3.18.30 \(\int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}-\frac {17 (1-2 x)^{3/2}}{126 (3 x+2)^4}-\frac {17 \sqrt {1-2 x}}{12348 (3 x+2)}-\frac {17 \sqrt {1-2 x}}{5292 (3 x+2)^2}+\frac {17 \sqrt {1-2 x}}{378 (3 x+2)^3}-\frac {17 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{6174 \sqrt {21}} \]

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Rubi [A]  time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \begin {gather*} \frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}-\frac {17 (1-2 x)^{3/2}}{126 (3 x+2)^4}-\frac {17 \sqrt {1-2 x}}{12348 (3 x+2)}-\frac {17 \sqrt {1-2 x}}{5292 (3 x+2)^2}+\frac {17 \sqrt {1-2 x}}{378 (3 x+2)^3}-\frac {17 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{6174 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

(1 - 2*x)^(5/2)/(105*(2 + 3*x)^5) - (17*(1 - 2*x)^(3/2))/(126*(2 + 3*x)^4) + (17*Sqrt[1 - 2*x])/(378*(2 + 3*x)
^3) - (17*Sqrt[1 - 2*x])/(5292*(2 + 3*x)^2) - (17*Sqrt[1 - 2*x])/(12348*(2 + 3*x)) - (17*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/(6174*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 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)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}+\frac {34}{21} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}-\frac {17}{42} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac {17 \sqrt {1-2 x}}{378 (2+3 x)^3}+\frac {17}{378} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac {17 \sqrt {1-2 x}}{378 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{5292 (2+3 x)^2}+\frac {17 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1764}\\ &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac {17 \sqrt {1-2 x}}{378 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{5292 (2+3 x)^2}-\frac {17 \sqrt {1-2 x}}{12348 (2+3 x)}+\frac {17 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{12348}\\ &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac {17 \sqrt {1-2 x}}{378 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{5292 (2+3 x)^2}-\frac {17 \sqrt {1-2 x}}{12348 (2+3 x)}-\frac {17 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{12348}\\ &=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac {17 \sqrt {1-2 x}}{378 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{5292 (2+3 x)^2}-\frac {17 \sqrt {1-2 x}}{12348 (2+3 x)}-\frac {17 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{6174 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 42, normalized size = 0.33 \begin {gather*} \frac {(1-2 x)^{5/2} \left (\frac {16807}{(3 x+2)^5}-1088 \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{1764735} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

((1 - 2*x)^(5/2)*(16807/(2 + 3*x)^5 - 1088*Hypergeometric2F1[5/2, 5, 7/2, 3/7 - (6*x)/7]))/1764735

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IntegrateAlgebraic [A]  time = 0.35, size = 88, normalized size = 0.69 \begin {gather*} \frac {\left (6885 (1-2 x)^4-74970 (1-2 x)^3-9408 (1-2 x)^2+408170 (1-2 x)-204085\right ) \sqrt {1-2 x}}{30870 (3 (1-2 x)-7)^5}-\frac {17 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{6174 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

((-204085 + 408170*(1 - 2*x) - 9408*(1 - 2*x)^2 - 74970*(1 - 2*x)^3 + 6885*(1 - 2*x)^4)*Sqrt[1 - 2*x])/(30870*
(-7 + 3*(1 - 2*x))^5) - (17*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6174*Sqrt[21])

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fricas [A]  time = 1.73, size = 114, normalized size = 0.89 \begin {gather*} \frac {85 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (6885 \, x^{4} + 23715 \, x^{3} - 48252 \, x^{2} - 23998 \, x + 7912\right )} \sqrt {-2 \, x + 1}}{1296540 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/1296540*(85*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x + 1
) - 5)/(3*x + 2)) - 21*(6885*x^4 + 23715*x^3 - 48252*x^2 - 23998*x + 7912)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 0.96, size = 116, normalized size = 0.91 \begin {gather*} \frac {17}{259308} \, \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 {6885 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 74970 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 9408 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 408170 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 204085 \, \sqrt {-2 \, x + 1}}{987840 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^6,x, algorithm="giac")

[Out]

17/259308*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/987840*(6885
*(2*x - 1)^4*sqrt(-2*x + 1) + 74970*(2*x - 1)^3*sqrt(-2*x + 1) - 9408*(2*x - 1)^2*sqrt(-2*x + 1) + 408170*(-2*
x + 1)^(3/2) - 204085*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.01, size = 75, normalized size = 0.59 \begin {gather*} -\frac {17 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{129654}+\frac {\frac {153 \left (-2 x +1\right )^{\frac {9}{2}}}{686}-\frac {17 \left (-2 x +1\right )^{\frac {7}{2}}}{7}-\frac {32 \left (-2 x +1\right )^{\frac {5}{2}}}{105}+\frac {119 \left (-2 x +1\right )^{\frac {3}{2}}}{9}-\frac {119 \sqrt {-2 x +1}}{18}}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(5*x+3)/(3*x+2)^6,x)

[Out]

7776*(17/592704*(-2*x+1)^(9/2)-17/54432*(-2*x+1)^(7/2)-1/25515*(-2*x+1)^(5/2)+119/69984*(-2*x+1)^(3/2)-119/139
968*(-2*x+1)^(1/2))/(-6*x-4)^5-17/129654*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.41, size = 128, normalized size = 1.00 \begin {gather*} \frac {17}{259308} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {6885 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 74970 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 9408 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 408170 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 204085 \, \sqrt {-2 \, x + 1}}{30870 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^6,x, algorithm="maxima")

[Out]

17/259308*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/30870*(6885*(-2*x + 1
)^(9/2) - 74970*(-2*x + 1)^(7/2) - 9408*(-2*x + 1)^(5/2) + 408170*(-2*x + 1)^(3/2) - 204085*sqrt(-2*x + 1))/(2
43*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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mupad [B]  time = 1.21, size = 107, normalized size = 0.84 \begin {gather*} \frac {\frac {119\,\sqrt {1-2\,x}}{4374}-\frac {119\,{\left (1-2\,x\right )}^{3/2}}{2187}+\frac {32\,{\left (1-2\,x\right )}^{5/2}}{25515}+\frac {17\,{\left (1-2\,x\right )}^{7/2}}{1701}-\frac {17\,{\left (1-2\,x\right )}^{9/2}}{18522}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}}-\frac {17\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{129654} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3))/(3*x + 2)^6,x)

[Out]

((119*(1 - 2*x)^(1/2))/4374 - (119*(1 - 2*x)^(3/2))/2187 + (32*(1 - 2*x)^(5/2))/25515 + (17*(1 - 2*x)^(7/2))/1
701 - (17*(1 - 2*x)^(9/2))/18522)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^
4)/3 + (2*x - 1)^5 - 19208/243) - (17*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/129654

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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)**(3/2)*(3+5*x)/(2+3*x)**6,x)

[Out]

Timed out

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