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

Optimal. Leaf size=100 \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac {\sqrt {1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac {42995 \sqrt {1-2 x}}{74088 (3 x+2)}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \]

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Rubi [A]  time = 0.03, antiderivative size = 100, 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 = {98, 145, 51, 63, 206} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac {\sqrt {1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac {42995 \sqrt {1-2 x}}{74088 (3 x+2)}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-42995*Sqrt[1 - 2*x])/(74088*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^2)/(84*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(3168
+ 4955*x))/(10584*(2 + 3*x)^3) - (42995*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

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 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 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)^5} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}-\frac {1}{84} \int \frac {(-389-685 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}+\frac {42995 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{10584}\\ &=-\frac {42995 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}+\frac {42995 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{74088}\\ &=-\frac {42995 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}-\frac {42995 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{74088}\\ &=-\frac {42995 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 52, normalized size = 0.52 \begin {gather*} \frac {\sqrt {1-2 x} \left (\frac {343 \left (31500 x^2+41823 x+13885\right )}{(3 x+2)^4}-343960 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{777924} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*((343*(13885 + 41823*x + 31500*x^2))/(2 + 3*x)^4 - 343960*Hypergeometric2F1[1/2, 3, 3/2, 3/7 -
(6*x)/7]))/777924

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IntegrateAlgebraic [A]  time = 0.29, size = 79, normalized size = 0.79 \begin {gather*} \frac {\left (1160865 (1-2 x)^3-7873845 (1-2 x)^2+17806943 (1-2 x)-13427323\right ) \sqrt {1-2 x}}{37044 (3 (1-2 x)-7)^4}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-13427323 + 17806943*(1 - 2*x) - 7873845*(1 - 2*x)^2 + 1160865*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(37044*(-7 + 3*(1
- 2*x))^4) - (42995*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

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fricas [A]  time = 1.12, size = 99, normalized size = 0.99 \begin {gather*} \frac {42995 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1160865 \, x^{3} + 2195625 \, x^{2} + 1385462 \, x + 291670\right )} \sqrt {-2 \, x + 1}}{1555848 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1555848*(42995*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*
x + 2)) - 21*(1160865*x^3 + 2195625*x^2 + 1385462*x + 291670)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96
*x + 16)

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giac [A]  time = 1.02, size = 100, normalized size = 1.00 \begin {gather*} \frac {42995}{1555848} \, \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 {1160865 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 7873845 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 17806943 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 13427323 \, \sqrt {-2 \, x + 1}}{592704 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

42995/1555848*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/592704*(
1160865*(2*x - 1)^3*sqrt(-2*x + 1) + 7873845*(2*x - 1)^2*sqrt(-2*x + 1) - 17806943*(-2*x + 1)^(3/2) + 13427323
*sqrt(-2*x + 1))/(3*x + 2)^4

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maple [A]  time = 0.02, size = 66, normalized size = 0.66 \begin {gather*} -\frac {42995 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{777924}-\frac {324 \left (-\frac {42995 \left (-2 x +1\right )^{\frac {7}{2}}}{444528}+\frac {374945 \left (-2 x +1\right )^{\frac {5}{2}}}{571536}-\frac {363407 \left (-2 x +1\right )^{\frac {3}{2}}}{244944}+\frac {274027 \sqrt {-2 x +1}}{244944}\right )}{\left (-6 x -4\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-324*(-42995/444528*(-2*x+1)^(7/2)+374945/571536*(-2*x+1)^(5/2)-363407/244944*(-2*x+1)^(3/2)+274027/244944*(-2
*x+1)^(1/2))/(-6*x-4)^4-42995/777924*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.28, size = 110, normalized size = 1.10 \begin {gather*} \frac {42995}{1555848} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1160865 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 7873845 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 17806943 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13427323 \, \sqrt {-2 \, x + 1}}{37044 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

42995/1555848*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/37044*(1160865*(-
2*x + 1)^(7/2) - 7873845*(-2*x + 1)^(5/2) + 17806943*(-2*x + 1)^(3/2) - 13427323*sqrt(-2*x + 1))/(81*(2*x - 1)
^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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mupad [B]  time = 1.19, size = 90, normalized size = 0.90 \begin {gather*} -\frac {42995\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{777924}-\frac {\frac {274027\,\sqrt {1-2\,x}}{61236}-\frac {363407\,{\left (1-2\,x\right )}^{3/2}}{61236}+\frac {374945\,{\left (1-2\,x\right )}^{5/2}}{142884}-\frac {42995\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \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)^5),x)

[Out]

- (42995*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/777924 - ((274027*(1 - 2*x)^(1/2))/61236 - (363407*(1 -
2*x)^(3/2))/61236 + (374945*(1 - 2*x)^(5/2))/142884 - (42995*(1 - 2*x)^(7/2))/111132)/((2744*x)/27 + (98*(2*x
- 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

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sympy [F(-1)]  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)**5/(1-2*x)**(1/2),x)

[Out]

Timed out

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