3.21.31 \(\int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx\) [2031]

Optimal. Leaf size=80 \[ \frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]

[Out]

2381/9261*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/42*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^2-1/882*(8329+1242
5*x)*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 151, 65, 212} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac {\sqrt {1-2 x} (12425 x+8329)}{882 (3 x+2)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(42*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(8329 + 12425*x))/(882*(2 + 3*x)) + (2381*ArcTan
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), 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*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

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}{\sqrt {1-2 x} (2+3 x)^3} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {1}{42} \int \frac {(-191-355 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}-\frac {2381}{882} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}+\frac {2381}{882} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {1-2 x} \left (16469+49207 x+36750 x^2\right )}{882 (2+3 x)^2}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/882*(Sqrt[1 - 2*x]*(16469 + 49207*x + 36750*x^2))/(2 + 3*x)^2 + (2381*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(44
1*Sqrt[21])

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Maple [A]
time = 0.11, size = 57, normalized size = 0.71

method	result	size
risch	\(\frac {73500 x^{3}+61664 x^{2}-16269 x -16469}{882 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {2381 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\)	\(51\)
derivativedivides	\(-\frac {125 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {69 \left (1-2 x \right )^{\frac {3}{2}}}{98}+\frac {205 \sqrt {1-2 x}}{126}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {2381 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\)	\(57\)
default	\(-\frac {125 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {69 \left (1-2 x \right )^{\frac {3}{2}}}{98}+\frac {205 \sqrt {1-2 x}}{126}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {2381 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\)	\(57\)
trager	\(-\frac {\left (36750 x^{2}+49207 x +16469\right ) \sqrt {1-2 x}}{882 \left (2+3 x \right )^{2}}-\frac {2381 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{18522}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/27*(1-2*x)^(1/2)-2/3*(-69/98*(1-2*x)^(3/2)+205/126*(1-2*x)^(1/2))/(-4-6*x)^2+2381/9261*arctanh(1/7*21^(1/
2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.52, size = 83, normalized size = 1.04 \begin {gather*} -\frac {2381}{18522} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {125}{27} \, \sqrt {-2 \, x + 1} + \frac {621 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1435 \, \sqrt {-2 \, x + 1}}{1323 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2381/18522*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/27*sqrt(-2*x + 1)
+ 1/1323*(621*(-2*x + 1)^(3/2) - 1435*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]
time = 0.90, size = 75, normalized size = 0.94 \begin {gather*} \frac {2381 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (36750 \, x^{2} + 49207 \, x + 16469\right )} \sqrt {-2 \, x + 1}}{18522 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18522*(2381*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(36750*x^2 +
49207*x + 16469)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.03, size = 77, normalized size = 0.96 \begin {gather*} -\frac {2381}{18522} \, \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 {125}{27} \, \sqrt {-2 \, x + 1} + \frac {621 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1435 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2381/18522*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/27*sqrt(
-2*x + 1) + 1/5292*(621*(-2*x + 1)^(3/2) - 1435*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.06, size = 63, normalized size = 0.79 \begin {gather*} \frac {2381\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9261}-\frac {125\,\sqrt {1-2\,x}}{27}-\frac {\frac {205\,\sqrt {1-2\,x}}{1701}-\frac {23\,{\left (1-2\,x\right )}^{3/2}}{441}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2381*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/9261 - (125*(1 - 2*x)^(1/2))/27 - ((205*(1 - 2*x)^(1/2))/1
701 - (23*(1 - 2*x)^(3/2))/441)/((28*x)/3 + (2*x - 1)^2 + 7/9)

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