3.20.8 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx\) [1908]

Optimal. Leaf size=148 \[ \frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac {324 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]

[Out]

-4016/48125*(1-2*x)^(3/2)*(2+3*x)^2+38/4125*(1-2*x)^(3/2)*(2+3*x)^3+39/275*(1-2*x)^(3/2)*(2+3*x)^4-1/5*(1-2*x)
^(3/2)*(2+3*x)^5/(3+5*x)-2/515625*(1-2*x)^(3/2)*(298462+204777*x)-324/390625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/
2))*55^(1/2)+324/78125*(1-2*x)^(1/2)

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Rubi [A]
time = 0.04, 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 = {99, 158, 152, 52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac {39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac {38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac {4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac {2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac {324 \sqrt {1-2 x}}{78125}-\frac {324 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(324*Sqrt[1 - 2*x])/78125 - (4016*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/48125 + (38*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/4125 +
(39*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/275 - ((1 - 2*x)^(3/2)*(2 + 3*x)^5)/(5*(3 + 5*x)) - (2*(1 - 2*x)^(3/2)*(2984
62 + 204777*x))/515625 - (324*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

Rule 52

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 65

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 99

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 152

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 158

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 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} (2+3 x)^5}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}+\frac {1}{5} \int \frac {(9-39 x) \sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx\\ &=\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {1}{275} \int \frac {\sqrt {1-2 x} (2+3 x)^3 (-288+114 x)}{3+5 x} \, dx\\ &=\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}+\frac {\int \frac {\sqrt {1-2 x} (2+3 x)^2 (24894+36144 x)}{3+5 x} \, dx}{12375}\\ &=-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {\int \frac {(-1742580-2866878 x) \sqrt {1-2 x} (2+3 x)}{3+5 x} \, dx}{433125}\\ &=-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}+\frac {162 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}+\frac {1782 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac {1782 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac {324 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 78, normalized size = 0.53 \begin {gather*} \frac {-\frac {5 \sqrt {1-2 x} \left (23061496-2532130 x-135193430 x^2-76760550 x^3+181738125 x^4+270112500 x^5+106312500 x^6\right )}{3+5 x}-24948 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{30078125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*Sqrt[1 - 2*x]*(23061496 - 2532130*x - 135193430*x^2 - 76760550*x^3 + 181738125*x^4 + 270112500*x^5 + 1063
12500*x^6))/(3 + 5*x) - 24948*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/30078125

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Maple [A]
time = 0.11, size = 90, normalized size = 0.61

method	result	size
risch	\(\frac {212625000 x^{7}[phone] x^{6}[phone] x^{5}-335259225 x^{4}-193626310 x^{3}[phone] x^{2}[phone] x -23061496}{6015625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {324 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\)	\(71\)
derivativedivides	\(\frac {243 \left (1-2 x \right )^{\frac {11}{2}}}{2200}-\frac {981 \left (1-2 x \right )^{\frac {9}{2}}}{1000}+\frac {107109 \left (1-2 x \right )^{\frac {7}{2}}}{35000}-\frac {434043 \left (1-2 x \right )^{\frac {5}{2}}}{125000}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {326 \sqrt {1-2 x}}{78125}+\frac {22 \sqrt {1-2 x}}{390625 \left (-\frac {6}{5}-2 x \right )}-\frac {324 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\)	\(90\)
default	\(\frac {243 \left (1-2 x \right )^{\frac {11}{2}}}{2200}-\frac {981 \left (1-2 x \right )^{\frac {9}{2}}}{1000}+\frac {107109 \left (1-2 x \right )^{\frac {7}{2}}}{35000}-\frac {434043 \left (1-2 x \right )^{\frac {5}{2}}}{125000}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {326 \sqrt {1-2 x}}{78125}+\frac {22 \sqrt {1-2 x}}{390625 \left (-\frac {6}{5}-2 x \right )}-\frac {324 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\)	\(90\)
trager	\(-\frac {\left (106312500 x^{6}[phone] x^{5}[phone] x^{4}-76760550 x^{3}-135193430 x^{2}-2532130 x [phone]\right ) \sqrt {1-2 x}}{6015625 \left (3+5 x \right )}-\frac {162 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{390625}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(2+3*x)^5/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

243/2200*(1-2*x)^(11/2)-981/1000*(1-2*x)^(9/2)+107109/35000*(1-2*x)^(7/2)-434043/125000*(1-2*x)^(5/2)+2/3125*(
1-2*x)^(3/2)+326/78125*(1-2*x)^(1/2)+22/390625*(1-2*x)^(1/2)/(-6/5-2*x)-324/390625*arctanh(1/11*55^(1/2)*(1-2*
x)^(1/2))*55^(1/2)

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Maxima [A]
time = 0.51, size = 107, normalized size = 0.72 \begin {gather*} \frac {243}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {981}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {107109}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {434043}{125000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {162}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {326}{78125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/2200*(-2*x + 1)^(11/2) - 981/1000*(-2*x + 1)^(9/2) + 107109/35000*(-2*x + 1)^(7/2) - 434043/125000*(-2*x +
1)^(5/2) + 2/3125*(-2*x + 1)^(3/2) + 162/390625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) + 326/78125*sqrt(-2*x + 1) - 11/78125*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]
time = 0.66, size = 90, normalized size = 0.61 \begin {gather*} \frac {12474 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (106312500 \, x^{6} + 270112500 \, x^{5} + 181738125 \, x^{4} - 76760550 \, x^{3} - 135193430 \, x^{2} - 2532130 \, x + 23061496\right )} \sqrt {-2 \, x + 1}}{30078125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30078125*(12474*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 5*(1
06312500*x^6 + 270112500*x^5 + 181738125*x^4 - 76760550*x^3 - 135193430*x^2 - 2532130*x + 23061496)*sqrt(-2*x
+ 1))/(5*x + 3)

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Sympy [F(-1)] Timed out
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)*(2+3*x)**5/(3+5*x)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.56, size = 138, normalized size = 0.93 \begin {gather*} -\frac {243}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {981}{1000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {107109}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {434043}{125000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {162}{390625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {326}{78125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/2200*(2*x - 1)^5*sqrt(-2*x + 1) - 981/1000*(2*x - 1)^4*sqrt(-2*x + 1) - 107109/35000*(2*x - 1)^3*sqrt(-2*
x + 1) - 434043/125000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/3125*(-2*x + 1)^(3/2) + 162/390625*sqrt(55)*log(1/2*abs(
-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 326/78125*sqrt(-2*x + 1) - 11/78125*sqrt(-2*
x + 1)/(5*x + 3)

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Mupad [B]
time = 0.07, size = 91, normalized size = 0.61 \begin {gather*} \frac {326\,\sqrt {1-2\,x}}{78125}-\frac {22\,\sqrt {1-2\,x}}{390625\,\left (2\,x+\frac {6}{5}\right )}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {434043\,{\left (1-2\,x\right )}^{5/2}}{125000}+\frac {107109\,{\left (1-2\,x\right )}^{7/2}}{35000}-\frac {981\,{\left (1-2\,x\right )}^{9/2}}{1000}+\frac {243\,{\left (1-2\,x\right )}^{11/2}}{2200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,324{}\mathrm {i}}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*324i)/390625 - (22*(1 - 2*x)^(1/2))/(390625*(2*x + 6/5)) + (3
26*(1 - 2*x)^(1/2))/78125 + (2*(1 - 2*x)^(3/2))/3125 - (434043*(1 - 2*x)^(5/2))/125000 + (107109*(1 - 2*x)^(7/
2))/35000 - (981*(1 - 2*x)^(9/2))/1000 + (243*(1 - 2*x)^(11/2))/2200

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