3.2059 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac{201 \sqrt{1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac{1512 \sqrt{1-2 x} (3 x+2)^2}{75625}-\frac{189 \sqrt{1-2 x} (2875 x+8976)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

[Out]

(-1512*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 - (Sqrt[1 - 2*x]*(2 + 3*x)^4)/(110*(3 + 5*x)^2) - (201*Sqrt[1 - 2*x]*(
2 + 3*x)^3)/(6050*(3 + 5*x)) - (189*Sqrt[1 - 2*x]*(8976 + 2875*x))/756250 - (22113*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/(378125*Sqrt[55])

________________________________________________________________________________________

Rubi [A]  time = 0.0402599, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac{201 \sqrt{1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac{1512 \sqrt{1-2 x} (3 x+2)^2}{75625}-\frac{189 \sqrt{1-2 x} (2875 x+8976)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1512*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 - (Sqrt[1 - 2*x]*(2 + 3*x)^4)/(110*(3 + 5*x)^2) - (201*Sqrt[1 - 2*x]*(
2 + 3*x)^3)/(6050*(3 + 5*x)) - (189*Sqrt[1 - 2*x]*(8976 + 2875*x))/756250 - (22113*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/(378125*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 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^5}{\sqrt{1-2 x} (3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{1}{110} \int \frac{(-150-183 x) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{\int \frac{(-6237-3024 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}+\frac{\int \frac{(2+3 x) (348138+543375 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{151250}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{189 \sqrt{1-2 x} (8976+2875 x)}{756250}+\frac{22113 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{756250}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{189 \sqrt{1-2 x} (8976+2875 x)}{756250}-\frac{22113 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{756250}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{189 \sqrt{1-2 x} (8976+2875 x)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0669388, size = 68, normalized size = 0.57 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (7350750 x^4+32506650 x^3+76970520 x^2+63610155 x+16525496\right )}{(5 x+3)^2}-44226 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{41593750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*Sqrt[1 - 2*x]*(16525496 + 63610155*x + 76970520*x^2 + 32506650*x^3 + 7350750*x^4))/(3 + 5*x)^2 - 44226*S
qrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/41593750

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -{\frac{243}{2500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{513}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{39393}{12500}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{333}{2420} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{67}{220}\sqrt{1-2\,x}} \right ) }-{\frac{22113\,\sqrt{55}}{20796875}{\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)^5/(3+5*x)^3/(1-2*x)^(1/2),x)

[Out]

-243/2500*(1-2*x)^(5/2)+513/625*(1-2*x)^(3/2)-39393/12500*(1-2*x)^(1/2)+4/125*(333/2420*(1-2*x)^(3/2)-67/220*(
1-2*x)^(1/2))/(-10*x-6)^2-22113/20796875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.51166, size = 136, normalized size = 1.13 \begin{align*} -\frac{243}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{513}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{22113}{41593750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{39393}{12500} \, \sqrt{-2 \, x + 1} + \frac{333 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{75625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/2500*(-2*x + 1)^(5/2) + 513/625*(-2*x + 1)^(3/2) + 22113/41593750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 39393/12500*sqrt(-2*x + 1) + 1/75625*(333*(-2*x + 1)^(3/2) - 737*sqrt(-2
*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

________________________________________________________________________________________

Fricas [A]  time = 1.62659, size = 281, normalized size = 2.34 \begin{align*} \frac{22113 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (7350750 \, x^{4} + 32506650 \, x^{3} + 76970520 \, x^{2} + 63610155 \, x + 16525496\right )} \sqrt{-2 \, x + 1}}{41593750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/41593750*(22113*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(735075
0*x^4 + 32506650*x^3 + 76970520*x^2 + 63610155*x + 16525496)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [A]  time = 2.72346, size = 138, normalized size = 1.15 \begin{align*} -\frac{243}{2500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{513}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{22113}{41593750} \, \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{39393}{12500} \, \sqrt{-2 \, x + 1} + \frac{333 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{302500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/2500*(2*x - 1)^2*sqrt(-2*x + 1) + 513/625*(-2*x + 1)^(3/2) + 22113/41593750*sqrt(55)*log(1/2*abs(-2*sqrt(
55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 39393/12500*sqrt(-2*x + 1) + 1/302500*(333*(-2*x + 1
)^(3/2) - 737*sqrt(-2*x + 1))/(5*x + 3)^2