3.19.21 \(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2} \, dx\) [1821]

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

[Out]

-212/1701*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+7/9*(3+5*x)^2*(1-2*x)^(1/2)-1/3*(3+5*x)^3*(1-2*x)^(1/2)
/(2+3*x)-2/81*(211+170*x)*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, 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 = {99, 158, 152, 65, 212} \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 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 {\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} \text {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 = 0.11, size = 63, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1-2 x} \left (-439-110 x+1725 x^2+1350 x^3\right )}{81 (2+3 x)}-\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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Maple [A]
time = 0.11, size = 63, normalized size = 0.68

method	result	size
risch	\(-\frac {2700 x^{4}+2100 x^{3}-1945 x^{2}-768 x +439}{81 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {212 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1701}\)	\(56\)
derivativedivides	\(\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{18}-\frac {725 \left (1-2 x \right )^{\frac {3}{2}}}{162}+\frac {10 \sqrt {1-2 x}}{27}-\frac {2 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}-\frac {212 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1701}\)	\(63\)
default	\(\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{18}-\frac {725 \left (1-2 x \right )^{\frac {3}{2}}}{162}+\frac {10 \sqrt {1-2 x}}{27}-\frac {2 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}-\frac {212 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1701}\)	\(63\)
trager	\(\frac {\left (1350 x^{3}+1725 x^{2}-110 x -439\right ) \sqrt {1-2 x}}{162+243 x}-\frac {106 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1701}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, 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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Fricas [A]
time = 0.98, 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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Sympy [A]
time = 62.74, size = 216, normalized size = 2.32 \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}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{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}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{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, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/81 + 214*P
iecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, x < -2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/
21, x > -2/3))/81

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Giac [A]
time = 1.66, 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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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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