3.2034 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=120 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac{5293 \sqrt{1-2 x}}{43218 (3 x+2)}-\frac{5293 \sqrt{1-2 x}}{18522 (3 x+2)^2}-\frac{5293 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609 \sqrt{21}} \]

[Out]

(-5293*Sqrt[1 - 2*x])/(18522*(2 + 3*x)^2) - (5293*Sqrt[1 - 2*x])/(43218*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^
2)/(105*(2 + 3*x)^5) + (Sqrt[1 - 2*x]*(1255 + 1971*x))/(6615*(2 + 3*x)^4) - (5293*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(21609*Sqrt[21])

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Rubi [A]  time = 0.0323284, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 145, 51, 63, 206} \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac{5293 \sqrt{1-2 x}}{43218 (3 x+2)}-\frac{5293 \sqrt{1-2 x}}{18522 (3 x+2)^2}-\frac{5293 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

(-5293*Sqrt[1 - 2*x])/(18522*(2 + 3*x)^2) - (5293*Sqrt[1 - 2*x])/(43218*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^
2)/(105*(2 + 3*x)^5) + (Sqrt[1 - 2*x]*(1255 + 1971*x))/(6615*(2 + 3*x)^4) - (5293*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(21609*Sqrt[21])

Rule 98

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 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 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{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^6} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^2}{105 (2+3 x)^5}-\frac{1}{105} \int \frac{(-488-850 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac{\sqrt{1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac{5293 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{1323}\\ &=-\frac{5293 \sqrt{1-2 x}}{18522 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac{\sqrt{1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac{5293 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{6174}\\ &=-\frac{5293 \sqrt{1-2 x}}{18522 (2+3 x)^2}-\frac{5293 \sqrt{1-2 x}}{43218 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac{\sqrt{1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac{5293 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{43218}\\ &=-\frac{5293 \sqrt{1-2 x}}{18522 (2+3 x)^2}-\frac{5293 \sqrt{1-2 x}}{43218 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac{\sqrt{1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac{5293 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{43218}\\ &=-\frac{5293 \sqrt{1-2 x}}{18522 (2+3 x)^2}-\frac{5293 \sqrt{1-2 x}}{43218 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac{\sqrt{1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac{5293 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.026927, size = 52, normalized size = 0.43 \[ \frac{\sqrt{1-2 x} \left (\frac{343 \left (18375 x^2+24371 x+8083\right )}{(3 x+2)^5}-84688 \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{756315} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

(Sqrt[1 - 2*x]*((343*(8083 + 24371*x + 18375*x^2))/(2 + 3*x)^5 - 84688*Hypergeometric2F1[1/2, 4, 3/2, 3/7 - (6
*x)/7]))/756315

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Maple [A]  time = 0.011, size = 75, normalized size = 0.6 \begin{align*} 1944\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{5293\, \left ( 1-2\,x \right ) ^{9/2}}{518616}}-{\frac{5293\, \left ( 1-2\,x \right ) ^{7/2}}{47628}}+{\frac{78563\, \left ( 1-2\,x \right ) ^{5/2}}{178605}}-{\frac{324347\, \left ( 1-2\,x \right ) ^{3/2}}{428652}}+{\frac{58781\,\sqrt{1-2\,x}}{122472}} \right ) }-{\frac{5293\,\sqrt{21}}{453789}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^3/(2+3*x)^6/(1-2*x)^(1/2),x)

[Out]

1944*(5293/518616*(1-2*x)^(9/2)-5293/47628*(1-2*x)^(7/2)+78563/178605*(1-2*x)^(5/2)-324347/428652*(1-2*x)^(3/2
)+58781/122472*(1-2*x)^(1/2))/(-6*x-4)^5-5293/453789*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.70023, size = 173, normalized size = 1.44 \begin{align*} \frac{5293}{907578} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2143665 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 23342130 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 92390088 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 158930030 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 100809415 \, \sqrt{-2 \, x + 1}}{108045 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

5293/907578*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/108045*(2143665*(-2
*x + 1)^(9/2) - 23342130*(-2*x + 1)^(7/2) + 92390088*(-2*x + 1)^(5/2) - 158930030*(-2*x + 1)^(3/2) + 100809415
*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208
)

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Fricas [A]  time = 1.60568, size = 365, normalized size = 3.04 \begin{align*} \frac{26465 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (2143665 \, x^{4} + 7383735 \, x^{3} + 8806422 \, x^{2} + 4450198 \, x + 816938\right )} \sqrt{-2 \, x + 1}}{4537890 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/4537890*(26465*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x
+ 1) - 5)/(3*x + 2)) - 21*(2143665*x^4 + 7383735*x^3 + 8806422*x^2 + 4450198*x + 816938)*sqrt(-2*x + 1))/(243*
x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(2+3*x)**6/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.47777, size = 157, normalized size = 1.31 \begin{align*} \frac{5293}{907578} \, \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{2143665 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 23342130 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 92390088 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 158930030 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 100809415 \, \sqrt{-2 \, x + 1}}{3457440 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

5293/907578*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3457440*(2
143665*(2*x - 1)^4*sqrt(-2*x + 1) + 23342130*(2*x - 1)^3*sqrt(-2*x + 1) + 92390088*(2*x - 1)^2*sqrt(-2*x + 1)
- 158930030*(-2*x + 1)^(3/2) + 100809415*sqrt(-2*x + 1))/(3*x + 2)^5