3.17.81 \(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{2+3 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac {125}{84} (1-2 x)^{7/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {5135}{324} (1-2 x)^{3/2}-\frac {2}{81} \sqrt {1-2 x}+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \begin {gather*} -\frac {125}{84} (1-2 x)^{7/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {5135}{324} (1-2 x)^{3/2}-\frac {2}{81} \sqrt {1-2 x}+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/81 - (5135*(1 - 2*x)^(3/2))/324 + (80*(1 - 2*x)^(5/2))/9 - (125*(1 - 2*x)^(7/2))/84 + (2*Sq
rt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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 88

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 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 {\sqrt {1-2 x} (3+5 x)^3}{2+3 x} \, dx &=\int \left (\frac {5135}{108} \sqrt {1-2 x}-\frac {400}{9} (1-2 x)^{3/2}+\frac {125}{12} (1-2 x)^{5/2}-\frac {\sqrt {1-2 x}}{27 (2+3 x)}\right ) \, dx\\ &=-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}-\frac {1}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}-\frac {7}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}+\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 58, normalized size = 0.71 \begin {gather*} \frac {1}{567} \sqrt {1-2 x} \left (6750 x^3+10035 x^2+2875 x-4804\right )+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-4804 + 2875*x + 10035*x^2 + 6750*x^3))/567 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/8
1

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IntegrateAlgebraic [A]  time = 0.07, size = 70, normalized size = 0.85 \begin {gather*} \frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {\left (3375 (1-2 x)^3-20160 (1-2 x)^2+35945 (1-2 x)+56\right ) \sqrt {1-2 x}}{2268} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2268*((56 + 35945*(1 - 2*x) - 20160*(1 - 2*x)^2 + 3375*(1 - 2*x)^3)*Sqrt[1 - 2*x]) + (2*Sqrt[7/3]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/81

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fricas [A]  time = 1.67, size = 62, normalized size = 0.76 \begin {gather*} \frac {1}{243} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{567} \, {\left (6750 \, x^{3} + 10035 \, x^{2} + 2875 \, x - 4804\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/243*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 1/567*(6750*x^3 + 10035*x^2
+ 2875*x - 4804)*sqrt(-2*x + 1)

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giac [A]  time = 1.21, size = 90, normalized size = 1.10 \begin {gather*} \frac {125}{84} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {80}{9} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {5135}{324} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{243} \, \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 {2}{81} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/84*(2*x - 1)^3*sqrt(-2*x + 1) + 80/9*(2*x - 1)^2*sqrt(-2*x + 1) - 5135/324*(-2*x + 1)^(3/2) - 1/243*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/81*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 56, normalized size = 0.68 \begin {gather*} \frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{243}-\frac {5135 \left (-2 x +1\right )^{\frac {3}{2}}}{324}+\frac {80 \left (-2 x +1\right )^{\frac {5}{2}}}{9}-\frac {125 \left (-2 x +1\right )^{\frac {7}{2}}}{84}-\frac {2 \sqrt {-2 x +1}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5135/324*(-2*x+1)^(3/2)+80/9*(-2*x+1)^(5/2)-125/84*(-2*x+1)^(7/2)+2/243*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*
21^(1/2)-2/81*(-2*x+1)^(1/2)

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maxima [A]  time = 1.20, size = 73, normalized size = 0.89 \begin {gather*} -\frac {125}{84} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {80}{9} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {5135}{324} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2}{81} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/84*(-2*x + 1)^(7/2) + 80/9*(-2*x + 1)^(5/2) - 5135/324*(-2*x + 1)^(3/2) - 1/243*sqrt(21)*log(-(sqrt(21) -
3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/81*sqrt(-2*x + 1)

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mupad [B]  time = 1.18, size = 57, normalized size = 0.70 \begin {gather*} \frac {80\,{\left (1-2\,x\right )}^{5/2}}{9}-\frac {5135\,{\left (1-2\,x\right )}^{3/2}}{324}-\frac {2\,\sqrt {1-2\,x}}{81}-\frac {125\,{\left (1-2\,x\right )}^{7/2}}{84}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2{}\mathrm {i}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(80*(1 - 2*x)^(5/2))/9 - (2*(1 - 2*x)^(1/2))/81 - (5135*(1 - 2*x)^(3/2))/324 - (21^(1/2)*atan((21^(1/2)*(1 - 2
*x)^(1/2)*1i)/7)*2i)/243 - (125*(1 - 2*x)^(7/2))/84

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sympy [A]  time = 8.05, size = 114, normalized size = 1.39 \begin {gather*} - \frac {125 \left (1 - 2 x\right )^{\frac {7}{2}}}{84} + \frac {80 \left (1 - 2 x\right )^{\frac {5}{2}}}{9} - \frac {5135 \left (1 - 2 x\right )^{\frac {3}{2}}}{324} - \frac {2 \sqrt {1 - 2 x}}{81} - \frac {14 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*(1 - 2*x)**(7/2)/84 + 80*(1 - 2*x)**(5/2)/9 - 5135*(1 - 2*x)**(3/2)/324 - 2*sqrt(1 - 2*x)/81 - 14*Piecewi
se((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/
21, 2*x - 1 > -7/3))/81

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