3.2047 \(\int \frac{(2+3 x)^6}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{55 (5 x+3)}-\frac{8}{275} \sqrt{1-2 x} (3 x+2)^4-\frac{1717 \sqrt{1-2 x} (3 x+2)^3}{9625}-\frac{26352 \sqrt{1-2 x} (3 x+2)^2}{34375}-\frac{3 \sqrt{1-2 x} (615875 x+1847824)}{171875}-\frac{398 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{171875 \sqrt{55}} \]

[Out]

(-26352*Sqrt[1 - 2*x]*(2 + 3*x)^2)/34375 - (1717*Sqrt[1 - 2*x]*(2 + 3*x)^3)/9625 - (8*Sqrt[1 - 2*x]*(2 + 3*x)^
4)/275 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(55*(3 + 5*x)) - (3*Sqrt[1 - 2*x]*(1847824 + 615875*x))/171875 - (398*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(171875*Sqrt[55])

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Rubi [A]  time = 0.0491521, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{55 (5 x+3)}-\frac{8}{275} \sqrt{1-2 x} (3 x+2)^4-\frac{1717 \sqrt{1-2 x} (3 x+2)^3}{9625}-\frac{26352 \sqrt{1-2 x} (3 x+2)^2}{34375}-\frac{3 \sqrt{1-2 x} (615875 x+1847824)}{171875}-\frac{398 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{171875 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-26352*Sqrt[1 - 2*x]*(2 + 3*x)^2)/34375 - (1717*Sqrt[1 - 2*x]*(2 + 3*x)^3)/9625 - (8*Sqrt[1 - 2*x]*(2 + 3*x)^
4)/275 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(55*(3 + 5*x)) - (3*Sqrt[1 - 2*x]*(1847824 + 615875*x))/171875 - (398*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(171875*Sqrt[55])

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 153

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 147

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 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{(2+3 x)^6}{\sqrt{1-2 x} (3+5 x)^2} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac{1}{55} \int \frac{(-83-72 x) (2+3 x)^4}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{8}{275} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}+\frac{\int \frac{(2+3 x)^3 (9630+15453 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{2475}\\ &=-\frac{1717 \sqrt{1-2 x} (2+3 x)^3}{9625}-\frac{8}{275} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac{\int \frac{(-998613-1660176 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx}{86625}\\ &=-\frac{26352 \sqrt{1-2 x} (2+3 x)^2}{34375}-\frac{1717 \sqrt{1-2 x} (2+3 x)^3}{9625}-\frac{8}{275} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}+\frac{\int \frac{(2+3 x) (69852762+116400375 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{2165625}\\ &=-\frac{26352 \sqrt{1-2 x} (2+3 x)^2}{34375}-\frac{1717 \sqrt{1-2 x} (2+3 x)^3}{9625}-\frac{8}{275} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac{3 \sqrt{1-2 x} (1847824+615875 x)}{171875}+\frac{199 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{171875}\\ &=-\frac{26352 \sqrt{1-2 x} (2+3 x)^2}{34375}-\frac{1717 \sqrt{1-2 x} (2+3 x)^3}{9625}-\frac{8}{275} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac{3 \sqrt{1-2 x} (1847824+615875 x)}{171875}-\frac{199 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{171875}\\ &=-\frac{26352 \sqrt{1-2 x} (2+3 x)^2}{34375}-\frac{1717 \sqrt{1-2 x} (2+3 x)^3}{9625}-\frac{8}{275} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac{3 \sqrt{1-2 x} (1847824+615875 x)}{171875}-\frac{398 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{171875 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0703031, size = 73, normalized size = 0.55 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (19490625 x^5+92998125 x^4+200942775 x^3+273540465 x^2+334366065 x+135011752\right )}{5 x+3}-2786 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{66171875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*Sqrt[1 - 2*x]*(135011752 + 334366065*x + 273540465*x^2 + 200942775*x^3 + 92998125*x^4 + 19490625*x^5))/(
3 + 5*x) - 2786*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/66171875

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Maple [A]  time = 0.01, size = 81, normalized size = 0.6 \begin{align*} -{\frac{81}{400} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{2187}{875} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{315171}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{105228}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{607689}{10000}\sqrt{1-2\,x}}+{\frac{2}{859375}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{398\,\sqrt{55}}{9453125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/400*(1-2*x)^(9/2)+2187/875*(1-2*x)^(7/2)-315171/25000*(1-2*x)^(5/2)+105228/3125*(1-2*x)^(3/2)-607689/10000
*(1-2*x)^(1/2)+2/859375*(1-2*x)^(1/2)/(-2*x-6/5)-398/9453125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.55753, size = 132, normalized size = 0.99 \begin{align*} -\frac{81}{400} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{2187}{875} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{315171}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{105228}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{199}{9453125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{607689}{10000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{171875 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/400*(-2*x + 1)^(9/2) + 2187/875*(-2*x + 1)^(7/2) - 315171/25000*(-2*x + 1)^(5/2) + 105228/3125*(-2*x + 1)^
(3/2) + 199/9453125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 607689/10000*
sqrt(-2*x + 1) - 1/171875*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 1.59278, size = 279, normalized size = 2.1 \begin{align*} \frac{1393 \, \sqrt{55}{\left (5 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (19490625 \, x^{5} + 92998125 \, x^{4} + 200942775 \, x^{3} + 273540465 \, x^{2} + 334366065 \, x + 135011752\right )} \sqrt{-2 \, x + 1}}{66171875 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/66171875*(1393*sqrt(55)*(5*x + 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(19490625*x^5 + 92
998125*x^4 + 200942775*x^3 + 273540465*x^2 + 334366065*x + 135011752)*sqrt(-2*x + 1))/(5*x + 3)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.01909, size = 165, normalized size = 1.24 \begin{align*} -\frac{81}{400} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{2187}{875} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{315171}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{105228}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{199}{9453125} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{607689}{10000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{171875 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/400*(2*x - 1)^4*sqrt(-2*x + 1) - 2187/875*(2*x - 1)^3*sqrt(-2*x + 1) - 315171/25000*(2*x - 1)^2*sqrt(-2*x
+ 1) + 105228/3125*(-2*x + 1)^(3/2) + 199/9453125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(
55) + 5*sqrt(-2*x + 1))) - 607689/10000*sqrt(-2*x + 1) - 1/171875*sqrt(-2*x + 1)/(5*x + 3)