3.2060 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{110 (5 x+3)^2}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{3025 (5 x+3)}-\frac{63 \sqrt{1-2 x} (75 x+352)}{30250}-\frac{2667 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}} \]

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(110*(3 + 5*x)^2) - (84*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(3025*(3 + 5*x)) - (63*Sqrt[1
- 2*x]*(352 + 75*x))/30250 - (2667*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(15125*Sqrt[55])

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Rubi [A]  time = 0.0286408, antiderivative size = 100, normalized size of antiderivative = 1., 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, 149, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{110 (5 x+3)^2}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{3025 (5 x+3)}-\frac{63 \sqrt{1-2 x} (75 x+352)}{30250}-\frac{2667 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(110*(3 + 5*x)^2) - (84*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(3025*(3 + 5*x)) - (63*Sqrt[1
- 2*x]*(352 + 75*x))/30250 - (2667*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(15125*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 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)^4}{\sqrt{1-2 x} (3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac{1}{110} \int \frac{(-147-189 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac{\int \frac{(-5502-4725 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac{63 \sqrt{1-2 x} (352+75 x)}{30250}+\frac{2667 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{30250}\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac{63 \sqrt{1-2 x} (352+75 x)}{30250}-\frac{2667 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{30250}\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac{63 \sqrt{1-2 x} (352+75 x)}{30250}-\frac{2667 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0621411, size = 63, normalized size = 0.63 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (163350 x^3+784080 x^2+764745 x+211864\right )}{(5 x+3)^2}-5334 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1663750} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]

[Out]

((-55*Sqrt[1 - 2*x]*(211864 + 764745*x + 784080*x^2 + 163350*x^3))/(3 + 5*x)^2 - 5334*Sqrt[55]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/1663750

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Maple [A]  time = 0.01, size = 66, normalized size = 0.7 \begin{align*}{\frac{27}{250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1107}{1250}\sqrt{1-2\,x}}+{\frac{4}{25\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{267}{2420} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{269}{1100}\sqrt{1-2\,x}} \right ) }-{\frac{2667\,\sqrt{55}}{831875}{\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)^4/(3+5*x)^3/(1-2*x)^(1/2),x)

[Out]

27/250*(1-2*x)^(3/2)-1107/1250*(1-2*x)^(1/2)+4/25*(267/2420*(1-2*x)^(3/2)-269/1100*(1-2*x)^(1/2))/(-10*x-6)^2-
2667/831875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 3.69743, size = 124, normalized size = 1.24 \begin{align*} \frac{27}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2667}{1663750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1107}{1250} \, \sqrt{-2 \, x + 1} + \frac{1335 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2959 \, \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)^4/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

27/250*(-2*x + 1)^(3/2) + 2667/1663750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 1107/1250*sqrt(-2*x + 1) + 1/75625*(1335*(-2*x + 1)^(3/2) - 2959*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x
+ 11)

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Fricas [A]  time = 1.64598, size = 248, normalized size = 2.48 \begin{align*} \frac{2667 \, \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 (163350 \, x^{3} + 784080 \, x^{2} + 764745 \, x + 211864\right )} \sqrt{-2 \, x + 1}}{1663750 \,{\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)^4/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/1663750*(2667*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(163350*x
^3 + 784080*x^2 + 764745*x + 211864)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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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)**4/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.83172, size = 116, normalized size = 1.16 \begin{align*} \frac{27}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2667}{1663750} \, \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{1107}{1250} \, \sqrt{-2 \, x + 1} + \frac{1335 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2959 \, \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)^4/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

27/250*(-2*x + 1)^(3/2) + 2667/1663750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 1107/1250*sqrt(-2*x + 1) + 1/302500*(1335*(-2*x + 1)^(3/2) - 2959*sqrt(-2*x + 1))/(5*x + 3)^2