3.20.70 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx\) [1970]

Optimal. Leaf size=95 \[ \frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]

[Out]

22/1875*(1-2*x)^(3/2)+2/625*(1-2*x)^(5/2)-111/350*(1-2*x)^(7/2)+1/10*(1-2*x)^(9/2)-242/15625*arctanh(1/11*55^(
1/2)*(1-2*x)^(1/2))*55^(1/2)+242/3125*(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212} \begin {gather*} \frac {1}{10} (1-2 x)^{9/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {2}{625} (1-2 x)^{5/2}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {242 \sqrt {1-2 x}}{3125}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(242*Sqrt[1 - 2*x])/3125 + (22*(1 - 2*x)^(3/2))/1875 + (2*(1 - 2*x)^(5/2))/625 - (111*(1 - 2*x)^(7/2))/350 + (
1 - 2*x)^(9/2)/10 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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 90

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

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)^{5/2} (2+3 x)^2}{3+5 x} \, dx &=\int \left (\frac {111}{50} (1-2 x)^{5/2}-\frac {9}{10} (1-2 x)^{7/2}+\frac {(1-2 x)^{5/2}}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {1}{25} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {11}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {121}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}-\frac {1331 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 61, normalized size = 0.64 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (-8188+69995 x-91410 x^2-43500 x^3+105000 x^4\right )-5082 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(-8188 + 69995*x - 91410*x^2 - 43500*x^3 + 105000*x^4) - 5082*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]])/328125

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Maple [A]
time = 0.10, size = 65, normalized size = 0.68

method	result	size
risch	\(-\frac {\left (105000 x^{4}-43500 x^{3}-91410 x^{2}+69995 x -8188\right ) \left (-1+2 x \right )}{65625 \sqrt {1-2 x}}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\)	\(54\)
derivativedivides	\(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {111 \left (1-2 x \right )^{\frac {7}{2}}}{350}+\frac {\left (1-2 x \right )^{\frac {9}{2}}}{10}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {242 \sqrt {1-2 x}}{3125}\)	\(65\)
default	\(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {111 \left (1-2 x \right )^{\frac {7}{2}}}{350}+\frac {\left (1-2 x \right )^{\frac {9}{2}}}{10}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {242 \sqrt {1-2 x}}{3125}\)	\(65\)
trager	\(\left (\frac {8}{5} x^{4}-\frac {116}{175} x^{3}-\frac {6094}{4375} x^{2}+\frac {13999}{13125} x -\frac {8188}{65625}\right ) \sqrt {1-2 x}-\frac {121 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{15625}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22/1875*(1-2*x)^(3/2)+2/625*(1-2*x)^(5/2)-111/350*(1-2*x)^(7/2)+1/10*(1-2*x)^(9/2)-242/15625*arctanh(1/11*55^(
1/2)*(1-2*x)^(1/2))*55^(1/2)+242/3125*(1-2*x)^(1/2)

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Maxima [A]
time = 0.49, size = 82, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {111}{350} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(-2*x + 1)^(9/2) - 111/350*(-2*x + 1)^(7/2) + 2/625*(-2*x + 1)^(5/2) + 22/1875*(-2*x + 1)^(3/2) + 121/156
25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/3125*sqrt(-2*x + 1)

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Fricas [A]
time = 0.94, size = 66, normalized size = 0.69 \begin {gather*} \frac {121}{15625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{65625} \, {\left (105000 \, x^{4} - 43500 \, x^{3} - 91410 \, x^{2} + 69995 \, x - 8188\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/15625*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1/65625*(105000*x^4 -
43500*x^3 - 91410*x^2 + 69995*x - 8188)*sqrt(-2*x + 1)

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Sympy [A]
time = 27.43, size = 117, normalized size = 1.23 \begin {gather*} \frac {\left (1 - 2 x\right )^{\frac {9}{2}}}{10} - \frac {111 \left (1 - 2 x\right )^{\frac {7}{2}}}{350} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {242 \sqrt {1 - 2 x}}{3125} + \frac {2662 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - 2*x)**(9/2)/10 - 111*(1 - 2*x)**(7/2)/350 + 2*(1 - 2*x)**(5/2)/625 + 22*(1 - 2*x)**(3/2)/1875 + 242*sqrt(
1 - 2*x)/3125 + 2662*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, x < -3/5), (-sqrt(55)*atanh(sqr
t(55)*sqrt(1 - 2*x)/11)/55, x > -3/5))/3125

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Giac [A]
time = 1.27, size = 106, normalized size = 1.12 \begin {gather*} \frac {1}{10} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {111}{350} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{15625} \, \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 {242}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*x - 1)^4*sqrt(-2*x + 1) + 111/350*(2*x - 1)^3*sqrt(-2*x + 1) + 2/625*(2*x - 1)^2*sqrt(-2*x + 1) + 22/1
875*(-2*x + 1)^(3/2) + 121/15625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 242/3125*sqrt(-2*x + 1)

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Mupad [B]
time = 1.17, size = 66, normalized size = 0.69 \begin {gather*} \frac {242\,\sqrt {1-2\,x}}{3125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {111\,{\left (1-2\,x\right )}^{7/2}}{350}+\frac {{\left (1-2\,x\right )}^{9/2}}{10}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*242i)/15625 + (242*(1 - 2*x)^(1/2))/3125 + (22*(1 - 2*x)^(3/2
))/1875 + (2*(1 - 2*x)^(5/2))/625 - (111*(1 - 2*x)^(7/2))/350 + (1 - 2*x)^(9/2)/10

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