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

Optimal. Leaf size=93 \[ -\frac {\sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)}+\frac {7}{9} \sqrt {1-2 x} (5 x+3)^2-\frac {2}{81} \sqrt {1-2 x} (170 x+211)-\frac {212 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]

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Rubi [A]  time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {97, 153, 147, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)}+\frac {7}{9} \sqrt {1-2 x} (5 x+3)^2-\frac {2}{81} \sqrt {1-2 x} (170 x+211)-\frac {212 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[1 - 2*x]*(3 + 5*x)^2)/9 - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(3*(2 + 3*x)) - (2*Sqrt[1 - 2*x]*(211 + 170*x))/
81 - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*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 153

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 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)^2} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{3 (2+3 x)}+\frac {1}{3} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {7}{9} \sqrt {1-2 x} (3+5 x)^2-\frac {\sqrt {1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac {1}{45} \int \frac {(-50-340 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {7}{9} \sqrt {1-2 x} (3+5 x)^2-\frac {\sqrt {1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac {2}{81} \sqrt {1-2 x} (211+170 x)+\frac {106}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {7}{9} \sqrt {1-2 x} (3+5 x)^2-\frac {\sqrt {1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac {2}{81} \sqrt {1-2 x} (211+170 x)-\frac {106}{81} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7}{9} \sqrt {1-2 x} (3+5 x)^2-\frac {\sqrt {1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac {2}{81} \sqrt {1-2 x} (211+170 x)-\frac {212 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 63, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1-2 x} \left (1350 x^3+1725 x^2-110 x-439\right )}{81 (3 x+2)}-\frac {212 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-439 - 110*x + 1725*x^2 + 1350*x^3))/(81*(2 + 3*x)) - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
81*Sqrt[21])

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IntegrateAlgebraic [A]  time = 0.12, size = 79, normalized size = 0.85 \begin {gather*} \frac {\left (675 (1-2 x)^3-3750 (1-2 x)^2+5255 (1-2 x)-424\right ) \sqrt {1-2 x}}{162 (3 (1-2 x)-7)}-\frac {212 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-424 + 5255*(1 - 2*x) - 3750*(1 - 2*x)^2 + 675*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(162*(-7 + 3*(1 - 2*x))) - (212*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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fricas [A]  time = 1.62, size = 69, normalized size = 0.74 \begin {gather*} \frac {106 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1350 \, x^{3} + 1725 \, x^{2} - 110 \, x - 439\right )} \sqrt {-2 \, x + 1}}{1701 \, {\left (3 \, x + 2\right )}} \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)^2,x, algorithm="fricas")

[Out]

1/1701*(106*sqrt(21)*(3*x + 2)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1350*x^3 + 1725*x^2 -
110*x - 439)*sqrt(-2*x + 1))/(3*x + 2)

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giac [A]  time = 1.25, size = 90, normalized size = 0.97 \begin {gather*} \frac {25}{18} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {725}{162} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {106}{1701} \, \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 {10}{27} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \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)^2,x, algorithm="giac")

[Out]

25/18*(2*x - 1)^2*sqrt(-2*x + 1) - 725/162*(-2*x + 1)^(3/2) + 106/1701*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq
rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 10/27*sqrt(-2*x + 1) + 1/81*sqrt(-2*x + 1)/(3*x + 2)

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maple [A]  time = 0.01, size = 63, normalized size = 0.68 \begin {gather*} -\frac {212 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1701}+\frac {25 \left (-2 x +1\right )^{\frac {5}{2}}}{18}-\frac {725 \left (-2 x +1\right )^{\frac {3}{2}}}{162}+\frac {10 \sqrt {-2 x +1}}{27}-\frac {2 \sqrt {-2 x +1}}{243 \left (-2 x -\frac {4}{3}\right )} \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)^2,x)

[Out]

25/18*(-2*x+1)^(5/2)-725/162*(-2*x+1)^(3/2)+10/27*(-2*x+1)^(1/2)-2/243*(-2*x+1)^(1/2)/(-2*x-4/3)-212/1701*arct
anh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.05, size = 80, normalized size = 0.86 \begin {gather*} \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {725}{162} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {106}{1701} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {10}{27} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \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)^2,x, algorithm="maxima")

[Out]

25/18*(-2*x + 1)^(5/2) - 725/162*(-2*x + 1)^(3/2) + 106/1701*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) + 10/27*sqrt(-2*x + 1) + 1/81*sqrt(-2*x + 1)/(3*x + 2)

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mupad [B]  time = 0.05, size = 64, normalized size = 0.69 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{243\,\left (2\,x+\frac {4}{3}\right )}+\frac {10\,\sqrt {1-2\,x}}{27}-\frac {725\,{\left (1-2\,x\right )}^{3/2}}{162}+\frac {25\,{\left (1-2\,x\right )}^{5/2}}{18}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,212{}\mathrm {i}}{1701} \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)^2,x)

[Out]

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*212i)/1701 + (2*(1 - 2*x)^(1/2))/(243*(2*x + 4/3)) + (10*(1 -
2*x)^(1/2))/27 - (725*(1 - 2*x)^(3/2))/162 + (25*(1 - 2*x)^(5/2))/18

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sympy [A]  time = 132.19, size = 202, normalized size = 2.17 \begin {gather*} \frac {25 \left (1 - 2 x\right )^{\frac {5}{2}}}{18} - \frac {725 \left (1 - 2 x\right )^{\frac {3}{2}}}{162} + \frac {10 \sqrt {1 - 2 x}}{27} + \frac {28 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {2}{3} \end {cases}\right )}{81} + \frac {214 \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)**2,x)

[Out]

25*(1 - 2*x)**(5/2)/18 - 725*(1 - 2*x)**(3/2)/162 + 10*sqrt(1 - 2*x)/27 + 28*Piecewise((sqrt(21)*(-log(sqrt(21
)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(
sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/81 + 214*Piecewise((-sqrt(21)*acoth(sqrt(21)*sq
rt(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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