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

Optimal. Leaf size=125 \[ \frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac {251 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac {5}{567} \sqrt {1-2 x} (7265 x+2323)-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]

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Rubi [A]  time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {97, 149, 147, 63, 206} \begin {gather*} \frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac {251 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac {5}{567} \sqrt {1-2 x} (7265 x+2323)-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(251*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(9*(2 + 3*x)^3) + (2*Sqrt[1 - 2
*x]*(3 + 5*x)^3)/(2 + 3*x)^2 - (5*Sqrt[1 - 2*x]*(2323 + 7265*x))/567 - (36038*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/(567*Sqrt[21])

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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^3}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {1}{54} \int \frac {(306-1800 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {\int \frac {(20934-130770 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134}\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)+\frac {18019}{567} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)-\frac {18019}{567} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 59, normalized size = 0.47 \begin {gather*} \frac {(1-2 x)^{5/2} \left (72076 (3 x+2)^3 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )-245 \left (18375 x^2+24657 x+8269\right )\right )}{324135 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(5/2)*(-245*(8269 + 24657*x + 18375*x^2) + 72076*(2 + 3*x)^3*Hypergeometric2F1[2, 5/2, 7/2, 3/7 - (
6*x)/7]))/(324135*(2 + 3*x)^3)

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IntegrateAlgebraic [A]  time = 0.24, size = 88, normalized size = 0.70 \begin {gather*} \frac {2 \left (7875 (1-2 x)^4+9450 (1-2 x)^3-335214 (1-2 x)^2+1009064 (1-2 x)-882931\right ) \sqrt {1-2 x}}{567 (3 (1-2 x)-7)^3}-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-882931 + 1009064*(1 - 2*x) - 335214*(1 - 2*x)^2 + 9450*(1 - 2*x)^3 + 7875*(1 - 2*x)^4)*Sqrt[1 - 2*x])/(56
7*(-7 + 3*(1 - 2*x))^3) - (36038*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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fricas [A]  time = 2.01, size = 94, normalized size = 0.75 \begin {gather*} \frac {18019 \, \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 (31500 \, x^{4} - 81900 \, x^{3} - 259614 \, x^{2} - 199243 \, x - 47939\right )} \sqrt {-2 \, x + 1}}{11907 \, {\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)^(3/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/11907*(18019*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*(
31500*x^4 - 81900*x^3 - 259614*x^2 - 199243*x - 47939)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 1.07, size = 102, normalized size = 0.82 \begin {gather*} \frac {250}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {18019}{11907} \, \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 {2050}{243} \, \sqrt {-2 \, x + 1} + \frac {17721 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 81571 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 93884 \, \sqrt {-2 \, x + 1}}{3402 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

250/243*(-2*x + 1)^(3/2) + 18019/11907*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt
(-2*x + 1))) + 2050/243*sqrt(-2*x + 1) + 1/3402*(17721*(2*x - 1)^2*sqrt(-2*x + 1) - 81571*(-2*x + 1)^(3/2) + 9
3884*sqrt(-2*x + 1))/(3*x + 2)^3

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maple [A]  time = 0.01, size = 75, normalized size = 0.60 \begin {gather*} -\frac {36038 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{11907}+\frac {250 \left (-2 x +1\right )^{\frac {3}{2}}}{243}+\frac {2050 \sqrt {-2 x +1}}{243}+\frac {-\frac {7876 \left (-2 x +1\right )^{\frac {5}{2}}}{189}+\frac {46612 \left (-2 x +1\right )^{\frac {3}{2}}}{243}-\frac {53648 \sqrt {-2 x +1}}{243}}{\left (-6 x -4\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(5*x+3)^3/(3*x+2)^4,x)

[Out]

250/243*(-2*x+1)^(3/2)+2050/243*(-2*x+1)^(1/2)+2/9*(-3938/21*(-2*x+1)^(5/2)+23306/27*(-2*x+1)^(3/2)-26824/27*(
-2*x+1)^(1/2))/(-6*x-4)^3-36038/11907*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.41, size = 110, normalized size = 0.88 \begin {gather*} \frac {250}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {18019}{11907} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2050}{243} \, \sqrt {-2 \, x + 1} + \frac {4 \, {\left (17721 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 81571 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 93884 \, \sqrt {-2 \, x + 1}\right )}}{1701 \, {\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)^(3/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

250/243*(-2*x + 1)^(3/2) + 18019/11907*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 2050/243*sqrt(-2*x + 1) + 4/1701*(17721*(-2*x + 1)^(5/2) - 81571*(-2*x + 1)^(3/2) + 93884*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 = 91, normalized size = 0.73 \begin {gather*} \frac {2050\,\sqrt {1-2\,x}}{243}+\frac {250\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {\frac {53648\,\sqrt {1-2\,x}}{6561}-\frac {46612\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {7876\,{\left (1-2\,x\right )}^{5/2}}{5103}}{\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 )\,36038{}\mathrm {i}}{11907} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^3)/(3*x + 2)^4,x)

[Out]

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*36038i)/11907 + (2050*(1 - 2*x)^(1/2))/243 + (250*(1 - 2*x)^(3
/2))/243 + ((53648*(1 - 2*x)^(1/2))/6561 - (46612*(1 - 2*x)^(3/2))/6561 + (7876*(1 - 2*x)^(5/2))/5103)/((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)**(3/2)*(3+5*x)**3/(2+3*x)**4,x)

[Out]

Timed out

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