3.21.99 \(\int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)} \, dx\) [2099]

Optimal. Leaf size=67 \[ \frac {1331}{28 \sqrt {1-2 x}}+\frac {400}{9} \sqrt {1-2 x}-\frac {125}{36} (1-2 x)^{3/2}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \]

[Out]

-125/36*(1-2*x)^(3/2)+2/1323*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1331/28/(1-2*x)^(1/2)+400/9*(1-2*x)^
(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 45, 65, 212} \begin {gather*} -\frac {125}{36} (1-2 x)^{3/2}+\frac {400}{9} \sqrt {1-2 x}+\frac {1331}{28 \sqrt {1-2 x}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(28*Sqrt[1 - 2*x]) + (400*Sqrt[1 - 2*x])/9 - (125*(1 - 2*x)^(3/2))/36 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(63*Sqrt[21])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 89

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], (c + d*x)^n*((e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

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 {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\int \left (\frac {1331}{28 (1-2 x)^{3/2}}-\frac {1225}{36 \sqrt {1-2 x}}-\frac {125 x}{6 \sqrt {1-2 x}}-\frac {1}{63 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac {1331}{28 \sqrt {1-2 x}}+\frac {1225}{36} \sqrt {1-2 x}-\frac {1}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {125}{6} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {1331}{28 \sqrt {1-2 x}}+\frac {1225}{36} \sqrt {1-2 x}+\frac {1}{63} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {125}{6} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {1331}{28 \sqrt {1-2 x}}+\frac {400}{9} \sqrt {1-2 x}-\frac {125}{36} (1-2 x)^{3/2}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 55, normalized size = 0.82 \begin {gather*} \frac {-21 \left (-5576+4725 x+875 x^2\right )+2 \sqrt {21-42 x} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*(-5576 + 4725*x + 875*x^2) + 2*Sqrt[21 - 42*x]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1323*Sqrt[1 - 2*x])

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Maple [A]
time = 0.13, size = 47, normalized size = 0.70

method	result	size
risch	\(-\frac {875 x^{2}+4725 x -5576}{63 \sqrt {1-2 x}}+\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\)	\(39\)
derivativedivides	\(-\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}+\frac {1331}{28 \sqrt {1-2 x}}+\frac {400 \sqrt {1-2 x}}{9}\)	\(47\)
default	\(-\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}+\frac {1331}{28 \sqrt {1-2 x}}+\frac {400 \sqrt {1-2 x}}{9}\)	\(47\)
trager	\(\frac {\left (875 x^{2}+4725 x -5576\right ) \sqrt {1-2 x}}{-63+126 x}+\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1323}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/36*(1-2*x)^(3/2)+2/1323*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1331/28/(1-2*x)^(1/2)+400/9*(1-2*x)^
(1/2)

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Maxima [A]
time = 0.51, size = 64, normalized size = 0.96 \begin {gather*} -\frac {125}{36} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{1323} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {400}{9} \, \sqrt {-2 \, x + 1} + \frac {1331}{28 \, \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)^(3/2)/(2+3*x),x, algorithm="maxima")

[Out]

-125/36*(-2*x + 1)^(3/2) - 1/1323*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) +
400/9*sqrt(-2*x + 1) + 1331/28/sqrt(-2*x + 1)

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Fricas [A]
time = 0.72, size = 64, normalized size = 0.96 \begin {gather*} \frac {\sqrt {21} {\left (2 \, x - 1\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (875 \, x^{2} + 4725 \, x - 5576\right )} \sqrt {-2 \, x + 1}}{1323 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1323*(sqrt(21)*(2*x - 1)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(875*x^2 + 4725*x - 5576)*s
qrt(-2*x + 1))/(2*x - 1)

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Sympy [A]
time = 29.85, size = 95, normalized size = 1.42 \begin {gather*} - \frac {125 \left (1 - 2 x\right )^{\frac {3}{2}}}{36} + \frac {400 \sqrt {1 - 2 x}}{9} - \frac {2 \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 )}{63} + \frac {1331}{28 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*(1 - 2*x)**(3/2)/36 + 400*sqrt(1 - 2*x)/9 - 2*Piecewise((-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))/63 + 1331/(28*sqrt(1 - 2*x))

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Giac [A]
time = 0.85, size = 67, normalized size = 1.00 \begin {gather*} -\frac {125}{36} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{1323} \, \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 {400}{9} \, \sqrt {-2 \, x + 1} + \frac {1331}{28 \, \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)^(3/2)/(2+3*x),x, algorithm="giac")

[Out]

-125/36*(-2*x + 1)^(3/2) - 1/1323*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 400/9*sqrt(-2*x + 1) + 1331/28/sqrt(-2*x + 1)

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Mupad [B]
time = 0.06, size = 48, normalized size = 0.72 \begin {gather*} \frac {1331}{28\,\sqrt {1-2\,x}}+\frac {400\,\sqrt {1-2\,x}}{9}-\frac {125\,{\left (1-2\,x\right )}^{3/2}}{36}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2{}\mathrm {i}}{1323} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1331/(28*(1 - 2*x)^(1/2)) - (21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*2i)/1323 + (400*(1 - 2*x)^(1/2))/9
- (125*(1 - 2*x)^(3/2))/36

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