 x)^{5/2}}{882 (2+3 x)}-\frac {4435}{378} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {4435}{567} \sqrt {1-2 x}-\frac {4435 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac {887 (1-2 x)^{5/2}}{882 (2+3 x)}-\frac {4435}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {4435}{567} \sqrt {1-2 x}-\frac {4435 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac {887 (1-2 x)^{5/2}}{882 (2+3 x)}+\frac {4435}{162} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {4435}{567} \sqrt {1-2 x}-\frac {4435 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac {211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac {887 (1-2 x)^{5/2}}{882 (2+3 x)}+\frac {4435 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 54, normalized size = 0.47 \begin {gather*} \frac {(1-2 x)^{7/2} \left (343 (211 x+136)-10644 (3 x+2)^3 \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{302526 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

((1 - 2*x)^(7/2)*(343*(136 + 211*x) - 10644*(2 + 3*x)^3*Hypergeometric2F1[2, 7/2, 9/2, 3/7 - (6*x)/7]))/(30252
6*(2 + 3*x)^3)

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IntegrateAlgebraic [A]  time = 0.25, size = 88, normalized size = 0.77 \begin {gather*} \frac {4435 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}}-\frac {\left (900 (1-2 x)^4+7020 (1-2 x)^3-87813 (1-2 x)^2+248360 (1-2 x)-217315\right ) \sqrt {1-2 x}}{81 (3 (1-2 x)-7)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

-1/81*((-217315 + 248360*(1 - 2*x) - 87813*(1 - 2*x)^2 + 7020*(1 - 2*x)^3 + 900*(1 - 2*x)^4)*Sqrt[1 - 2*x])/(-
7 + 3*(1 - 2*x))^3 + (4435*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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fricas [A]  time = 1.37, size = 95, normalized size = 0.83 \begin {gather*} \frac {4435 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (3600 \, x^{4} - 21240 \, x^{3} - 61353 \, x^{2} - 48697 \, x - 12212\right )} \sqrt {-2 \, x + 1}}{3402 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

1/3402*(4435*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(36
00*x^4 - 21240*x^3 - 61353*x^2 - 48697*x - 12212)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.85, size = 102, normalized size = 0.89 \begin {gather*} -\frac {100}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {4435}{3402} \, \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 {1480}{243} \, \sqrt {-2 \, x + 1} - \frac {27819 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 126700 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 144305 \, \sqrt {-2 \, x + 1}}{1944 \, {\left (3 \, x + 2\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

-100/243*(-2*x + 1)^(3/2) - 4435/3402*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 1480/243*sqrt(-2*x + 1) - 1/1944*(27819*(2*x - 1)^2*sqrt(-2*x + 1) - 126700*(-2*x + 1)^(3/2) + 1
44305*sqrt(-2*x + 1))/(3*x + 2)^3

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maple [A]  time = 0.01, size = 75, normalized size = 0.66 \begin {gather*} \frac {4435 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1701}-\frac {100 \left (-2 x +1\right )^{\frac {3}{2}}}{243}-\frac {1480 \sqrt {-2 x +1}}{243}-\frac {4 \left (-\frac {3091 \left (-2 x +1\right )^{\frac {5}{2}}}{12}+\frac {31675 \left (-2 x +1\right )^{\frac {3}{2}}}{27}-\frac {144305 \sqrt {-2 x +1}}{108}\right )}{9 \left (-6 x -4\right )^{3}} \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)^4,x)

[Out]

-100/243*(-2*x+1)^(3/2)-1480/243*(-2*x+1)^(1/2)-4/9*(-3091/12*(-2*x+1)^(5/2)+31675/27*(-2*x+1)^(3/2)-144305/10
8*(-2*x+1)^(1/2))/(-6*x-4)^3+4435/1701*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.34, size = 110, normalized size = 0.96 \begin {gather*} -\frac {100}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {4435}{3402} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1480}{243} \, \sqrt {-2 \, x + 1} - \frac {27819 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 126700 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 144305 \, \sqrt {-2 \, x + 1}}{243 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\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)^4,x, algorithm="maxima")

[Out]

-100/243*(-2*x + 1)^(3/2) - 4435/3402*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 1480/243*sqrt(-2*x + 1) - 1/243*(27819*(-2*x + 1)^(5/2) - 126700*(-2*x + 1)^(3/2) + 144305*sqrt(-2*x + 1)
)/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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mupad [B]  time = 0.07, size = 92, normalized size = 0.81 \begin {gather*} -\frac {1480\,\sqrt {1-2\,x}}{243}-\frac {100\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {\frac {144305\,\sqrt {1-2\,x}}{6561}-\frac {126700\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {3091\,{\left (1-2\,x\right )}^{5/2}}{729}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,4435{}\mathrm {i}}{1701} \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)^4,x)

[Out]

- (21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*4435i)/1701 - (1480*(1 - 2*x)^(1/2))/243 - (100*(1 - 2*x)^(3
/2))/243 - ((144305*(1 - 2*x)^(1/2))/6561 - (126700*(1 - 2*x)^(3/2))/6561 + (3091*(1 - 2*x)^(5/2))/729)/((98*x
)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)

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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)**4,x)

[Out]

Timed out

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